Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

GUARANTEED LO WER BOUNDS F OR COST FUNCTIONALS OF TIME-PERIODIC P ARABOLIC OPTIMIZA TION PROBLEMS MONIKA W OLFMA YR Abstract. I n this pa p er, a new technique is sho wn for deriving computable, guarant eed low er bounds of functional type (minoran ts) f or tw o di ﬀeren t cost functionals sub ject to a parabolic time-p erio dic b oundary v a lue problem. T ogether with previous results on upp er b ounds (ma jo- ran ts) for one of the c ost functionals, b oth minoran ts and ma jorant s lead to tw o-sided estimates of funct ional t yp e for the optimal cont rol problem. Both upp er and low er b ounds are deriv ed for the secon d new cost functional sub ject to the same parabolic PDE- constra ints, but where the target is a desired gradient . The time-p eriodic optimal con trol problems are discret ized b y the multiha rmonic ﬁnite elemen t method leading to large systems of li near equations having a sad- dle p oint struc ture. The deriv ation of preconditioners for the minimal residual method for the new optimization problem is discussed in m ore detail. Finally , seve ral n umerical experiment s for both optimal con trol pro blems are presen ted conﬁrming the theoretical results obtained. This wo rk pro vides the basis for an adaptive sc heme for time-perio dic optimization problems. 1. Introduction In this work, we deriv e fully computable, guar ant eed low er b ounds (minorant s) for cos t function- als o f pa rab olic optimization problems w ith given time-p erio dic conditions. The fully computable upper bounds (ma jorants) for one of the cost functionals were derived in [26]. The seco nd cost functional is new and for this one b oth upp er and lower b ounds a re present ed. The motiv ation for the s econd problem lies in applications, where the target is the gr adien t or ﬂux instead of the state function. Low er b ounds for cost functionals of time-per iod ic parab olic optimal con trol problems hav e not b een discussed y et. How ever, optimal control probl e ms are highly impor ta n t for diﬀer- ent a pplica tions (see e.g. [36] as well as the o r iginal w ork [29]). These applications a lso include time-per io dic problems, s ee for instance [1] and [15] considering pro blems in e le c tr omagnetics and bio c hemistry , resp ectively . F or these t yp es of problems the multiharmonic (or har monic-balanced) ﬁnite element metho d (short MhFEM) is a natural choice. The functio ns a re approximated b y truncated F ourier series in time and by the ﬁnite element metho d (FEM) in spa ce – more precisely , the F ourier co eﬃcients. W e refer to the application o f this discretization technique alre a dy in [45] as well as later in [3, 4, 5, 9] for non-linear time-harmo nic eddy current pro blems. Moreov er, time-per io dic optimal control problems a nd the MhFEM were discussed in e.g . [16, 17, 2 7] and [18, 19]. Recent works on r obust preconditioners for time-p erio dic parab olic and eddy-current optimal co n trol problems a r e discussed in [28] and [2], resp ectively . In this w ork, a standard ﬁ- nite element discretization is used with cont inuous, piecewise linear ﬁnite elements and a regular grid as discussed, e.g., in [6, 7, 43]. How ever, the metho d is wider a pplicable using also other ﬁnite elements or also ﬁnite diﬀerences (for instances, if the given domain is geometrically rather non-complex). F unctional a poster io ri estimation provides a useful machinery to derive computable and guar- anteed q uan tities for the desired unknown solution, see, e.g., [39, 12] on parab olic problems. Recent works on new estimates for parab olic problems and parab olic optimal co n trol problems can b e found in [32 ] and [42], resp ectively . A po steriori estimates of functiona l type for elliptic optimal con trol pr o blems ca n b e found in [10, 11, 4 0, 31]. Firs t functional type estimates for in- verse problems, which are related to optimal c o n trol problems, can be found in [41, 8]. Moreov er, recent results on guaranteed computable estimates for conv ection-dominated diﬀusion pr oblems This researc h w as supported b y the A cadem y of Finl and, gr ant 295897. 1 2 MONIKA WOLFMA YR are presented in [33]. In [26], ma jorants for one cost functional o f a time-p erio dic parab olic op- timal co n trol a nd for the cor resp onding optimality system were presented. This work presents the corres ponding minorants for this cost functional using the new technique presented in [4 4], which makes use o f ideas deriv ed b y Mikhlin [34] but genera lized for the class of optimal con trol problems. W e mention here that [3 0] presents a diﬀeren t a pproach for the deriv atio n of a lower bo und for a class o f elliptic optimal control problems. W e extend the analysis in this pap er and consider a seco nd cos t functional with r espe c t to the same par abo lic time-p erio dic b oundary v alue problem. In the seco nd optimal control problem, the target is a given desired g radient. Problems of that type hav e bee n earlier discussed in [11]. The results on computable low er b ounds tog ether with the upper bo unds lead to tw o- sided estimates which can b e used to der iv e ma jorants for the discretization err or in state and control. These ma jora n ts and minorants provide a new form ulation of the optimization pro blems since they can, in principle, be used as ob jects of direct minimization on their diﬀerence. The ma jorants and minora n ts c a n b e used in order to derive an adaptive scheme in time and in s pace. The linearity of the optimal control pr oblems leads to decoupled problems in the F ourier modes including decoupling in the ma jor ant s and minorants, whic h is shown in this w ork. The ov erall estimators provide the mo des/mo de num b ers which are necess ary for computations. The problems for the diﬀeren t mo des can then b e computed on diﬀeren t g rids, for whic h estimators in space can be used with respec t to the ﬁnite element discretized F ourier co eﬃcients. Altogether we derive a space-time a daptiv e metho d. Its idea has b een for the ﬁrst time in tro duced in [26] a nd has b een called the adaptive m ultiharmonic ﬁnite element method (AMhFEM). In this work, robust preco nditioners for the preconditioned minimal r esidual (MinRes) metho d (see [37]) are discussed for the second optimization problem, which are new for this case. Also the practical perfo rmance o f the AMLI preconditioner MinRes s olver is pr esen ted in v a rious n umerical exp erimen ts for b oth o ptimization pro blems. F or additional numerical tests regarding the AMLI precondititioned MinRes solver used in this work and its p erformance for diﬀerent cases of given data in time-p erio dic parab olic optimal control problems, we r efer the reader to [27]. The w ork is arr a nged in the following sectio ns : In Section 2, the t wo types of cos t functionals ar e presented including so me preliminary r esults. W e denote the problems by o ptimization problem I and I I. Moreover, the for mer result on the ma jorant for problem I is summarized there. The new minorant fo r optimization problem I is derived in Section 3 follow ed b y the discussion of the ma jorant a nd minorant for optimization problem I I in Sec tio n 4. In Section 5 , the MhFEM for bo th optimization pr oblems is presented. Robust preconditioner s for applying the preconditioned MinRes metho d on the problems discretized b y the MhFEM are pr e s en ted in Section 6. Section 7 discusses detailed a set of v ario us numerical exp eriments for both optimization pr oblems I and II. A few ﬁnal rema rks are drawn in Section 8. 2. Time-periodic p arabolic PDE, t he two cost functionals and preliminar y resul ts W e deno te b y Ω ⊂ R d with poss ible dimensions d = { 1 , 2 , 3 } the s patial b ounded Lipsch itz domain with b oundary Γ := ∂ Ω . Also we denote by Q := Ω × (0 , T ) the space - time domain and Σ := Γ × (0 , T ) its lateral surface, where (0 , T ) is the g iv en time interv al. The optimization problems are b oth sub ject to the following parab olic PDE with given ho mogeneous Dirichlet bo undary conditions and time-p erio dical condition: σ ∂ t y − ∇ · ( ν ∇ y ) = u in Q, (1) y = 0 on Σ , (2) y (0) = y ( T ) in Ω . (3) The function y is the state and u will b e the control function. W e assume that the co eﬃcient functions σ and ν are p ositive and b ounded satisfying 0 < σ ≤ σ ( x ) ≤ σ and 0 < ν ≤ ν ( x ) ≤ ν for x ∈ Ω with constants σ , σ , ν a nd ν . The time-perio dic problems in this pap er a re motived by real- life applications such as computational electro magnetics, where these parameters corr espo nd to the reluctivit y and conductivity b eing usually piecewise constant b ecause of the v a rious materia ls of the electrical devices . GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 3 2.1. Prelimi naries. In the following, we pr esen t a prop er functional space setting fo r time- per io dic pr oblems which starts by deﬁning the Hilb ert spa ces H 1 , 0 ( Q ) = { u ∈ L 2 ( Q ) : ∇ u ∈ [ L 2 ( Q )] d } , H 1 , 0 0 ( Q ) = { u ∈ H 1 , 0 ( Q ) : u = 0 o n Σ } , H 0 , 1 ( Q ) = { u ∈ L 2 ( Q ) : ∂ t u ∈ L 2 ( Q ) } , H 0 , 1 per ( Q ) = { u ∈ H 0 , 1 ( Q ) : u (0) = u ( T ) in Ω } , H 1 , 1 ( Q ) = { u ∈ L 2 ( Q ) : ∇ u ∈ [ L 2 ( Q )] d , ∂ t u ∈ L 2 ( Q ) } , (see also [23, 2 4]). F or instance, the no rm in H 1 , 1 is given b y k u k 1 , 1 :=  Z Q  u ( x , t ) 2 + |∇ u ( x , t ) | 2 + | ∂ t u ( x , t ) | 2  d x dt  1 / 2 . In the following, w e skip the subindex for the nor ms and inner pr oducts in L 2 ( Q ) and deno te them b y k · k and h· , ·i . F or L 2 (Ω) and H 1 (Ω) , we denote them by k · k Ω and h· , ·i Ω and k · k 1 , Ω and h· , ·i 1 , Ω , resp ectively . Time-p e rio dic functions whic h are a t least from the space L 2 can be naturally r epresented b y F ourier se r ies as u ( x , t ) = u c 0 ( x ) + ∞ X k =1 ( u c k ( x ) cos ( kω t ) + u s k ( x ) sin ( k ω t )) for ω = 2 π / T b eing the frequency , T the perio d and with the F ourier coﬃcients u c 0 ( x ) = 1 T Z T 0 u ( x , t ) dt, u c k ( x ) = 2 T Z T 0 u ( x , t ) cos( k ω t ) dt, u s k ( x ) = 2 T Z T 0 u ( x , t ) sin( k ω t ) dt. W e deﬁne the norm in F ourier space a s follows   ∂ 1 / 2 t u   2 := | u | 2 0 , 1 2 := T 2 ∞ X k =1 k ω k u k k 2 Ω (4) as well as the s pa ces H 0 , 1 2 per ( Q ) = { u ∈ L 2 ( Q ) :   ∂ 1 / 2 t u   < ∞} , H 1 , 1 2 per ( Q ) = { u ∈ H 1 , 0 ( Q ) :   ∂ 1 / 2 t u   < ∞} , H 1 , 1 2 0 ,per ( Q ) = { u ∈ H 1 , 1 2 per ( Q ) : u = 0 on Σ } , where u k = ( u c k , u s k ) T , k ∈ N . W e also intro duce the o rthogonal vector u ⊥ k = ( − u s k , u c k ) T . The inner pro ducts (including also a σ -weigh ted v ersion) in these spaces are deﬁned by h ∂ 1 / 2 t u, ∂ 1 / 2 t v i := T 2 P ∞ k =1 k ω h u k , v k i Ω , and h σ ∂ 1 / 2 t u, ∂ 1 / 2 t v i := T 2 P ∞ k =1 k ω h σ u k , v k i Ω . The H 1 , 1 2 per ( Q ) -seminorm is deﬁned by | u | 2 1 , 1 2 = T k∇ u c 0 k 2 Ω + T 2 ∞ X k =1  k ω k u k k 2 Ω + k∇ u k k 2 Ω  = k∇ u k 2 + k ∂ 1 / 2 t u k 2 and the corres ponding nor m is k u k 2 1 , 1 2 := k u k 2 + | u | 2 1 , 1 2 . Finally , we a lso deﬁne the pro duct h κ, ∂ 1 / 2 t u i := T 2 P ∞ k =1 ( k ω ) 1 / 2 h κ k , u k i Ω as well a s the orthog o nal F o urier series u ⊥ ( x , t ) := ∞ X k =1 ( − u ⊥ k ) T · (cos( k ω t ) , s in( kω t )) T . Using this notation we can pro ve that ( u ⊥ ) ⊥ = -u,   u ⊥   =   u   and   ∂ 1 / 2 t u ⊥   =   ∂ 1 / 2 t u   for all u ∈ H 0 , 1 2 per ( Q ) and also that k u ⊥ k k Ω = k u k k Ω . Brieﬂy , we rec a ll from [25], the following iden tities h σ ∂ 1 / 2 t u, ∂ 1 / 2 t v i = h σ ∂ t u, v ⊥ i , h σ ∂ 1 / 2 t u, ∂ 1 / 2 t v ⊥ i = h σ ∂ t u, v i , ∀ u ∈ H 0 , 1 per ( Q ) ∀ v ∈ H 0 , 1 2 per ( Q ) , (5) as well a s orthogona lit y r elations h σ ∂ t u, u i = 0 and h σ u ⊥ , u i = 0 ∀ u ∈ H 0 , 1 per ( Q ) , (6) h σ ∂ 1 / 2 t u, ∂ 1 / 2 t u ⊥ i = 0 and h ν ∇ u, ∇ u ⊥ i = 0 ∀ u ∈ H 1 , 1 2 per ( Q ) . (7) 4 MONIKA WOLFMA YR Note that the following ident ity is v alid (in F ourier series sense) Z Q κ ∂ 1 / 2 t u ⊥ d x dt = − Z Q ∂ 1 / 2 t κ ⊥ u d x dt ∀ κ, u ∈ H 0 , 1 2 per ( Q ) . (8) F riedrichs’ inequality in Q can be prov ed by using standard F riedr ic hs’ inequa lit y on the F ourier co eﬃcien ts with res pect to the spatial domain Ω . W e have that k∇ u k 2 ≥ 1 C 2 F k u k 2 , where C F > 0 is F riedrichs’ constant. In the following, the parameter λ > 0 denotes the regular ization o r cost parameter. 2.2. Optimi zation probl em I. In the ﬁr st case, we w ant to minimize the following cost func- tional with resp ect to the unkno wn state y and control u : J ( y , u ) := 1 2 k y − y d k 2 + λ 2 k u k 2 (9) sub ject to the time-perio dic b oundary v a lue pr o blem (1)– (3). The g iv en desired state y d ∈ L 2 ( Q ) do es not ha ve to be time-p erio dic. It only has to b e from the space L 2 ( Q ) . The cost functional J deﬁned in (9) can be wr itten as J ( y , u ) = T J 0 ( y c 0 , u c 0 ) + T 2 ∞ X k =1 J k ( y k , u k ) , where J 0 ( y c 0 , u c 0 ) = 1 2 k y c 0 − y c d 0 k 2 Ω + λ 2 k u c 0 k 2 Ω and J k ( y k , u k ) = 1 2 k y k − y d k k 2 Ω + λ 2 k u k k 2 Ω . In [2 6], the corresp onding optimality system is der iv ed, which is g iven in weak formulation as follows: Giv en y d ∈ L 2 ( Q ) , ﬁnd y , p ∈ H 1 , 1 2 0 ,per ( Q ) such that Z Q  y z − ν ∇ p · ∇ z + σ ∂ 1 / 2 t p ∂ 1 / 2 t z ⊥  d x dt = Z Q y d z d x dt, ∀ z ∈ H 1 , 1 2 0 ,per ( Q ) , (10) Z Q  ν ∇ y · ∇ q + σ ∂ 1 / 2 t y ∂ 1 / 2 t q ⊥ + λ − 1 p q  d x dt = 0 , ∀ q ∈ H 1 , 1 2 0 ,per ( Q ) . (11) The reduced optimalit y system (1 0)–(11) i.a. results from using the condition u = − λ − 1 p , since no b ox constra in ts are impo sed on the co n trol function u in this pap er. This a lso leads to the space of admissible controls being giv en by H 1 , 1 2 0 ,per ( Q ) . Remark 1. Sinc e the optimality system is derive d ﬁrst, and then the disc r etization is p erforme d later, we c an say that the ﬁr st optimize, then discretize appr o ach is applie d her e (as discusse d, e.g., e arlier in [14] ). Let us deﬁne V := H 1 0 (Ω) and V := V × V . Expanding all functions in to F ourier se r ies in (10)– (11) together with using the orthogona lit y of the cosine and sine functions yie lds the following problems corresp onding to the k th and 0th F our ier co eﬃcients and whic h a re decoupled due to the linearity of the optimal control problem: Find y k , p k ∈ V so that Z Ω  y k · z k − ν ∇ p k · ∇ z k + k ω σ p k · z ⊥ k  d x = Z Ω y d k · z k d x , ∀ z k ∈ V , (12) Z Ω  ν ∇ y k · ∇ q k + k ω σ y k · q ⊥ k + λ − 1 p k · q k  d x = 0 , ∀ q k ∈ V . (13) and y c 0 , p c 0 ∈ V so that Z Ω  y c 0 · z c 0 − ν ∇ p c 0 · ∇ z c 0  d x = Z Ω y c d 0 · z c 0 d x , ∀ z c 0 ∈ V , (14) Z Ω  ν ∇ y c 0 · ∇ q c 0 + λ − 1 p c 0 · q c 0  d x = 0 , ∀ q c 0 ∈ V . (15) Both problems (12)–(1 3) and (14)–(15) are uniquely solv able (see [27]). GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 5 2.3. Ma joran t for cost functional (9) . Here, the re s ults of [26] o n upp er bounds for optimiza- tion pro blem I are summerized, since they ar e needed later to derive the tw o-sided estimate for the cost functional (9), which deepe ns and extends the a p o steriori error analysis fo r optimization problem I. Let y = y ( v ) b e the corresp onding state to an a r bitrary control v . The following upp e r bo und can b e prov ed: J ( y ( v ) , v ) ≤ J ⊕ ( α, β ; η , τ , v ) ∀ v ∈ L 2 ( Q ) , for arbitrar y α, β > 0 , η ∈ H 1 , 1 0 ,per ( Q ) and τ ∈ H ( div , Q ) := { τ ∈ [ L 2 ( Q )] d : ∇ · τ ( · , t ) ∈ L 2 (Ω) for a.e. t ∈ (0 , T ) } , where, for any τ ∈ H ( div , Q ) , the identit y Z Ω ∇ · τ w d x = − Z Ω τ · ∇ w d x ∀ w ∈ V is v alid. The guaranteed and fully co mputable ma jor ant is given b y J ⊕ ( α, β ; η , τ , v ) := 1 + α 2 k η − y d k 2 + γ ( kR 2 ( η , τ ) k 2 + C 2 F β kR 1 ( η , τ , v ) k 2 ) + λ 2 k v k 2 , (16) where µ 1 = 1 √ 2 min { ν , σ } and w e hav e set γ := (1+ α )(1+ β ) C 2 F 2 αµ 1 2 . The parameters α, β > 0 hav e b een in tro duced in or der to obtain a quadra tic functional b y applying Y oung ’s inequalit y . Remark 2. The arbitr ary functions η ∈ H 1 , 1 0 ,per ( Q ) and v ∈ L 2 ( Q ) c an b e taken later as the appr oximate solutions of the optimal c ontr ol pr oblem (9) su bje ct to (1) – (3) and τ ∈ H ( div , Q ) r epr esents the image of the ﬂux ν ∇ η . Note again that ﬁrst o ptimize, then discretize is appli e d in this p ap er, se e also R emark 1. F or the der iv ation of (16), the follo wing estimate for the a pproximation error has b een use d: | y ( v ) − η | 1 , 1 2 ≤ 1 µ 1 ( C F kR 1 ( η , τ , v ) k + kR 2 ( η , τ ) k ) , (17) where R 1 ( η , τ , v ) := σ ∂ t η − ∇ · τ − v and R 2 ( η , τ ) := τ − ν ∇ η . (18) The deriv a tion of es timate (17) can be found in [2 5]. The function J ⊕ ( α, β ; η , τ , v ) is a sharp upper bound on J ( y ( v ) , v ) for arbitrary but ﬁxed v as well as on the o ptimal v alue J ( y ( u ) , u ) inf η ∈ H 1 , 1 0 ,per ( Q ) , τ ∈ H ( div ,Q ) v ∈ L 2 ( Q ) ,α,β > 0 J ⊕ ( α, β ; η , τ , v ) = J ( y ( u ) , u ) , (19) since the inﬁm um is attained for the o ptimal con trol u , its corresp onding s tate y ( u ) and its exact ﬂux ν ∇ y ( u ) , and for α going to zer o . Therefore, w e hav e the estimate J ( y ( u ) , u ) ≤ J ⊕ ( α, β ; η , τ , v ) ∀ η ∈ H 1 , 1 0 ,per ( Q ) , τ ∈ H ( div , Q ) , v ∈ L 2 ( Q ) , α, β > 0 . (20) 2.4. Optimi zation problem I I. In the s econd case, we want to minimize the following cost functional with res pect to the unknown state y and con trol u : ˜ J ( y , u ) := 1 2 k∇ y − g d k 2 + λ 2 k u k 2 (21) sub ject to the time-p erio dic b oundary v a lue pro blem (1)– (3), where g d ∈ [ L 2 ( Q )] d is the given desired gradient. The optimality system can analogously be derived as for optimization pro blem I using the La grange functional ˜ L ( y , u, p ) = ˜ J ( y , u ) − Z Q ( σ ∂ t y − ∇ · ( ν ∇ y ) − u ) p d x dt. 6 MONIKA WOLFMA YR Optimality equations ∂ u ˜ L ( y , u, p ) = 0 and ∂ p ˜ L ( y , u, p ) = 0 are similar . Equation ∂ y ˜ L ( y , u, p ) = 0 is diﬀerent. The optimalit y conditions ar e given in weak form as follows: Given g d ∈ [ L 2 ( Q )] d , ﬁnd y , p ∈ H 1 , 1 2 0 ,per ( Q ) such that Z Q  ∇ y · ∇ z − ν ∇ p · ∇ z + σ ∂ 1 / 2 t p ∂ 1 / 2 t z ⊥  d x dt = Z Q g d · ∇ z d x dt, ∀ z ∈ H 1 , 1 2 0 ,per ( Q ) , (22) Z Q  ν ∇ y · ∇ q + σ ∂ 1 / 2 t y ∂ 1 / 2 t q ⊥ + λ − 1 p q  d x dt = 0 , ∀ q ∈ H 1 , 1 2 0 ,per ( Q ) . (23) The optimality systems co rresp onding to every mo de k are analogo usly derived as for optimization problem I (similar to (12)–(1 3) and (14)–(15)). In Section 4, we will derive new t wo-sided b ounds for optimization problem I I. 3. Guaranteed lower bounds leading to tw o-sided bounds f or optimiza tion pr oblem I In this w ork, we complemen t the guaranteed upper bo unds for the discretization error in state and control of minimizing co st functional J deﬁned in (9) sub ject to (1)–(3). This is done by obtaining fully computable lower bounds for J following the tec hnique fro m [44] (der ived for elliptic problems) leading to tw o-sided b ounds for the c o st functional (9). 3.1. Minorant for cost f unctional (9 ) . Let y = y ( u ) be the optimal state c orresp onding to the optimal control function u ∈ L 2 ( Q ) , which is connected with the a djoin t state p = p ( u ) by the iden tity u = − λ − 1 p ( u ) . Then y = y ( u ) is the solution of the v ariatio na l fo r m ulation Z Q  ν ∇ y · ∇ q + σ ∂ 1 / 2 t y ∂ 1 / 2 t q ⊥ + λ − 1 p q  d x dt = 0 ∀ q ∈ H 1 , 1 2 0 ,per ( Q ) (24) of the b oundary v alue problem (1)–(3) (see als o (10 )– (11)). F or any arbitray function η ∈ H 1 , 1 0 ,per ( Q ) , one can obtain that J ( y ( u ) , u ) = 1 2 k y − η k 2 + Z Q ( y − η ) ( η − y d ) d x dt + 1 2 k η − y d k 2 + λ 2 k u k 2 . Since 1 2 k y − η k 2 ≥ 0 and using the iden tit y u = − λ − 1 p ( u ) , J can b e estimated from below b y J ( y ( u ) , u ) = J ( y ( u ) , − λ − 1 p ( u )) ≥ 1 2 k η − y d k 2 + 1 2 λ k p k 2 + Z Q ( y − η ) ( η − y d ) d x dt. (25) F or η ∈ H 1 , 1 0 ,per ( Q ) , let p η , ˜ p η ∈ H 1 , 1 2 0 ,per ( Q ) b e the so lutions to the equations Z Q  ν ∇ p η · ∇ z − σ ∂ 1 / 2 t p η ∂ 1 / 2 t z ⊥  d x dt = Z Q ( η − y d ) z d x dt, ∀ z ∈ H 1 , 1 2 0 ,per ( Q ) , (26) Z Q  ν ∇ η · ∇ q + σ ∂ 1 / 2 t η ∂ 1 / 2 t q ⊥ + λ − 1 ˜ p η q  d x dt = 0 , ∀ q ∈ H 1 , 1 2 0 ,per ( Q ) . (27) Remark 3. Note that we assume d that η ∈ H 1 , 1 0 ,per ( Q ) ac c or ding to the derivation of t he major ant, but so far the assumption η ∈ H 1 , 1 2 0 ,per ( Q ) would b e enough. A dding and s ubtracting p η in (25) together with 1 2 λ k p − p η k 2 ≥ 0 yields the estimate J ( y ( u ) , u ) ≥ 1 2 k η − y d k 2 + 1 2 λ k p η k 2 + Z Q ( y − η ) ( η − y d ) d x dt + Z Q λ − 1 ( p − p η ) p η d x dt. GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 7 Using equation (26) and iden tity (8) leads to the estimate                                              J ( y ( u ) , u ) ≥ 1 2 k η − y d k 2 + 1 2 λ k p η k 2 + Z Q λ − 1 ( p − p η ) p η d x dt + Z Q  ν ∇ p η · ∇ ( y − η ) − σ ∂ 1 / 2 t p η ∂ 1 / 2 t ( y − η ) ⊥  d x dt = 1 2 k η − y d k 2 + 1 2 λ k p η k 2 + Z Q λ − 1 ( p − p η ) p η d x dt + Z Q  ( ν ∇ y − ν ∇ η ) · ∇ p η +  σ ∂ 1 / 2 t y − σ ∂ 1 / 2 t η  ∂ 1 / 2 t p ⊥ η  d x dt = 1 2 k η − y d k 2 + 1 2 λ k p η k 2 + Z Q  ν ∇ y · ∇ p η + σ ∂ 1 / 2 t y ∂ 1 / 2 t p ⊥ η + λ − 1 p p η  d x dt − Z Q  ν ∇ η · ∇ p η + σ ∂ 1 / 2 t η ∂ 1 / 2 t p ⊥ η + λ − 1 p η p η  d x dt. (28) By applying equations (24) and (27), it follows that J ( y ( u ) , u ) ≥ 1 2 k η − y d k 2 + 1 2 λ k p η k 2 + Z Q λ − 1 ( ˜ p η − p η ) p η d x dt. W e in tr oduce no w the arbitrary function ζ ∈ H 1 , 1 0 ,per ( Q ) . Note that at the momen t ζ ∈ H 1 , 1 2 0 ,per ( Q ) would be e no ugh, but the higher regular it y in time will be needed in another step. This go es along with the higher regula rit y assumption on η (see Remark 3). Since 1 2 λ k p η − ζ k 2 ≥ 0 , we have that J ( y ( u ) , u ) ≥ 1 2 k η − y d k 2 + 1 2 λ k ζ k 2 + Z Q λ − 1  p η ζ − ζ 2 + ˜ p η p η − p 2 η  d x dt. Now we add and subtract λ − 1 ˜ p η ζ in the la st integral as well a s use eq uation (27) a gain. More over, we exploit the fact that w e ha ve assumed that η ∈ H 1 , 1 0 ,per ( Q ) , hence, w e ca n apply the iden tities (5). Altogether these steps yield the estimate J ( y ( u ) , u ) ≥ 1 2 k η − y d k 2 + 1 2 λ k ζ k 2 − Z Q  ν ∇ η · ∇ ζ + σ ∂ t η ζ + λ − 1 ζ 2  d x dt + Z Q λ − 1 ( ζ − p η )( p η − ˜ p η ) d x dt. (29) In the following, we need to estimate the la st in tegra l of this expressio n in order to formulate a computable low er bo und for the co st functional. F or that let us ﬁrst prove a c o mputable upper bo und for the error in the adjoint state, whic h is presented i n the following theorem. Note that here we w ill need the higher reg ularity assumption (in time) on ζ . Theorem 1. L et y d ∈ L 2 ( Q ) b e given. L et p η ∈ H 1 , 1 2 0 ,per ( Q ) solve (26) for an arbitr ary η ∈ H 1 , 1 0 ,per ( Q ) . The fol lowing estimate hol ds: k∇ ( p η − ζ ) k ≤ 1 µ 1 ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) (30) for any ζ ∈ H 1 , 1 0 ,per ( Q ) , wher e µ 1 = 1 √ 2 min { ν , σ } , R 3 ( ζ , ρ , η ) = η − y d + ∇ · ρ + σ ∂ t ζ and R 4 ( ζ , ρ ) = ρ − ν ∇ ζ with ρ ∈ H ( div , Q ) and C F > 0 . Pr o of. Let us co nsider the adjoint equation (2 6). Adding a nd subtracting ζ ∈ H 1 , 1 0 ,per ( Q ) in the left-hand side of the equation leads to Z Q  ν ∇ ( p η − ζ ) · ∇ z − σ ∂ 1 / 2 t ( p η − ζ ) ∂ 1 / 2 t z ⊥  d x dt = Z Q ( η − y d ) z d x dt − Z Q ν ∇ ζ · ∇ z d x dt + Z Q σ ∂ 1 / 2 t ζ ∂ 1 / 2 t z ⊥ d x dt. (31) 8 MONIKA WOLFMA YR Next we intro duce the auxiliary v ariable ρ ∈ H ( div , Q ) . T ogether with using that ζ ∈ H 1 , 1 0 ,per ( Q ) as well as a pplying Cauch y–Sch warz’ and F r iedr ic hs’ inequalities, the following estimate for the right-hand side of (31) can b e obtained: sup 0 6 = z ∈ H 1 , 1 2 0 ,per ( Q ) R Q ( η − y d + ∇ · ρ + σ ∂ t ζ ) z d x dt + R Q ( ρ − ν ∇ ζ ) · ∇ z d x dt | z | 1 , 1 2 ≤ sup 0 6 = z ∈ H 1 , 1 2 0 ,per ( Q ) k η − y d + ∇ · ρ + σ ∂ t ζ kk z k + k ρ − ν ∇ ζ kk∇ z k | z | 1 , 1 2 ≤ sup 0 6 = z ∈ H 1 , 1 2 0 ,per ( Q ) C F k η − y d + ∇ · ρ + σ ∂ t ζ kk∇ z k + k ρ − ν ∇ ζ kk∇ z k k∇ z k ≤ C F k η − y d + ∇ · ρ + σ ∂ t ζ k + k ρ − ν ∇ ζ k . Using the b oundedness o f the co eﬃcien ts σ and ν , the ortho g onality rela tions (7) and applying that k z ⊥ k = k z k , we ca n prov e the estimate from below for the left-hand side o f (31), which is sup 0 6 = z ∈ H 1 , 1 2 0 ,per ( Q ) R Q  ν ∇ ( p η − ζ ) · ∇ z − σ ∂ 1 / 2 t ( p η − ζ ) ∂ 1 / 2 t z ⊥  d x dt | z | 1 , 1 2 . First, we estimate the supremum from b elow with choos ing z = ( p η − ζ ) + ( p η − ζ ) ⊥ , for which | z | 1 , 1 2 = | ( p η − ζ ) + ( p η − ζ ) ⊥ | 1 , 1 2 = √ 2 | p η − ζ | 1 , 1 2 , using the orthog onality relations (7). Next, applying the second equation in (7) g iv es h ν ∇ ( p η − ζ ) , ∇ z i = h ν ∇ ( p η − ζ ) , ∇ (( p η − ζ ) + ( p η − ζ ) ⊥ ) i = h ν ∇ ( p η − ζ ) , ∇ ( p η − ζ ) i + h ν ∇ ( p η − ζ ) , ∇ ( p η − ζ ) ⊥ i = h ν ∇ ( p η − ζ ) , ∇ ( p η − ζ ) i , and applying the ﬁrst equation in (7) as well as using the iden tit y (( p η − ζ ) ⊥ ) ⊥ = − ( p η − ζ ) gives h σ ∂ 1 / 2 t ( p η − ζ ) , ∂ 1 / 2 t z ⊥ i = h σ ∂ 1 / 2 t ( p η − ζ ) , ∂ 1 / 2 t (( p η − ζ ) + ( p η − ζ ) ⊥ ) ⊥ i = h σ ∂ 1 / 2 t ( p η − ζ ) , ∂ 1 / 2 t ( p η − ζ ) ⊥ i + h σ∂ 1 / 2 t ( p η − ζ ) , ∂ 1 / 2 t (( p η − ζ ) ⊥ ) ⊥ i = −h σ ∂ 1 / 2 t ( p η − ζ ) , ∂ 1 / 2 t ( p η − ζ ) i leading to the estimate sup 0 6 = z ∈ H 1 , 1 2 0 ,per ( Q ) R Q  ν ∇ ( p η − ζ ) · ∇ z − σ ∂ 1 / 2 t ( p η − ζ ) ∂ 1 / 2 t z ⊥  d x dt | z | 1 , 1 2 ≥ h ν ∇ ( p η − ζ ) , ∇ ( p η − ζ ) i + h σ ∂ 1 / 2 t ( p η − ζ ) , ∂ 1 / 2 t ( p η − ζ ) i | ( p η − ζ ) + ( p η − ζ ) ⊥ | 1 , 1 2 = h ν ∇ ( p η − ζ ) , ∇ ( p η − ζ ) i + h σ ∂ 1 / 2 t ( p η − ζ ) , ∂ 1 / 2 t ( p η − ζ ) i √ 2 | p η − ζ | 1 , 1 2 ≥ ν k∇ ( p η − ζ ) k 2 + σ k ∂ 1 / 2 t ( p η − ζ ) k 2 √ 2 | p η − ζ | 1 , 1 2 ≥ µ 1 | p η − ζ | 1 , 1 2 , where µ 1 = 1 √ 2 min { ν , σ } . Combining no w b oth estimates tog ether with k∇ ( p η − ζ ) k ≤ | p η − ζ | 1 , 1 2 we ﬁnally deriv e (30).  GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 9 Now we hav e all the to ols in order to estimate the last term of (29). W e o btain as follows Z Q λ − 1 ( ζ − p η )( p η − ζ + ζ − ˜ p η ) d x dt = Z Q ( λ − 1 ( ζ − p η )( p η − ζ ) + λ − 1 ( ζ − p η )( ζ − ˜ p η )) d x dt = − λ − 1 k ζ − p η k 2 + Z Q ( λ − 1 ( ζ − p η )( ζ − ˜ p η )) d x dt = − λ − 1 k ζ − p η k 2 + Z Q  ν ∇ η · ∇ ( ζ − p η ) + σ∂ 1 / 2 t η ∂ 1 / 2 t ( ζ − p η ) ⊥  d x dt + Z Q λ − 1 ζ ( ζ − p η ) d x dt = − λ − 1 k ζ − p η k 2 + Z Q  σ ∂ t η − ∇ · τ + λ − 1 ζ  ( ζ − p η ) d x dt + Z Q ( ν ∇ η − τ ) · ∇ ( ζ − p η ) d x dt ≥ − λ − 1 C 2 F k∇ ( ζ − p η ) k 2 − ( C F kR 1 ( η , τ , − λ − 1 ζ ) k + kR 2 ( η , τ ) k ) k∇ ( ζ − p η ) k leading to Z Q λ − 1 ( ζ − p η )( p η − ζ + ζ − ˜ p η ) d x dt ≥ − C 2 F µ 1 2 λ ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) 2 − 1 µ 1 ( C F kR 1 ( η , τ , − λ − 1 ζ ) k + kR 2 ( η , τ ) k ) ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) , (32) where τ , ρ ∈ H ( div , Q ) and we hav e used equation (27), relations (6)–(7), Cauch y–Sch warz’ and F riedrichs’ inequalities, estimate (30) and that η ∈ H 1 , 1 0 ,per ( Q ) . Finally , we obtain the following e s timate from b elow: J ( y ( u ) , u ) ≥ J ⊖ ( η , ζ , τ , ρ ) ∀ η , ζ ∈ H 1 , 1 0 ,per ( Q ) , τ , ρ ∈ H ( div , Q ) , (33) where the (fully computable) minoran t is given b y J ⊖ ( η , ζ , τ , ρ ) = 1 2 k η − y d k 2 + 1 2 λ k ζ k 2 − Z Q  ν ∇ η · ∇ ζ + σ ∂ t η ζ + λ − 1 ζ 2  d x dt − C 2 F µ 1 2 λ ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) 2 − 1 µ 1 ( C F kR 1 ( η , τ , − λ − 1 ζ ) k + kR 2 ( η , τ ) k ) ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) . (34) Theorem 2. The su pr emum of the minor ant J ⊖ as deﬁne d in (34) is attaine d for the m inimum of the c ost functional (9) subje ct to (1) – (3) , which is e quivalent to the optimal value of the optima lity system (10) – (11) as fol lows sup η, ζ ∈ H 1 , 1 0 ,per ( Q ) , τ , ρ ∈ H ( div ,Q ) J ⊖ ( η , ζ , τ , ρ ) = J ( y ( u ) , u ) . (35) Pr o of. The estimate is shar p for the exa ct control u , η = y ( u ) , ζ = p ( u ) , τ = ν ∇ y ( u ) and ρ = ν ∇ p ( u ) . Hence, J ⊖ ( y ( u ) , p ( u ) , ν ∇ y ( u )) , ν ∇ p ( u )) = 1 2 k y − y d k 2 + 1 2 λ k p k 2 = 1 2 k y − y d k 2 + λ 2 k u k 2 = J ( y ( u ) , u ) .  Remark 4 . It has b e en shown in [10] that if we cho ose ﬁnite di mensional su bsp ac es that ar e limit dense in t he sp ac es of the exact solution y ( u ) , of its adjoint p ( u ) and of t heir ﬂux es, ν ∇ y ( u ) and ν ∇ p ( u ) , whic h ar e H 1 , 1 0 ,per ( Q ) and H ( div , Q ) , and we cho ose se qu enc es of the fun ctions ( η , ζ , τ , ρ ) in these ﬁnite di mensional subsp ac es, so for instanc e ( η i , ζ i , τ i , ρ i ) , t hen they c onver ge (let i → ∞ ) to the exact solution, its ad joint and their ﬂuxes. The c orr esp onding major ants J ⊕ i and minor ants J ⊖ i c onver ge to the exact value of the c ost functional J . As shown in [10] , major ants c an b e use d in or der to pr o duc e se quenc es of state and c ontr ols with values of the c ost functional b eing as close to the ex act c ost functional value as one ne e ds it. 10 MONIKA WOLFMA YR 3.2. Guaran teed upp er b ounds for the discretization errors of the contr ol and the state. Here, we present a p oster iori err or estimates for con trol and state measured b y the norm ||| u − v ||| 2 := 1 2 k y ( u ) − y ( v ) k 2 + λ 2 k u − v k 2 . The next theorem was proved for the elliptic case (together w ith con trol co nstraints) in [44]. The nor m ||| · ||| can be r epresented in terms of the state and the adjoint state (instead of the con trol), since there are no control cons tr ain ts impo sed. Hence, u = − λ − 1 p ( u ) , v = − λ − 1 p ( v ) , and ||| u − v ||| 2 = 1 2 k y ( u ) − y ( v ) k 2 + 1 2 λ k p ( u ) − p ( v ) k 2 . Theorem 3. W e obtain the fol lowing identity for an arbitr ary c ontr ol v ∈ L 2 ( Q ) : ||| u − v ||| 2 = J ( y ( v ) , v ) − J ( y ( u ) , u ) . (36) Pr o of. T og ether with u = − λ − 1 p ( u ) and v = − λ − 1 p ( v ) the diﬀerence can be computed a s J ( y ( v ) , v ) − J ( y ( u ) , u ) = 1 2 k y ( v ) − y d k 2 − 1 2 k y ( u ) − y d k 2 + λ 2 k v k 2 − λ 2 k u k 2 = 1 2 Z Q ( y ( v ) − y ( u ) + 2 y ( u ) − 2 y d )( y ( v ) − y ( u )) d x dt + λ 2 Z Q ( v − u + 2 u )( v − u ) d x dt = 1 2 k y ( u ) − y ( v ) k 2 + Z Q ( y ( u ) − y d )( y ( v ) − y ( u )) d x dt + λ 2 k u − v k 2 + λ − 1 Z Q p ( u )( p ( v ) − p ( u )) d x dt. Since the adjoint sta tes p ( u ) , p ( v ) ∈ H 1 , 1 0 ,per ( Q ) fulﬁll (10)–(11) for the co rresp onding states y ( u ) , y ( v ) ∈ H 1 , 1 0 ,per ( Q ) , we obtain J ( y ( v ) , v ) − J ( y ( u ) , u ) = 1 2 k y ( u ) − y ( v ) k 2 + λ 2 k u − v k 2 + λ − 1 Z Q p ( u )( p ( v ) − p ( u )) d x dt + Z Q ( ν ∇ p ( u )( ∇ y ( v ) − ∇ y ( u )) − σ ∂ 1 / 2 t p ( u ) ∂ 1 / 2 t ( y ( v ) − y ( u )) ⊥ ) d x dt = 1 2 k y ( u ) − y ( v ) k 2 + λ 2 k u − v k 2 + λ − 1 Z Q p ( u ) p ( v ) d x dt + Z Q ( ν ∇ y ( v ) · ∇ p ( u ) + σ∂ 1 / 2 t y ( v ) ∂ 1 / 2 t p ( u ) ⊥ ) d x dt = 1 2 k y ( u ) − y ( v ) k 2 + λ 2 k u − v k 2 . This proves now the equalit y relation (36) by applying the equations (10)–(11) for ( u, y ( u ) , p ( u )) as well a s ( v , y ( v ) , p ( v )) .  Using the result of Theore m 3, we can derive the ma jorant for the discretiza tion e rrors of cont rol and state in the norm ||| · ||| . Theorem 4. The functional M ⊕ ( α, β ; η , ζ , τ , ρ , v ) = α 2 k η − y d k 2 + γ ( kR 2 ( η , τ ) k 2 + C 2 F β kR 1 ( η , τ , v ) k 2 ) + λ 2 k v k 2 − 1 2 λ k ζ k 2 + Z Q  ν ∇ η · ∇ ζ + σ ∂ t η ζ + λ − 1 ζ 2  d x dt + C 2 F µ 1 2 λ ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) 2 + 1 µ 1 ( C F kR 1 ( η , τ , − λ − 1 ζ ) k + kR 2 ( η , τ ) k ) ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) for an arbitr ary v ∈ L 2 ( Q ) , η , ζ ∈ H 1 , 1 0 ,per ( Q ) , τ , ρ ∈ H ( div , Q ) and α, β > 0 , and wher e µ 1 = 1 √ 2 min { ν , σ } , is a major ant for the discr etization err or ||| u − v ||| 2 ≤ M ⊕ ( α, β ; η , ζ , τ , ρ , v ) = J ⊕ ( α, β ; η , τ , v ) − J ⊖ ( η , ζ , τ , ρ ) . (37) Pr o of. Applying (36) together with (20) and (33) yields estimate (37).  GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 11 Prop osition 1. The inﬁmum of the major ant M ⊕ ( α, β ; η , ζ , τ , ρ , v ) in (37 ) is attaine d for t he minimum of the optimiza tion pr oblem I, which is e quivalent to the solution of the optimality system (10) – (11) ( v = u , η = y ( u ) , ζ = p ( u ) = − λu , τ = ν ∇ y ( u ) , ρ = ν ∇ p ( u )) as fol lows inf η, ζ ∈ H 1 , 1 0 ,per ( Q ) , τ , ρ ∈ H ( di v ,Q ) , v ∈ L 2 ( Q ) ,α,β > 0 M ⊕ ( α, β ; η , ζ , τ , ρ , v ) = 0 . Pr o of. The ma jorant M ⊕ ( α, β ; y ( u ) , p ( u ) , ν ∇ y ( u ) , ν ∇ p ( u ) , u ) = α 2 k y ( u ) − y d k 2 equals zero if α go es to zero.  The ma jor ant M ⊕ is a sharp, guara n teed and fully co mputable upper b ound for the control- state er ror in ||| · ||| . Howev er , it ov erestimates the L 2 -norm ||| · ||| which is of o rder h 2 , since the ma jorant only decreases with order h . F ollowing the idea fro m [44] a weighted H 1 -norm is intro duced decre asing with the same or der as the ma jora n t. F or that, we deﬁne the norm ||| u − v ||| 2 1 := 1 2 k y ( u ) − y ( v ) k 2 + λµ 1 2 2 C 2 F | y ( u ) − y ( v ) | 2 1 , 1 2 . Theorem 5. The fol lowing estimate: ||| u − v ||| 2 1 ≤ J ( y ( v ) , v ) − J ( y ( u ) , u ) + 3 λ 4 C 2 F ( C F kR 1 ( η , τ , v ) k + k R 2 ( η , τ ) k ) 2 (38) is valid for an arbitr ary c ontr ol function v ∈ L 2 ( Q ) and with R 1 ( η , τ , v ) and R 2 ( η , τ ) deﬁne d as in (18) . Pr o of. Let the parameter δ > 0 b e ar bitrary but ﬁxed. W e add and subtract η , a pply tria ngle inequality o n µ 1 2 C 2 F δ | y ( u ) − y ( v ) | 2 1 , 1 2 and obtain µ 1 2 C 2 F δ | y ( u ) − y ( v ) | 2 1 , 1 2 ≤ µ 1 2 2 C 2 F δ  | y ( u ) − η | 2 1 , 1 2 + | y ( v ) − η | 2 1 , 1 2  , where we further add and subtract v . Then we apply triangle inequalit y tw o times leading to µ 1 2 C 2 F δ | y ( u ) − y ( v ) | 2 1 , 1 2 ≤ 1 2 C 2 F δ   k τ − ν ∇ η k + C F k σ ∂ t η − ∇ · τ − v k + C F k u − v k  2 + ( k τ − ν ∇ η k + C F k σ ∂ t η − ∇ · τ − v k ) 2  ≤ 3 2 C 2 F δ ( k τ − ν ∇ η k + C F k σ ∂ t η − ∇ · τ − v k ) 2 + 1 δ k u − v k 2 . T ogether with (36 ) this yields ||| u − v ||| 2 + µ 1 2 C 2 F δ | y ( u ) − y ( v ) | 2 1 , 1 2 − 1 δ k u − v k 2 = 1 2 k y ( u ) − y ( v ) k 2 + µ 1 2 C 2 F δ | y ( u ) − y ( v ) | 2 1 , 1 2 +  λ 2 − 1 δ  k u − v k 2 ≤ J ( y ( v ) , v ) − J ( y ( u ) , u ) + 3 2 C 2 F δ ( C F kR 1 ( η , τ , v ) k + kR 2 ( η , τ ) k ) 2 . W e see that the c hoice δ = 2 /λ ﬁnally provides estimate (38).  This theorem directly leads to the following tw o results presented in Prop ositions 2 and 3. Prop osition 2. The fol lowing err or major ant for any c ontr ol v ∈ L 2 ( Q ) is obtaine d: ||| u − v ||| 2 1 ≤ M ⊕ 1 ( α, β ; η , ζ , τ , ρ , v ) := J ⊕ ( α, β ; η , τ , v ) − J ⊖ ( η , ζ , τ , ρ ) + 3 λ 4 C 2 F ( C F kR 1 ( η , τ , v ) k + kR 2 ( η , τ ) k ) 2 (39) 12 MONIKA WOLFMA YR with M ⊕ 1 ( α, β ; η , ζ , τ , ρ , v ) = α 2 k η − y d k 2 + γ ( kR 2 ( η , τ ) k 2 + C 2 F β kR 1 ( η , τ , v ) k 2 ) + λ 2 k v k 2 − 1 2 λ k ζ k 2 + Z Q  ν ∇ η · ∇ ζ + σ ∂ t η ζ + λ − 1 ζ 2  d x dt + C 2 F µ 1 2 λ ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) 2 + 1 µ 1 ( C F kR 1 ( η , τ , − λ − 1 ζ ) k + kR 2 ( η , τ ) k ) ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) + 3 λ 4 C 2 F ( C F kR 1 ( η , τ , v ) k + k R 2 ( η , τ ) k ) 2 wher e µ 1 = 1 √ 2 min { ν , σ } , α, β > 0 as wel l as arb itr ary η , ζ ∈ H 1 , 1 0 ,per ( Q ) and τ , ρ ∈ H ( div , Q ) . Prop osition 3. The inﬁmum of the major ant M ⊕ 1 ( α, β ; η , ζ , τ , ρ , v ) in (39 ) is attaine d for t he minimum of the optimization pr oblem I b eing e quivalent to solving the optimality system (10) – (11) ( v = u , η = y ( u ) , ζ = p ( u ) = − λu , τ = ν ∇ y ( u ) , ρ = ν ∇ p ( u )) as fol lows inf η, ζ ∈ H 1 , 1 0 ,per ( Q ) , τ , ρ ∈ H ( di v ,Q ) , v ∈ L 2 ( Q ) ,α,β > 0 M ⊕ 1 ( α, β ; η , ζ , τ , ρ , v ) = 0 , and α going to zer o. 4. Two- sided bounds for optimiza tion pr oblem I I In this section, w e analogous ly derive the ma jor ant s and mino rants for the sec o nd cost func- tional, how ever, sk ipping details which a re similar to the deriv ation in the case of o ptimization problem I. 4.1. Ma joran t fo r cost functional (21 ) . A dding and subtra cting ∇ η in the cost functional ˜ J ( y ( v ) , v ) , applying the tr iangle inequality a s w ell as using the estimate k∇ y ( v ) − ∇ η k 2 ≤ | y ( v ) − η | 2 1 , 1 2 = k ∇ y ( v ) − ∇ η k 2 + k ∂ 1 / 2 t y ( v ) − ∂ 1 / 2 t η k 2 , we co nclude that ˜ J ( y ( v ) , v ) ≤ 1 2  k∇ η − g d k + | y ( v ) − η | 1 , 1 2  2 + λ 2 k v k 2 . T ogether with (17 ) this leads to the estimate ˜ J ( y ( v ) , v ) ≤ 1 2  k∇ η − g d k + 1 µ 1 kR 2 ( η , τ ) k + C F µ 1 kR 1 ( η , τ , v ) k  2 + λ 2 k v k 2 , where a gain µ 1 = 1 √ 2 min { ν , σ } a s well as R 1 ( η , τ , v ) and R 2 ( η , τ ) are deﬁned as in (18). Finally , in tro ducing parameters α, β > 0 and applying Y oung’s inequality , we ca n refor m ulate the estimate such that the right -hand side is given b y a quadr atic functional as follows ˜ J ( y ( v ) , v ) ≤ ˜ J ⊕ ( α, β ; η , τ , v ) ∀ v ∈ L 2 ( Q ) , where ˜ J ⊕ ( α, β ; η , τ , v ) := 1 + α 2 k∇ η − g d k 2 + γ ( kR 2 ( η , τ ) k 2 + C 2 F β kR 1 ( η , τ , v ) k 2 ) + λ 2 k v k 2 . (40) The inﬁm um of the ma jora n t (40) is attained fo r the minimum of the cost functional (21) sub ject to (1)–(3), which is equiv alent to the o ptimal v alue of the optimality system (22)–(23). Analog ously to (19), we can show that inf η ∈ H 1 , 1 0 ,per ( Q ) , τ ∈ H ( div ,Q ) v ∈ L 2 ( Q ) ,α,β > 0 ˜ J ⊕ ( α, β ; η , τ , v ) = ˜ J ( y ( u ) , u ) , (41) and that ˜ J ( y ( u ) , u ) ≤ ˜ J ⊕ ( α, β ; η , τ , v ) ∀ η ∈ H 1 , 1 0 ,per ( Q ) , τ ∈ H ( div , Q ) , v ∈ L 2 ( Q ) , α, β > 0 . (42) GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 13 4.2. Minorant for cos t functional (21) . Let us derive now the minora n t. F or any η ∈ H 1 , 1 0 ,per ( Q ) , we have that ˜ J ( y ( v ) , v ) = 1 2 k∇ y − ∇ η k 2 + Z Q ( ∇ y − ∇ η ) · ( ∇ η − g d ) d x dt + 1 2 k∇ η − g d k 2 + λ 2 k v k 2 for all v ∈ L 2 ( Q ) . The ﬁr st norm is again gr eater or equal to zero, together with the identit y v = − λ − 1 p ( v ) , we can estimate ˜ J fro m b elow by ˜ J ( y ( v ) , v ) = ˜ J ( y ( v ) , − λ − 1 p ( v )) ≥ 1 2 k∇ η − g d k 2 + 1 2 λ k p k 2 + Z Q ( ∇ y − ∇ η ) · ( ∇ η − g d ) d x dt. Note that Remark 3 a pplies here as well. F or η ∈ H 1 , 1 0 ,per ( Q ) , let p η , ˜ p η ∈ H 1 , 1 2 0 ,per ( Q ) b e the solutions to the equations Z Q  ν ∇ p η · ∇ z − σ ∂ 1 / 2 t p η ∂ 1 / 2 t z ⊥  d x dt = Z Q ( ∇ η − g d ) · ∇ z d x dt, ∀ z ∈ H 1 , 1 2 0 ,per ( Q ) , (43) Z Q  ν ∇ η · ∇ q + σ ∂ 1 / 2 t η ∂ 1 / 2 t q ⊥ + λ − 1 ˜ p η q  d x dt = 0 , ∀ q ∈ H 1 , 1 2 0 ,per ( Q ) . (44) Deriving the minor an t for the second minimization functional uses similar ideas now as presented for pr oblem I (see Subsection 3.1). How ever, in the following we present the main steps whic h are impo r tan t for problem I I. Adding and subtracting p η together with 1 2 λ k p − p η k 2 ≥ 0 yields ˜ J ( y ( v ) , v ) = ˜ J ( y ( v ) , − λ − 1 p ( v )) ≥ 1 2 k∇ η − g d k 2 + 1 2 λ k p η k 2 + Z Q ( ∇ y − ∇ η ) · ( ∇ η − g d ) d x dt + Z Q λ − 1 ( p − p η ) p η d x dt. App ying equation (43), identit y (8) (a nalogously to (28)) and then using equa tio ns (2 4) and (44) provides the estimate ˜ J ( y ( u ) , u ) ≥ 1 2 k∇ η − g d k 2 + 1 2 λ k p η k 2 + Z Q λ − 1 ( ˜ p η − p η ) p η d x dt. T ogether with in tro ducing an arbitrary function ζ ∈ H 1 , 1 0 ,per ( Q ) , following (29), applying Theore m 1 and using (32), w e ﬁnally der iv e the estimate ˜ J ( y ( u ) , u ) ≥ ˜ J ⊖ ( η , ζ , τ , ρ ) ∀ η , ζ ∈ H 1 , 1 0 ,per ( Q ) , τ , ρ ∈ H ( div , Q ) (45) with the fully co mputable minorant ˜ J ⊖ ( η , ζ , τ , ρ ) = 1 2 k∇ η − g d k 2 + 1 2 λ k ζ k 2 − Z Q  ν ∇ η · ∇ ζ + σ ∂ t η ζ + λ − 1 ζ 2  d x dt − C 2 F µ 1 2 λ ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) 2 − 1 µ 1 ( C F kR 1 ( η , τ , − λ − 1 ζ ) k + kR 2 ( η , τ ) k ) ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) . (46) The following theor e m is analog ous to Theo rem 2 and so is its pro o f. Also Remark 4 can be applied for optimizatio n problem II. Theorem 6. The su pr emum of the minor ant ˜ J ⊖ as deﬁne d in (46) is attaine d for the m inimum of the c ost functional (21 ) subje ct to (1) – (3) , which is e quivalent to the optimal value of t he optimality system (22) – (23) as fol lows sup η, ζ ∈ H 1 , 1 0 ,per ( Q ) , τ , ρ ∈ H ( div ,Q ) ˜ J ⊖ ( η , ζ , τ , ρ ) = ˜ J ( y ( u ) , u ) . (47) 14 MONIKA WOLFMA YR 4.3. Guaran teed upp er b ounds for the discretization errors of the contr ol and the state. W e presen t the ma jora n ts for the control-state er rors measured in the norm |||| u − v |||| 2 := 1 2 k∇ y ( u ) − ∇ y ( v ) k 2 + λ 2 k u − v k 2 = 1 2 k∇ y ( u ) − ∇ y ( v ) k 2 + 1 2 λ k p ( u ) − p ( v ) k 2 . W e obtain the identit y |||| u − v |||| 2 = ˜ J ( y ( v ) , v ) − ˜ J ( y ( u ) , u ) for an arbitrary con trol v ∈ L 2 ( Q ) yielding the error ma jorant |||| u − v |||| 2 ≤ ˜ M ⊕ ( α, β ; η , ζ , τ , ρ , v ) := ˜ J ⊕ ( α, β ; η , τ , v ) − ˜ J ⊖ ( η , ζ , τ , ρ ) (48) for α, β > 0 , arbitrary η, ζ ∈ H 1 , 1 0 ,per ( Q ) , τ , ρ ∈ H ( div , Q ) with ˜ M ⊕ ( α, β ; η , ζ , τ , ρ , v ) = α 2 k∇ η − g d k 2 + γ ( kR 2 ( η , τ ) k 2 + C 2 F β kR 1 ( η , τ , v ) k 2 ) + λ 2 k v k 2 − 1 2 λ k ζ k 2 + Z Q  ν ∇ η · ∇ ζ + σ ∂ t η ζ + λ − 1 ζ 2  d x dt + C 2 F µ 1 2 λ ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) 2 + 1 µ 1 ( C F kR 1 ( η , τ , − λ − 1 ζ ) k + kR 2 ( η , τ ) k ) ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) , where µ 1 = 1 √ 2 min { ν , σ } . The inﬁmu m of this ma jora nt is attained for the minimum of the optimization problem I I being equiv alent to solving the optimality system (22)–(23) ( v = u , η = y ( u ) , ζ = p ( u ) = − λu , τ = ν ∇ y ( u ) and ρ = ν ∇ p ( u )) as follows inf η, ζ ∈ H 1 , 1 0 ,per ( Q ) , τ , ρ ∈ H ( di v ,Q ) , v ∈ L 2 ( Q ) ,α,β > 0 ˜ M ⊕ ( α, β ; η , ζ , τ , ρ , v ) = 0 . Analogously , deﬁning |||| u − v |||| 2 1 := 1 2 k∇ y ( u ) − ∇ y ( v ) k 2 + λµ 1 2 2 C 2 F | y ( u ) − y ( v ) | 2 1 , 1 2 we derive |||| u − v |||| 2 1 ≤ ˜ M ⊕ 1 ( α, β ; η , ζ , τ , ρ , v ) := ˜ J ⊕ ( α, β ; η , τ , v ) − ˜ J ⊖ ( η , ζ , τ , ρ ) + 3 λ 4 C 2 F ( C F kR 1 ( η , τ , v ) k + kR 2 ( η , τ ) k ) 2 , where now ˜ M ⊕ 1 ( α, β ; η , ζ , τ , ρ , v ) = α 2 k∇ η − g d k 2 + γ ( kR 2 ( η , τ ) k 2 + C 2 F β kR 1 ( η , τ , v ) k 2 ) + λ 2 k v k 2 − 1 2 λ k ζ k 2 + Z Q  ν ∇ η · ∇ ζ + σ ∂ t η ζ + λ − 1 ζ 2  d x dt + C 2 F µ 1 2 λ ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) 2 + 1 µ 1 ( C F kR 1 ( η , τ , − λ − 1 ζ ) k + kR 2 ( η , τ ) k ) ( C F kR 3 ( ζ , ρ , η ) k + kR 4 ( ζ , ρ ) k ) + 3 λ 4 C 2 F ( C F kR 1 ( η , τ , v ) k + kR 2 ( η , τ ) k ) 2 . All other results simila r to Propositio ns 2 and 3 follow completely . 5. Mul tiharmonic finite element (MhFE) discretiza tion The desir ed state y d and des ir ed gr adien t g d belo ng to L 2 ( Q ) and [ L 2 ( Q )] d , resp ectively . So they can be repr esen ted a s F ourier series ha ving F ourier coe ﬃcien ts from L 2 (Ω) . Mor eov er , we assume that η and ζ approximating the e xact state y and adjoint state p , resp ectively , as w ell as the vector- v alued functions τ and ρ are g iv en by trunca ted F o ur ier series. W e also hav e the m ultiharmonic time deriv ative deﬁned by ∂ t η ( x , t ) = P N k =1 ( k ω η s k ( x ) cos ( kω t ) − k ω η c k ( x ) sin( k ω t )) a s w ell as the gradient and div ergence by ∇ η ( x , t ) = ∇ η c 0 ( x ) + N X k =1 ( ∇ η c k ( x ) cos( k ω t ) + ∇ η s k ( x ) sin( k ω t )) , ∇ · τ ( x , t ) = ∇ · τ c 0 ( x ) + N X k =1 ( ∇ · τ c k ( x ) cos( k ω t ) + ∇ · τ s k ( x ) sin( k ω t )) . GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 15 F or the n umerical treatmen t, we truncate the F ourier series expansio ns of all appea ring functions at an index N ∈ N creating F ourier s eries approximations of the functions. Next, w e approximate the F ourier coeﬃcients y k = ( y c k , y s k ) T ∈ V and p k = ( p c k , p s k ) T ∈ V of the unknown state and adjoint state functions by ﬁnite element functions y kh = ( y c kh , y s kh ) T ∈ V h and p kh = ( p c kh , p s kh ) T ∈ V h . The ﬁnite element s pa ce V h = V h × V h ⊂ V with V h = span { φ 1 , . . . , φ n } and { φ i ( x ) : i = 1 , 2 , . . . , n h } is conforming. W e hav e deﬁned n = n h = dim V h = O ( h − d ) and h is the discretiza tio n parameter. The ba sis of the ﬁnite element space V h on the tria ngulation T h , which is r egular, consists of piecewise linear a nd contin uo us elemen ts. 5.1. Optimi zation problem I. The MhFE discretization (see also [27] fo r mo r e details on the m ultiharmonic ﬁnite element a na lysis for such an optimal control problem) yields the system of linear equations having a saddle p oint structure corresp onding to the decoupled problems (12)–(13) as follows     M h 0 − K h,ν k ω M h,σ 0 M h − k ω M h,σ − K h,ν − K h,ν − k ω M h,σ − λ − 1 M h 0 k ω M h,σ − K h,ν 0 − λ − 1 M h         y c k y s k p c k p s k     =     M h y c d k M h y s d k 0 0     ∀ k = 1 , . . . , N . (49) The functions y c kh ( x ) = P n j =1 y c k,j φ j ( x ) , y s kh ( x ) = P n j =1 y s k,j φ j ( x ) , p c kh ( x ) = P n j =1 p c k,j φ j ( x ) and p s kh ( x ) = P n j =1 p s k,j φ j ( x ) are deﬁned by the co rresp onding no dal function v a lues y c k = ( y c k,j ) j =1 ,...,n , y s k = ( y s k,j ) j =1 ,...,n , p c k = ( p c k,j ) j =1 ,...,n , p s k = ( p s k,j ) j =1 ,...,n ∈ R n . W e hav e deﬁned the stiﬀness matrix K h,ν as well a s the mass matrices M h and M h,σ b y their en tr ies K ij h,ν = Z Ω ν ∇ φ i · ∇ φ j d x , M ij h = Z Ω φ i φ j d x , M ij h,σ = Z Ω σ φ i φ j d x . The right-hand side can b e obtained by computing the v ectors M h y c d k =  Z Ω y c d k φ i d x  i =1 ,...,n and M h y s d k =  Z Ω y s d k φ i d x  i =1 ,...,n . Note that for the ca se k = 0 , hence for (14)–(15), we obta in  M h − K h,ν − K h,ν − λ − 1 M h   y c 0 p c 0  =  M h y c d 0 0  . (50) Solving all the systems of linear equations ﬁnally lead to the contributions for computing the MhFE approximations y N h ( x , t ) and p N h ( x , t ) given by y N h ( x , t ) = y c 0 h ( x ) + P N k =1 ( y c kh ( x ) cos( k ω t ) + y s kh ( x ) sin( k ω t )) , p N h ( x , t ) = p c 0 h ( x ) + P N k =1 ( p c kh ( x ) cos( k ω t ) + p s kh ( x ) sin( k ω t )) . F or some prop er fast solvers for these sys tems we r efer to [1 7, 21, 2 7]. Both, ma jor a n t (1 6) and mino- rant (34) o f the cost functional J ca n b e computed by c ho osing the MhFE approximations y N h , p N h and u N h = − λ − 1 p N h as η , ζ and v , r espectively . The arbitrary functions τ a nd ρ can also be repres en ted in for m of m ultiharmonic functions τ N h = τ c 0 h ( x ) + P N k =1 ( τ c kh ( x ) cos( k ω t ) + τ s kh ( x ) sin( k ω t )) and ρ N h = ρ c 0 h ( x ) + P N k =1 ( ρ c kh ( x ) cos ( kω t ) + ρ s kh ( x ) sin( k ω t )) . Hence, ma jo- rant (16) and minor ant (34) have a multiharmonic str uctur e to o. The linearity o f the problem again y ie lds the decoupling of the problems introducing α k , β k > 0 and resulting int o ma jo- rants J ⊕ k and minora n ts J ⊖ k corresp onding to ev ery F o ur ier mo de. W e start with the ma jora n ts J ⊕ 0 = J ⊕ 0 ( α 0 , β 0 ; y c 0 h , p c 0 h , τ c 0 h ) and J ⊕ k = J ⊕ k ( α k , β k ; y kh , p kh , τ kh ) tog e ther with deﬁning the parameter γ k := ((1 + α k )(1 + β k ) C 2 F ) / (2 α k µ 1 2 ) . W e ha ve that J ⊕ 0 = 1 + α 0 2 k y c 0 h − y d c 0 k 2 Ω + 1 2 λ k p c 0 h k 2 Ω + γ 0 ( kR 2 c 0 k 2 Ω + C 2 F β 0 kR 1 c 0 k 2 Ω ) (51) and J ⊕ k = 1 + α k 2 k y kh − y d k k 2 Ω + 1 2 λ k p kh k 2 Ω + γ k ( kR 2 k k 2 Ω + C 2 F β k kR 1 k k 2 Ω ) . (52) 16 MONIKA WOLFMA YR Deﬁning α N +1 = ( α 0 , . . . , α N +1 ) T and β N = ( β 0 , . . . , β N ) T , w e can wr ite the ov erall ma jor ant as J ⊕ ( α N +1 , β N ; y N h , p N h , τ N h ) = T J ⊕ 0 + T 2 N X k =1 J ⊕ k + 1 + α N +1 2 E N . (53) Here, the terms ar e R 1 c 0 = ∇ · τ c 0 h − λ − 1 p c 0 h , R 2 c 0 = τ c 0 h − ν ∇ y c 0 h , R 1 k = k ω σ y ⊥ kh + div τ kh − λ − 1 p kh = ( R 1 c k , R 1 s k ) T = ( − k ω σ y s kh + ∇ · τ c kh − λ − 1 p c kh , k ω σ y c kh + ∇ · τ s kh − λ − 1 p s kh ) T and R 2 k = τ kh − ν ∇ y kh = ( R 2 c k , R 2 s k ) T = ( τ c kh − ν ∇ y c kh , τ s kh − ν ∇ y s kh ) T . The trunca tion’s remainder term E N := k y d − y d N k 2 = T 2 P ∞ k = N +1 k y d k k 2 Ω = T 2 P ∞ k = N +1  k y c d k k 2 Ω + k y s d k k 2 Ω  can alwa ys be computed with any desired accurac y , since y d is known (see also [25]). The minorant (34) can b e written a s J ⊖ ( y N h , p N h , τ N h , ρ N h ) = T J ⊖ 0 + T 2 N X k =1 J ⊖ k + E N 2 , (54) where J ⊖ 0 = J ⊖ 0 ( y c 0 h , p c 0 h , τ c 0 h , ρ c 0 h ) and J ⊖ k = J ⊖ k ( y kh , p kh , τ kh , ρ kh ) are given b y J ⊖ 0 = 1 2 k y c 0 h − y d c 0 k 2 Ω + 1 2 λ k p c 0 h k 2 Ω − Z Ω  ν ∇ y c 0 h · ∇ p c 0 h + λ − 1 ( p c 0 h ) 2  d x − C 2 F µ 1 2 λ ( C F kR 3 c 0 k Ω + kR 4 c 0 k Ω ) 2 − 1 µ 1 ( C F kR 1 c 0 k Ω + kR 2 c 0 k Ω ) ( C F kR 3 c 0 k Ω + kR 4 c 0 k Ω ) (55) with R 3 c 0 = ∇ · ρ c 0 h + y c 0 h − y d c 0 , R 4 c 0 = ρ c 0 h − ν ∇ p c 0 h , and J ⊖ k = 1 2 k y kh − y d k k 2 Ω + 1 2 λ k p kh k 2 Ω − Z Ω  ν ∇ y kh · ∇ p kh − k ω σ y ⊥ kh · p kh + λ − 1 p 2 kh  d x − C 2 F µ 1 2 λ ( C F kR 3 k k Ω + kR 4 k k Ω ) 2 − 1 µ 1 ( C F kR 1 k k Ω + kR 2 k k Ω ) ( C F kR 3 k k Ω + kR 4 k k Ω ) (56) with R 3 k = k ω σ p ⊥ kh + ∇ · ρ kh + y kh − y d k = ( R 3 c k , R 3 s k ) T = ( − k ω σ p s kh + ∇ · ρ c kh + y c kh − y d c k , k ω σ p c kh + ∇ · ρ s kh + y s kh − y d s k ) T and R 4 k = ρ kh − ν ∇ p kh = ( R 4 c k , R 4 s k ) T = ( ρ c kh − ν ∇ p c kh , ρ s kh − ν ∇ p s kh ) T . Remark 5. F or any index ¯ N ∈ N , ¯ N > N , the t runc ate d re mainder term E N , ¯ N := T 2 ¯ N X k = N +1 k y d k k 2 Ω is a ful ly c omputable lower b ound for the r emainder term E N . F or any given y d ∈ L 2 ( Q ) , this pr ovides an arbi tr arily tight lower b ound for the minor ant (54) , which is in r eturn optimal. The ﬂuxes τ c 0 h , ρ c 0 h and τ kh , ρ kh for a ll k = 1 , . . . , N , denoted by τ kh = R ﬂux h ( ν ∇ y kh ) a nd ρ kh = R ﬂux h ( ν ∇ p kh ) a r e reconstr ucted b y low est-order Raviart-Thomas elements mapping L 2 - functions to H ( div , Ω) , see [38] as w ell as [25, 26], leading to τ N h = R ﬂux h ( ν ∇ y N h ) and ρ N h = R ﬂux h ( ν ∇ p N h ) . W e minimize J ⊕ with resp ect to the po sitiv e par ameters α k and β k leading to the optimized α N +1 and β N . Finally , the multiharmonic ma jor ant (53) and minorant (54) lead to upper and low er b ounds for J whic h are guaranteed and computable. The computation o f the inﬁm um of J ⊕ and the supremum of J ⊖ provide the minim um of J , see also [26]. 5.2. Optimi zation probl em I I. F or the s econd problem, w e only s umma r ize the main r e sults and changes. The MhFE discretization leads to the following discrete problem:     K h 0 − K h,ν k ω M h,σ 0 K h − k ω M h,σ − K h,ν − K h,ν − k ω M h,σ − λ − 1 M h 0 k ω M h,σ − K h,ν 0 − λ − 1 M h         y c k y s k p c k p s k     =     K h g c d k K h g s d k 0 0     (57) GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 17 and  K h − K h,ν − K h,ν − λ − 1 M h   y c 0 p c 0  =  K h g c d 0 0  . (58) Now, the righ t-hand side vectors are computed b y K h g c d k =  Z Ω g d c k · ∇ φ i d x  i =1 ,...,n and K h g s d k =  Z Ω g d s k · ∇ φ i d x  i =1 ,...,n . W e summarize the discrete ma jor a n t (40) and minorant (46) of the cost functional ˜ J computed b y choosing the MhFE approximations for all the (arbitrar y) functions. Deﬁning now ˜ E N := k g d − g d N k 2 = T 2 P ∞ k = N +1 k g d k k 2 Ω = T 2 P ∞ k = N +1  k g c d k k 2 Ω + k g s d k k 2 Ω  as truncatio n’s remainder term, we write the ma jorant (40) as ˜ J ⊕ ( α N +1 , β N ; y N h , p N h , τ N h ) = T ˜ J ⊕ 0 + T 2 N X k =1 ˜ J ⊕ k + 1 + α N +1 2 ˜ E N , (59) where ˜ J ⊕ 0 = ˜ J ⊕ 0 ( α 0 , β 0 ; y c 0 h , p c 0 h , τ c 0 h ) and ˜ J ⊕ k = ˜ J ⊕ k ( α k , β k s ; y kh , p kh , τ kh ) are g iven b y ˜ J ⊕ 0 = 1 + α 0 2 k∇ y c 0 h − g d c 0 k 2 Ω + 1 2 λ k p c 0 h k 2 Ω + γ 0 ( kR 2 c 0 k 2 Ω + C 2 F β 0 kR 1 c 0 k 2 Ω ) (60) and ˜ J ⊕ k = 1 + α k 2 k∇ y kh − g d k k 2 Ω + 1 2 λ k p kh k 2 Ω + γ k ( kR 2 k k 2 Ω + C 2 F β k kR 1 k k 2 Ω ) . (61) The minora nt (4 6) can b e written as ˜ J ⊖ ( y N h , p N h , τ N h , ρ N h ) = T ˜ J ⊖ 0 + T 2 N X k =1 ˜ J ⊖ k + ˜ E N 2 , (62) where ˜ J ⊖ 0 = ˜ J ⊖ 0 ( y c 0 h , p c 0 h , τ c 0 h , ρ c 0 h ) and ˜ J ⊖ k = ˜ J ⊖ k ( y kh , p kh , τ kh , ρ kh ) are given b y ˜ J ⊖ 0 = 1 2 k∇ y c 0 h − g d c 0 k 2 Ω + 1 2 λ k p c 0 h k 2 Ω − Z Ω  ν ∇ y c 0 h · ∇ p c 0 h + λ − 1 ( p c 0 h ) 2  d x − C 2 F µ 1 2 λ ( C F kR 3 c 0 k Ω + kR 4 c 0 k Ω ) 2 − 1 µ 1 ( C F kR 1 c 0 k Ω + kR 2 c 0 k Ω ) ( C F kR 3 c 0 k Ω + kR 4 c 0 k Ω ) (63) and ˜ J ⊖ k = 1 2 k∇ y kh − g d k k 2 Ω + 1 2 λ k p kh k 2 Ω − Z Ω  ν ∇ y kh · ∇ p kh − k ω σ y ⊥ kh · p kh + λ − 1 p 2 kh  d x − C 2 F µ 1 2 λ ( C F kR 3 k k Ω + kR 4 k k Ω ) 2 − 1 µ 1 ( C F kR 1 k k Ω + kR 2 k k Ω ) ( C F kR 3 k k Ω + kR 4 k k Ω ) . (64) 6. Robust p reconditioners for the minimal residual method The saddle point s ystems (49), (50), (57) and (58) can b e so lv ed b y using the preconditioned MinRes metho d, see [37]. A convergence r esult for the pr e conditioned MinRes metho d can b e found in [13] stating that the conv ergence rate of the preconditioned MinRes metho d dep ends on the condition num b er of the preconditioned sy stem. The deriv ation of preconditioners for problems (49) for k = 1 , . . . , N a nd (50) for k = 0 have already b een presented and discussed in [1 7, 27] given by P k = diag ( D k , D k , λ − 1 D k , λ − 1 D k ) and P 0 = dia g ( D 0 , λ − 1 D 0 ) , (65) resp ectively , where D k = √ λK h,ν + k ω √ λM h,σ + M h and D 0 = M h + √ λK h,ν . In [17], precondi- tioners are derived follo wing the technique in [46] based o n o per ator in terp olation theory . In this section, we present new robust pr econditioners for t he pro blem matr ic e s in (57) for k = 1 , . . . , N a nd in (58 ) for k = 0 in order to solve optimization problem I I. Here, we assume that σ and ν are consta n t, which we also c ho ose in the numerical results in Section 7 . Hence, M h,σ = σ M h 18 MONIKA WOLFMA YR and K h,ν = ν K h . The blo c k-diag onal pre c onditioners are pra ctically implemen ted by the version of the a lg ebraic m ultilev el itera tio n (AMLI) metho d from [2 0]. The AMLI preconditioned MinRes solver is ro bust and of o ptimal complexity which is prov ed in [21]. This can b e also o bserved in the n umerical results’ section. Let us cons ider the s addle po int system (57) for k ≥ N . The deriv ation of preconditioners for system (58) in case o f k = 0 is completely analogous. W e r efer the reader to [17] on details for the deriv ation of the preconditioners based on Sch ur co mplemen ts. Deﬁning the matrices and v ectors A = dia g ( K h , K h ) , B =  − ν K h − k ω σ M h k ω σ M h − ν K h  , C = diag ( λ − 1 M h , λ − 1 M h ) , (66) f = ( K h g c d k , K h g s d k ) T , y = ( y c k , y s k ) T and p = ( p c k , p s k ) T leads to the following problem structure  A B T B − C   y p  =  f 0  (67) with the symmetric and p ositive deﬁnite matrices A and C . W e deﬁne the nega tiv e Sch ur com- plemen ts S = C + B A − 1 B T and R = A + B T C − 1 B yielding the pr econditioners for k ≥ N as follows ˜ P = diag ( A, S ) and ˜ Q = diag ( R, C ) . (68) The negative Sc h ur complements are given by S = diag ( ν K h + λ − 1 M h + k 2 ω 2 σ 2 M h K − 1 h M h , ν K h + λ − 1 M h + k 2 ω 2 σ 2 M h K − 1 h M h ) (69) and R = diag ( K h + k 2 ω 2 σ 2 λM h + ν 2 λK h M − 1 h K h , K h + k 2 ω 2 σ 2 λM h + ν 2 λK h M − 1 h K h ) . (70) Let us deﬁne ˜ D S k = ν K h + λ − 1 M h + k 2 ω 2 σ 2 M h K − 1 h M h (71) and ˜ D R k = K h + k 2 ω 2 σ 2 λM h + ν 2 λK h M − 1 h K h . (72) Then S and R ca n be written as S = diag ( ˜ D S k , ˜ D S k ) and R = diag ( ˜ D R k , ˜ D R k ) , resp ectively . Remark 6. Both Schur c omplement pr e c onditioners ˜ P and ˜ Q in (68) c an b e chosen for c om- putations of optimization pr oblem II le ading to fast and r obust c onver genc e ra tes, se e [22] and [35] . Inserting now A and C from (66) and S a nd R fro m (69) and (70), resp ectively , the Sch ur complement preco nditioners in the form of ˜ P as pres en ted in (68) in the case of k ≥ N as well as for k = 0 are giv en by ˜ P k = dia g ( K h , K h , ˜ D S k , ˜ D S k ) and ˜ P 0 = dia g ( K h , ν K h + λ − 1 M h ) (73) Analogously in the fo rm of ˜ Q , they a re giv en by ˜ Q k = diag ( ˜ D R k , ˜ D R k , λ − 1 M h , λ − 1 M h ) and ˜ Q 0 = diag ( λ − 1 M h , K h + ν 2 λK h M − 1 h K h ) , (74) for k ≥ N and k = 0 for the saddle p oint sy stems (57) and (58), resp ectiv ely . W e refer aga in to [1 7] for further details o n the preconditioners’ deriv a tion. F or the numerical exp eriments of this w ork, w e simplify the preconditioner such that ˜ ˜ P k = diag ( ˜ ˜ D S k , ˜ ˜ D S k , λ − 1 ˜ ˜ D S k , λ − 1 ˜ ˜ D S k ) and ˜ ˜ P 0 = diag ( ˜ ˜ D S 0 , λ − 1 ˜ ˜ D S 0 ) , where ˜ ˜ D S k = ν √ λK h + (1+ k ω σ √ λ ) M h and ˜ ˜ D S 0 = M h + ν √ λK h in a similar form as (65) in order to apply the AMLI prec o nditioned MinRes method analogo usly as for optimizatio n problem I. GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 19 7. Numerical resul ts In this s ection, w e pres e nt num erica l results p erformed in C ++ for computing the minorants and ma jorants of the t wo optimal control problems with cost functionals (9) and (21) a nd for diﬀerent cases of given data denoted b y Examples 1–6. F or the n umerical results on the ma jorants of optimization problem I, we refer to [26] a nd for the numerical analys is including conv ergence results, to [17, 2 7]. How ever, all the numerical exp eriment s o n the ma jorants of optimization problem II as well as all on minorants for both problems I and II are new. W e p erform n umerical exp erimen ts for the same thre e cas es (applied on the problem data) a s for problem I but in Examples 4–6 they ar e applied on the desir e d gradients. The domain Ω is the tw o -dimensional unit squa re Ω = (0 , 1) 2 with F riedrichs’ consta n t C F = 1 / ( √ 2 π ) . The triangulatio n of the domain is p erformed regular leading to a uniform gr id. The ﬁnite elements are c hosen as described in Section 5. The co eﬃcients are chosen to b e ν = σ = 1 . In Examples 1, 2, 4 and 5, the c ost pa r ameter is chosen to be λ = 0 . 1 , T = 2 π/ ω and ω = 1 . In Examples 3 and 6, λ = 0 . 01 , T = 1 a nd ω = 2 π . The MhFE approximations for η , ζ and τ , ρ are chosen as w ell as the ﬂuxes are reco nstructed b y RT 0 -extensions (low est-order standard Raviart- Thomas) resulting in a veraged ﬂuxes which a re no w from H ( div , Ω) , see also [25, 26] for further details. The grid sizes range betw e e n 16 × 16 and 25 6 × 256 as well as 512 × 512 to obtain ﬁner grid solutions for Examples 3 and 6 as a reference for the exa ct solution. The pr econditioned MinRes iteration was stopped a fter 8 iteration steps in all computations using the AMLI preconditioner with 4 inner iterations. The numerical exp eriment s for Examples 1, 2 , 4 and 5 were per formed on a la ptop with Intel( R) Core (TM) i5-6267 U CPU @ 2.90GHz pro cessor and 1 6 GB 21 33 MHz LPDDR3 memory . The numerical experiments for Examples 3 and 6 were p erformed on a CPU server with a T um bleweed distribution having 64 co r es and 1 terabyte memory in order to pro vide enough memory for computing the ﬁner gr id solutions in addition. All computational times t sec are presented in seconds and include the CPU times also neede d to der ive the ma jorants and minorants. How ever, w e wan t to highlight that these times are muc h smaller compared to the rest. The computationa l times of Exa mples 3 and 6 exclude the computation of the solution on the ﬁner grid (51 2 × 512). 7.1. Numeri cal r esults for optimization problem I. 7.1.1. Example 1. The desired state is a time-perio dic and time-analytic function y d ( x , t ) = e t sin( t ) 0 . 1  12 + 4 π 4  sin 2 ( t ) − 6 cos( t )(cos( t ) − sin( t ))  sin( x 1 π ) sin ( x 2 π ) , (75) which is ho wev er no t time-harmonic. Note that the exact sta te function for this example is y ( x , t ) = e t sin( t ) 3 sin( x 1 π ) sin ( x 2 π ) . (76) The truncation index for the m ultiharmonic a pproximations is chosen as N = 8 here. T able 1 presents for diﬀerent grid sizes CPU times t sec , v alues fo r the ma jorants J ⊕ 0 and minorants J ⊖ 0 as deﬁned in (51) and (55), and eﬃciency indices I J ⊕ 0 eﬀ = J ⊕ 0 / J 0 , I J ⊖ 0 eﬀ = J ⊖ 0 / J 0 and I J , 0 eﬀ = J ⊕ 0 / J ⊖ 0 . Here, J 0 = J 0 ( y c 0 , u c 0 ) = 1 2 k y c 0 − y c d 0 k 2 Ω + λ 2 k u c 0 k 2 Ω as introduced in Subsection 2.2. In T able 2, the n umerical results for the F ourier mo de k = 1 are presen ted including J ⊕ k , J ⊖ k as deﬁned in (52) and (56) and the corresp onding eﬃciency indices I J ⊕ k eﬀ = J ⊕ k / J k , I J ⊖ k eﬀ = J ⊖ k / J k and I J ,k eﬀ = J ⊕ k / J ⊖ k . Moreov er, w e present the eﬃciency indices for M ⊕ 1 given fo r the modes by I M 1 , 0 eﬀ = s M ⊕ 1 , 0 ( α 0 , β 0 ; y c 0 h , p c 0 h , τ c 0 h , ρ c 0 h ) ||| y c 0 − y c 0 h ||| 2 1 , 0 and I M 1 ,k eﬀ = v u u t M ⊕ 1 ,k ( α k , β k ; y kh , p kh , τ kh , ρ kh ) ||| y k − y kh ||| 2 1 ,k . The erro r nor ms for the mo des are given by ||| y c 0 − y c 0 h ||| 2 1 , 0 = 1 2 k y c 0 − y c 0 h k 2 Ω + λµ 1 2 2 C 2 F k∇ y c 0 − ∇ y c 0 h k 2 Ω 20 MONIKA WOLFMA YR and ||| y k − y kh ||| 2 1 ,k = 1 2 + k ω λµ 1 2 2 C 2 F ! k y k − y kh k 2 Ω + λµ 1 2 2 C 2 F k∇ y k − ∇ y kh k 2 Ω leading the repr esen tation ||| u − v ||| 2 1 = T ||| y c 0 − y c 0 h ||| 2 1 , 0 + T 2 N X k =1 ||| y k − y kh ||| 2 1 ,k + F N (77) with the remainder term F N := T 2 P ∞ k = N +1 ||| y k ||| 2 1 ,k . F or the numerical exp eriments, we ca n estimate the eﬃciency index for M ⊕ 1 from ab ov e by estimating (77) fro m be low ignoring the remainder term F N leading to the o verall eﬃciency index fo r M ⊕ 1 I M 1 eﬀ = v u u t M ⊕ 1 ( α, β ; η , ζ , τ , ρ , v ) T ||| y c 0 − y c 0 h ||| 2 1 , 0 + T 2 P N k =1 ||| y k − y kh ||| 2 1 ,k . (78) The corresp onding ma jor ant s M ⊕ 1 , 0 = M ⊕ 1 , 0 ( α 0 , β 0 ; y c 0 h , p c 0 h , τ c 0 h , ρ c 0 h ) a nd M ⊕ 1 ,k = M ⊕ 1 ,k ( α k , β k ; y kh , p kh , τ kh , ρ kh ) are given b y M ⊕ 1 , 0 = J ⊕ 0 − J ⊖ 0 + 3 λ 4 C 2 F ( C F kR 1 c 0 k Ω + kR 2 c 0 k Ω ) 2 = α 0 2 k y c 0 h − y d c 0 k 2 Ω + γ 0 kR 2 c 0 k 2 Ω + γ 0 C 2 F β 0 kR 1 c 0 k 2 Ω + Z Ω  ν ∇ y c 0 h · ∇ p c 0 h + λ − 1 ( p c 0 h ) 2  d x + C 2 F µ 1 2 λ ( C F kR 3 c 0 k Ω + kR 4 c 0 k Ω ) 2 + 1 µ 1 ( C F kR 1 c 0 k Ω + kR 2 c 0 k Ω ) ( C F kR 3 c 0 k Ω + kR 4 c 0 k Ω ) + 3 λ 4 C 2 F ( C F kR 1 c 0 k Ω + kR 2 c 0 k Ω ) 2 and also M ⊕ 1 ,k = J ⊕ k − J ⊖ k + 3 λ 4 C 2 F ( C F kR 1 k k Ω + kR 2 k k Ω ) 2 . T able 3 sums up the n umerical r esults for Example 1 by presenting the minora n ts, ma jorants and eﬃciency indices on a grid of size 256 × 256 for all k up to N = 4 a nd then for k = 6 and k = 8 (since their results w ere similar as for k = 5 and k = 7 ). F or N = 3 and N = 8 , the trunca tion’s remainder terms ca n b e precomputed grid t sec J ⊖ 0 I J ⊖ 0 eﬀ J ⊕ 0 I J ⊕ 0 eﬀ I J , 0 eﬀ I M 1 , 0 eﬀ 16 × 16 0.02 1.13e+0 5 0.90 1.26e + 05 1.01 1.12 1.53 32 × 32 0.07 1.14e+0 5 0.90 1.27e + 05 1.00 1.11 1.47 64 × 64 0.24 1.14e+0 5 0.90 1.27e + 05 1.00 1.11 1.44 128 × 128 1.16 1.14e+0 5 0.90 1.27e+ 05 1.00 1.11 1.43 256 × 256 4.51 1.14e+0 5 0.90 1.27e+ 05 1.00 1.11 1.42 T able 1. E xample 1 . Minorant J ⊖ 0 , ma jo rant J ⊕ 0 and their eﬃciency indices computed on gr ids of diﬀere nt sizes. and are given by E 3 = 6 3694 . 86 and E 8 = 1 06 . 06 , r espectively . Since the ov er all eﬃciency indices in T able 3 stay all in approximately the same range, we obs e r ve that the metho d is robust. How ever, the eﬃciency indices for the co m bined error norm I M 1 eﬀ indicate tha t the mo des k = 1 and k = 4 are the most signiﬁcant to repres en t the so lution by its m ultiharmonic approximation. Compar ing the last t w o lines of T able 3 shows that the v alue for representing the cost functional of the exact solution is a lr eady suﬃciently ac c urate for a tr unca tion index N = 3 . One of the reasons for this is that the rema inder term E N can b e precomputed exactly . 7.1.2. Example 2. W e choo se the time-analytic, not time-p erio dic, desired state function y d ( x , t ) = e t 0 . 2  (5 + 2 π 4 ) sin( t ) − cos( t )  sin( x 1 π ) sin( x 2 π ) (79) having as ex act solution the state function y ( x , t ) = e t sin( t ) sin( x 1 π ) sin( x 2 π ) . (80) GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 21 grid t sec J ⊖ 1 I J ⊖ 1 eﬀ J ⊕ 1 I J ⊕ 1 eﬀ I J , 1 eﬀ I M 1 , 1 eﬀ 16 × 16 0.02 4.27e+0 5 0.90 4.74e + 05 1.00 1.11 6.18 32 × 32 0.07 4.31e+0 5 0.90 4.79e + 05 1.00 1.11 6.01 64 × 64 0.25 4.32e+0 5 0.90 4.80e + 05 1.00 1.11 5.93 128 × 128 1.13 4.32e+0 5 0.90 4.80e+ 05 1.00 1.11 5.90 256 × 256 4.66 4.32e+0 5 0.90 4.80e+ 05 1.00 1.11 5.89 T able 2. E xample 1 . Minorant J ⊖ 1 , ma jo rant J ⊕ 1 and their eﬃciency indices computed on gr ids of diﬀere nt sizes. mo de t sec J ⊖ I J ⊖ eﬀ J ⊕ I J ⊕ eﬀ I J eﬀ I M 1 eﬀ k = 0 4.51 1.14 e+05 0.90 1.27e + 05 1.0 0 1.11 1.42 k = 1 4.66 4.32 e+05 0.90 4.80e + 05 1.0 0 1.11 5.89 k = 2 4.75 1.79 e+05 0.90 1.99e + 05 1.0 0 1.11 1.64 k = 3 4.81 6.10 e+04 0.90 6.74e + 04 1.0 0 1.11 1.84 k = 4 4.74 7.68 e+03 0.91 8.42e + 03 1.0 0 1.10 1 1.06 k = 6 4.72 2.05 e+02 0.90 2.29e + 02 1.0 0 1.11 2.08 k = 8 4.81 1.97 e+01 0.93 2.19e + 01 1.0 4 1.12 1.22 ov erall ( N = 3 ) – 2.86e+ 06 0.90 3.17e+ 06 1.00 1.11 2.08 ov erall ( N = 8 ) – 2.86e+ 06 0.90 3.17e+ 06 1.00 1.11 2.09 T able 3. Exampl e 1 . Overall minora n t J ⊖ and overall ma jorant J ⊕ , their parts, and their eﬃciency indices computed on a grid o f size 256 × 2 56 . The approximations by the MhFEM are computed for the truncation index N = 10 . F or Example 2, it suﬃces to presen t her e o nly the o verall results as in T able 3, now pr esent ed in T a ble 4. W e compare the ov erall v alues o f ma jo r ant s and minorants for diﬀeren t tr uncation indices N = 6 and N = 10 , for which the co rresp onding truncation’s re ma inder terms ar e given by E 6 = 4 4094 . 84 and E 10 = 10597 . 20 . One can se e fr om the last tw o lines that the truncation index N = 6 s uﬃces already to provide an accurate enough approximate solution. Also eﬃciency indices b eing around 1 show that the ma jorants and mino rants p erform well for that example. mo de t sec J ⊖ I J ⊖ eﬀ J ⊕ I J ⊕ eﬀ I J eﬀ I M 1 eﬀ k = 0 4.55 3.20e+0 5 0.90 3.56e + 05 1.00 1.1 1 1.43 k = 1 4.53 1.02e+0 6 0.90 1.14e + 06 1.00 1.1 1 3.19 k = 2 4.62 2.57e+0 5 0.90 2.85e + 05 1.00 1.1 1 3.17 k = 3 4.80 6.07e+0 4 0.91 6.69e + 04 1.00 1.1 0 1.82 k = 4 4.75 1.99e+0 4 0.91 2.19e + 04 1.00 1.1 0 1.55 k = 6 4.83 4.05e+0 3 0.92 4.38e + 03 1.00 1.0 8 1.32 k = 8 4.90 1.30e+0 3 0.93 1.40e + 03 1.00 1.0 7 1.20 k = 10 4.72 5.40e+ 02 0.94 5.75 e+02 1.00 1 .06 1.14 ov erall ( N = 6 ) – 6.34e+06 0.90 7.06e+ 06 1.00 1.11 2.09 ov erall ( N = 10 ) – 6.34e+0 6 0.90 7.06 e + 06 1.00 1.1 1 2.09 T able 4. Exampl e 2 . Overall minora n t J ⊖ and overall ma jorant J ⊕ , their parts, and their eﬃciency indices computed on a grid o f size 256 × 2 56 . 7.1.3. Example 3. The desired state is c hosen to b e a spa c e-time non-smo oth function y d ( x , t ) = χ [ 1 2 , 1] 2 ( x ) χ [ 1 4 , 3 4 ] ( t ) . (81) 22 MONIKA WOLFMA YR Here, we denote by χ the space-time characteristic function. The desired state’s F ourier coﬃcients are analytica lly co mputed and given b y y c d 0 ( x ) = χ [ 1 2 , 1] 2 ( x ) / 2 and y c d k ( x ) = χ [ 1 2 , 1] 2 ( x )  sin( 3 kπ 2 ) − sin( kπ 2 )  k π and y s d k ( x ) = 0 ∀ k ∈ N . (82) The desired state has jumps in space and time. In this example, the exact solution cannot b e precomputed analytically . Hence, as approximation for it, we use a MhFE representation on the ﬁner grid with s ize 5 12 × 51 2 . Note a lso that the F our ier co eﬃcients a r e zero for the ev en F ourier mo des b esides for k = 0 . W e can o bserve in T able 5 that the eﬃciency indices, especia lly regar ding the combined no rm, ar e similar for higher mo des. W e hav e computed the results up to N = 11 and also added the results for the mo des k = 21 , k = 41 and k = 81 to the table a s examples. The ma jorants of the combined norm stay appr oxim ately in the same range for k ≥ 1 . The ma jorants and minor an ts for the cost functional ar e close to 1, which demonstr ates their eﬃciency a lso in this numerical example, where the given da ta has jumps in spa ce and time. mo de t sec J ⊖ k I J ⊖ k eﬀ J ⊕ k I J ⊕ k eﬀ I J , k eﬀ I M 1 ,k eﬀ k = 0 6.44 5.71e+ 04 0.9 2 8.20 e+04 1.32 1.4 4 6.48 k = 1 6.45 9.30e+ 04 0.9 2 1.31 e+05 1.30 1.4 1 3.59 k = 3 6.46 1.06e+ 04 0.9 5 1.35 e+04 1.20 1.2 7 2.86 k = 5 6.39 3.90e+ 03 0.9 7 4.63 e+03 1.15 1.1 9 3.16 k = 7 6.46 2.01e+ 03 0.9 8 2.28 e+03 1.11 1.1 3 3.28 k = 9 6.55 1.23e+ 03 0.9 8 1.35 e+03 1.08 1.1 0 3.30 k = 11 6.48 8 .23e+02 0.99 8.89e+0 2 1.07 1.08 3.2 3 k = 21 6.44 2 .28e+02 1.00 2.37e+0 2 1.03 1.04 3.8 8 k = 41 6.46 5 .99e+01 1.00 6.19e+0 1 1.03 1.03 4.8 1 k = 81 6.42 1 .53e+01 1.00 1.63e+0 1 1.06 1.06 3.1 9 T able 5. Example 3 . Minorants, ma jorants, and their eﬃciency indices as well as the eﬃciency indices of the combined norm computed on a gri d of size 256 × 256 . 7.2. Numeri cal res ults for optimization problem II. W e compute the n umerical results for the three same cas es as for problem I but now applied on the des ired gradient. 7.2.1. Example 4. W e set the desired gradient to b e time-p erio dic a nd time-a na lytic g d ( x , t ) = e t sin( t )( − 3 c o s( t )(cos( t ) + sin( t )) + (10 π 2 + 1 + 2 π 4 ) sin( t ) 2 ) 10 π  cos( x 1 π ) sin ( x 2 π ) sin( x 1 π ) cos( x 2 π )  . The exa ct solution for the state function is given by (76). Mor eov er , we present the eﬃciency indices for ˜ M ⊕ 1 given fo r the modes by I ˜ M 1 , 0 eﬀ = s ˜ M ⊕ 1 , 0 ( α 0 , β 0 ; y c 0 h , p c 0 h , τ c 0 h , ρ c 0 h ) |||| y c 0 − y c 0 h |||| 2 1 , 0 and I ˜ M 1 ,k eﬀ = v u u t ˜ M ⊕ 1 ,k ( α k , β k ; y kh , p kh , τ kh , ρ kh ) |||| y k − y kh |||| 2 1 ,k . The erro r nor ms for the mo des are given by |||| y c 0 − y c 0 h |||| 2 1 , 0 = 1 2 + λµ 1 2 2 C 2 F ! k∇ y c 0 − ∇ y c 0 h k 2 Ω and |||| y k − y kh |||| 2 1 ,k = k ω λµ 1 2 2 C 2 F k y k − y kh k 2 Ω + 1 2 + λµ 1 2 2 C 2 F ! k∇ y k − ∇ y kh k 2 Ω GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 23 leading the repr esen tation |||| u − v |||| 2 1 = T |||| y c 0 − y c 0 h |||| 2 1 , 0 + T 2 N X k =1 |||| y k − y kh |||| 2 1 ,k + ˜ F N (83) with the r emainder term ˜ F N := T 2 P ∞ k = N +1 |||| y k |||| 2 1 ,k . F or the numerical exper imen ts, the eﬃ- ciency index for ˜ M ⊕ 1 from ab ov e b y estimating (83) from below ignoring the r emainder term ˜ F N leading to the overall eﬃciency index for ˜ M ⊕ 1 given by I ˜ M 1 eﬀ = v u u t ˜ M ⊕ 1 ( α, β ; η , ζ , τ , ρ , v ) T |||| y c 0 − y c 0 h |||| 2 1 , 0 + T 2 P N k =1 |||| y k − y kh |||| 2 1 ,k . The corresp onding ma jorants are given b y ˜ M ⊕ 1 , 0 = ˜ M ⊕ 1 , 0 ( α 0 , β 0 ; y c 0 h , p c 0 h , τ c 0 h , ρ c 0 h ) = ˜ J ⊕ 0 − ˜ J ⊖ 0 + 3 λ 4 C 2 F ( C F kR 1 c 0 k Ω + kR 2 c 0 k Ω ) 2 and ˜ M ⊕ 1 ,k = ˜ M ⊕ 1 ,k ( α k , β k ; y kh , p kh , τ kh , ρ kh ) = ˜ J ⊕ k − ˜ J ⊖ k + 3 λ 4 C 2 F ( C F kR 1 k k Ω + kR 2 k k Ω ) 2 . W e present the n umerical results for the modes k = 0 and k = 1 for diﬀerent grid sizes in T a bles 6 a nd 7. The eﬃciency indices for the ma jora n ts and minorants a r e very close to 1.00 . Also the eﬃciency indices for ˜ M 1 , 0 show a go o d accuracy . T able 8 compares grid t sec ˜ J ⊖ 0 I ˜ J ⊖ 0 eﬀ ˜ J ⊕ 0 I ˜ J ⊕ 0 eﬀ I ˜ J , 0 eﬀ I ˜ M 1 , 0 eﬀ 16 × 16 0.02 8.92e+0 3 0.99 9.85e + 03 1.09 1.10 2.04 32 × 32 0.06 9.24e+0 3 0.99 9.95e + 03 1.07 1.08 1.97 64 × 64 0.24 9.32e+0 3 0.99 9.97e + 03 1.05 1.07 1.94 128 × 128 1.04 9.34e+0 3 0.98 9.98e+ 03 1.05 1.07 1.93 256 × 256 4.39 9.35e+0 3 0.98 9.98e+ 03 1.05 1.07 1.92 T able 6. E xample 4 . Minorant ˜ J ⊖ 0 , ma jo rant ˜ J ⊕ 0 and their eﬃciency indices computed on gr ids of diﬀere nt sizes. grid t sec ˜ J ⊖ 1 I ˜ J ⊖ 1 eﬀ ˜ J ⊕ 1 I ˜ J ⊕ 1 eﬀ I ˜ J , 1 eﬀ I ˜ M 1 , 1 eﬀ 16 × 16 0.02 3.22e+0 4 0.95 3.51e + 04 1.03 1.09 1.52 32 × 32 0.06 3.39e+0 4 0.96 3.58e + 04 1.02 1.05 1.33 64 × 64 0.26 3.45e+0 4 0.97 3.60e + 04 1.01 1.04 1.25 128 × 128 1.08 3.48e+0 4 0.97 3.62e+ 04 1.01 1.04 1.21 256 × 256 4.31 3.51e+0 4 0.98 3.65e+ 04 1.01 1.04 1.19 T able 7. E xample 4 . Minorant ˜ J ⊖ 1 , ma jo rant ˜ J ⊕ 1 and their eﬃciency indices computed on gr ids of diﬀere nt sizes. the r esults for diﬀerent F o urier mo des up to N = 8 computed on a grid of size 256 × 2 56 . Her e, the ov erall minorants, ma jorants and eﬃciency indices are presen ted, where the r emainder terms E N for N = 8 and also N = 6 hav e b een preco mputed exactly . The v alues of the eﬃciency in dices v a ry for diﬀeren t mo des k . F or example, the results for ˜ M 1 , 4 indicate that the mode k = 4 is essen tial to r epresent the solution accur ately . The v alues for I J ⊖ 7 eﬀ and I J ⊖ 8 eﬀ indicate that the minor ant s require a diﬀerent , higher reﬁnement for a more accurate representation of the ov erall so lution. An adaptive sch eme is the natura l choice. On the other ha nd, in these cases the ma jo rants g ive a go o d repr esen tation for the cost functional. Finally , comparing the las t tw o lines of T able 8 ag ain shows that the overall v alue fo r representing the cost functional of the exact solution is a lready suﬃcient ly accura te for a truncation index N = 6 . 24 MONIKA WOLFMA YR mo de t sec ˜ J ⊖ I ˜ J ⊖ eﬀ ˜ J ⊕ I ˜ J ⊕ eﬀ I ˜ J eﬀ I ˜ M 1 eﬀ k = 0 4.39 9.35 e+03 0.98 9.98e + 03 1.0 5 1.07 1.92 k = 1 4.31 3.51 e+04 0.98 3.65e + 04 1.0 1 1.04 1.19 k = 2 4.42 9.23 e+03 0.63 1.57e + 04 1.0 6 1.70 1.65 k = 3 4.43 2.85 e+03 0.58 5.06e + 03 1.0 3 1.78 1.08 k = 4 4.44 5.89 e+02 0.98 6.47e + 02 1.0 7 1.10 5.91 k = 6 4.36 1.74 e+01 0.97 2.82e + 01 1.5 8 1.62 2.32 k = 8 4.33 2.99e-0 1 0.23 1.37e+0 0 1.05 4.59 1.97 ov erall ( N = 6 ) – 2.09e+ 05 0.89 2.45e+ 05 1.04 1.17 2.46 ov erall ( N = 8 ) – 2.09e+ 05 0.89 2.45e+ 05 1.04 1.17 2.46 T able 8. Exampl e 4 . Overall minora n t ˜ J ⊖ and overall ma jorant ˜ J ⊕ , their parts, and their eﬃciency indices computed on a grid o f size 256 × 2 56 . 7.2.2. Example 5. W e choo se the non time-p erio dic but time-analytic desired g r adient g d ( x , t ) = − e t sin( t )(0 . 1 c o s( t ) − π 2 (1 + 2 π 2 0 . 1)) π  cos( x 1 π ) sin( x 2 π ) sin( x 1 π ) cos( x 2 π )  leading to the time-analytic, but not time-p erio dic exa ct state (80). W e compute the MhFE approximation of the desir ed gra dien t a nd so lv e the systems (49) and (50) for mo des up to N = 10 on a 256 × 2 56 -mesh and present the results in T able 9. The remainder terms for N = 6 and N = 10 a re E 6 = 4796 . 5 4 a nd E 10 = 1149 . 6 5 , resp ectively . The eﬃciency indices for the ov e rall ma jorant and minora nt show that a truncation index of N = 6 already gives a suﬃciently accura te approximation for the overall cost functiona l. Note that the eﬃciency index for ˜ M 1 , 2 indicates that the mo de k = 2 is e s sen tial for the mult iharmo nic approximation g iving an a ccurate representation of the solution. mo de t sec ˜ J ⊖ I ˜ J ⊖ eﬀ ˜ J ⊕ I ˜ J ⊕ eﬀ I ˜ J eﬀ I ˜ M 1 eﬀ k = 0 4.31 2.63e+0 4 1.00 2.79e + 04 1.06 1.0 6 1.36 k = 1 4.30 8.49e+0 4 1.00 8.60e + 04 1.02 1.0 1 1.00 k = 2 4.39 2.08e+0 4 0.98 2.21e + 04 1.04 1.0 6 2.83 k = 4 4.35 1.58e+0 3 0.96 1.75e + 03 1.06 1.1 1 1.74 k = 6 4.36 2.93e+0 2 0.87 3.53e + 02 1.05 1.2 0 1.63 k = 8 4.37 8.95e+0 1 0.82 1.20e + 02 1.10 1.3 4 1.08 k = 10 4.34 3.27e+ 01 0.71 5.23 e+01 1.14 1 .60 1.19 ov erall ( N = 6 ) – 5.23e+05 1.00 5.43e+ 05 1.04 1.04 2.00 ov erall ( N = 10 ) – 5.22e+0 5 1.00 5.42 e + 05 1.04 1.0 4 2.00 T able 9. Exampl e 5 . Overall minora n t ˜ J ⊖ and overall ma jorant ˜ J ⊕ , their parts, and eﬃciency indices of them a nd the com bined norm computed on a grid of size 25 6 × 256 . 7.2.3. Example 6. W e set the space-time non-smo o th desired gradient g d ( x , t ) = ( χ [ 1 2 , 1] 2 ( x ) χ [ 1 4 , 3 4 ] ( t ) , χ [ 1 2 , 1] 2 ( x ) χ [ 1 4 , 3 4 ] ( t )) T . (84) Also the co eﬃcients of the F ourier expansion as so ciated with g d can b e found analytically . They are as in Example 3 given b y (82) for each direction of the g radient (84). Again the e x act solution cannot b e computed analytically and hence we use its MhFE approximations o n the ﬁner mesh o f size 512 × 5 1 2 as a reference. T a ble 10 present s the results for mo des up to truncation index N = 11 as well a s for k = 2 1 , k = 4 1 and k = 81 a nalogously to Example 3 . T he results reﬂected by the eﬃciency indices show the g o o d re pr esent ation b y using the minorants and ma jor a n ts, especia lly , considering the eﬃciency indices in the last t wo columns o f T able 10. This aga in demonstrates GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 25 the eﬃciency of the minorant s and ma jorants for data having jumps in space and time but now for optimization problem I I. mo de t sec J ⊖ k I J ⊖ k eﬀ J ⊕ k I J ⊕ k eﬀ I J , k eﬀ I M 1 ,k eﬀ k = 0 8.12 1.55e+ 01 0.7 9 2.05 e+01 1.04 1.3 2 2.23 k = 1 8.03 7.70e+ 00 0.8 6 1.13 e+01 1.26 1.4 7 2.87 k = 3 8.51 1.46e+ 01 0.9 3 1.73 e+01 1.10 1.1 8 2.56 k = 5 8.25 1.44e+ 01 0.9 9 1.65 e+01 1.14 1.1 5 2.59 k = 7 8.08 8.92e+ 00 0.9 8 9.80 e+00 1.08 1.1 0 1.18 k = 9 8.35 4.51e+ 00 0.9 0 5.14 e+00 1.03 1.1 4 1.26 k = 11 8.36 2 .57e+00 0.96 3.13e+0 0 1.17 1.22 1.5 1 k = 21 8.51 1 .36e+00 0.99 2.27e+0 0 1.65 1.67 3.1 6 k = 41 7.78 4 .54e+00 0.86 6.26e+0 0 1.18 1.38 3.0 9 k = 81 7.81 3 .19e+00 0.79 7.09e+0 0 1.75 2.22 3.4 4 T able 10. E xample 6 . Minora n ts, ma jorants, and their e ﬃciency indices as well as the eﬃciency indices of the combined no rm computed on a grid of size 256 × 256 . 8. Conclusions and outloo k In this work, the a poster iori error analysis started in [26] has b e en extended no w by der iving new lo wer b ounds, called minor an ts, for the cost functional leading to an upper estimate for the error nor m of the state and control or equiv alently in state and adjoint state. These low er b ounds are guaranteed a nd computable. T og ether with using the results from [44] as w ell as [16] one can apply the metho d als o to time-perio dic optimal cont rol problems, where box constraints a re being impo s ed on the F ourier coeﬃcients of the control. The estimates are deriv ed for t wo diﬀeren t cost functionals, where the seco nd one is now new in this context. Since in the linear case the pr o blems are decoupled, the solutions on the F our ier co eﬃcien ts could easily b e computed on grids of diﬀerent sizes dep ending o n the a ccuracy needed, which could b e exactly determined b y using the a p osterior i e stimates presented in this w o rk leading to an a daptiv e metho d in time. T ogether with the adaptive ﬁnite element metho d we then o btain a space-time adaptive metho d, the adaptive multiharmonic ﬁnite elemen t method, which we call AMhFEM, as mentioned for the ﬁrst time in [26]. In this work, a ﬁr st deriv ation of preconditioners for the MinRes metho d for the second o p- timization problem ha s been presented as w ell as a preconditoner for a pplying AMLI has been suggested. Several numerical tests for optimization pro blem I and I I ha ve been present ed showing the eﬃciency of the upp er and especially – with rega rd to the a rticle – lo wer b ounds for the cos t functionals in practice. Ackno wledgments The author gratefully ackno wledges the ﬁnancia l suppo rt b y the Academ y o f Finland under the grant 2958 97. The author would like to thank the anonymous referees for their v a luable commen ts improving the article. References [1] K. Al tmann, S. Sti ngelin, and F. Tröl tzsch , On some optimal co ntr ol pr oblems for ele ctric al cir cuits , In t. J. Circuit Theory Appl., 42 (2014), pp. 808–830. [2] O. Axelsson a nd D. Lukáš , Pr e c onditioning met ho ds for e ddy-curr ent optimal ly c ontro l le d time-ha rmonic ele ct r omagnetic pr oblems , J. Numer. Math., 27 (2019), pp. 1–21. [3] F. Bachinger, M. Kal tenbacher, and S. Rei tzinger , An eﬃcient solution str ate gy for the HBFE metho d , Proceedings of the IGTE ’02 Symp osium Graz, Austria, 2 (2002), pp. 385–389 . [4] F. Bachinger, U. Langer, an d J. Schöberl , Numeric al analysis of nonline ar multiharmonic e ddy curr ent pr oblems , Numer. Math., 100 (2005), pp. 593–616. 26 MONIKA WOLFMA YR [5] , Eﬃcient solvers for nonline ar time-p erio dic e ddy curr ent pr oblems , Comput. Vis. Sci., 9 (2006), pp. 197–207. [6] D. Braess , Finite elements: The ory, fast solvers, and app lic ations in solid me chanics , Cam bridge Universit y Press, third edition, 2007. [7] P. G . Ciar let , The Finite Element M etho d for El liptic Pr oblems , vol. 4 of Studies in Mathemat ics and its Applications, North-Holland, Amsterdam, 1978. Republished b y SIAM in 2002. [8] C. Clason, B. Kal te nbacher, and D. W achsmuth , F unctional err or estimators for the adap tive dis- cr et ization of inverse pr oblems , In v erse Probl., 32 (2016), p. 104004. [9] D. M. Copelan d and U. La nger , Domain de co mp osition solvers for nonline ar multiharm onic ﬁnite element e quations , J. Numer. Math., 18 (2010), pp. 157–175. [10] A. Gaevska y a, R. H. W. Hoppe, and S. Repi n , A p osteriori estimates for c ost functionals of optimal c ontr ol pr oblems , Numerical Mathematics and Adv anced Applications, Pro ceedings of the ENUMA TH, 2005 (2006), pp. 308–316. [11] , F unctional appr o ach to a p osteriori err or estimation for e l liptic optimal c ontr ol pr oblems with dis- tribute d c ontr ol , J. M ath. Sci., 144 (2007), pp. 4535–4547. [12] A. V. Gaev ska y a and S. I. Repin , A p osteriori err or estimates for appr oximate solutions of line ar p ar ab olic pr oblems , Diﬀer. Equ., 41 (2005), pp. 970–983. [13] A. Greenbaum , Iter ative Metho ds for Solving Line ar Systems , F rontier s in Applied Mathematics 17, SIAM, Philadelphia, 1997. [14] M . Hinze and A. Rösch , Discr e t ization of optimal co ntr ol pr oblems , in Constrained Optimization and Optimal Con trol for Part ial Di ﬀeren tial Equations, Springer , 2012, pp. 391–430. [15] B. Houska, F. Logist, J . V an Impe, and M. Diehl , Appr oximate r obust optimization of time-p erio dic stationary states with appl ic ation to bio chemic al pr o c esses , in Pro ceedings of the 48th IEEE Conference on Decision and Con trol, Shanghai, China, 2009, pp. 6280–6285. [16] M . Kollmann a nd M. Kolmb auer , A pr e c onditione d MinRe s solver for time-p e rio dic p ar ab olic optimal c ontr ol pr oblems , Numer. Linear Al gebra Appl., 20 (2013), pp. 761–784. [17] M . K ollmann, M. Kolmb auer, U. Langer, M. Wolfma yr, and W. Zulehn er , A r obust ﬁnite element solver f or a multiha rmonic p ar ab olic optimal c ontr ol pr oblem , Comput. M ath. Appl., 65 (2013), pp. 469–486. [18] M . Kolmba uer and U. Langer , A r obust pr ec onditione d MinRes solver for distributed t ime-p erio dic e ddy curr ent optimal co ntr ol pr oblems , SIAM J. Sci. Comput., 34 (2012), pp. B785–B809. [19] , Eﬃcient solvers for some classes of t ime-p erio dic ed dy curr ent optimal c ontr ol pr oblems , Numerical Solution of Par tial Diﬀeren tial Equations: Theory , Algorithms, and Their Applications, 45 (2013), pp. 203–216. [20] J . Kra us , A dditive Schur c omplement appr oximation and applic ation to multilevel pr e co nditioning , SIAM J. Sci. Comput., 34 (2012) , pp. A2872–A289 5. [21] J . Kraus and M. Wolfma yr , On the r obustness and optimality of algebr aic multilevel metho ds for r e action- diﬀusion ty p e pr oblems , Comput. Vis. Sci., 16 (2013), pp. 15–32. [22] Y. A. Kuznetsov , Eﬃcient iter ativ e solvers for el liptic ﬁnite element pr oblems on nonm atching grids , Russ. J. Numer. Anal. Math. Model. , 10 (1995), pp. 187–211. [23] O. A. Ladyzhenska y a , The Boundary V alue Pr oblems of Mathematic al Physics , Nauk a, Mosco w, 1973. In Russian. T ranslated in App l. Math. Sc i . 49, Springer, 1985. [24] O. A. Ladyzhenska y a, V. A. Solonniko v, and N. N. Ural ’cev a , Line ar and Quasiline ar Equations of Par ab olic T yp e , AMS, Pro vidence, RI, 1968. [25] U. Lan ger, S. Repin , and M. Wolfma yr , F unctional a p osteriori e rr or estimates for p ar ab olic time -p erio dic b oundary value pr oblems , Comput . M etho ds Appl. Math., 15 (2015), pp. 353–372. [26] U. Lan ger, S. Repin , and M. Wolfma yr , F unctional a p osteriori e rr or estimates for time-p erio dic p ar ab olic optimal co ntr ol pr oblems , Num. F unc. Anal. Opt., 37 (2016), pp. 1267–12 94. [27] U. Langer and M. Wolfma yr , Multiharmonic ﬁnite element analysis of a time-p erio dic p ar ab olic optimal c ontr ol pr oblem , J. Numer. Math., 21 (2013), pp. 265–300. [28] Z. Z. Liang, O. Axelsson, an d M. Neytche v a , A r obust structur e d pr e c onditioner for time-harmo nic p ar ab olic optimal c ontr ol pr oblems , Numer. Algorithms, 79 (2018), pp. 575–596. [29] J . L. Lions , Optimal Contr ol of Systems Governe d by Partial Diﬀer ential Equations (Grund lehr en der Math- ematischen Wissenscha ften) , vol. 170, Springer Berlin, 1971. [30] O. Mali , Err or estimates for a class of el liptic optimal c ontr ol pr oblems , Num. F unc. Anal. Opt., 38 (2017), pp. 58–79. [31] O. Mali, P. Neitt aanmäki , and S. Repin , A ccu r acy V eriﬁc ation Me t ho ds. The ory and Algorithms , v ol. 32 of Computation al Methods in Applied Sciences, Springer, Netherlands, 2014. [32] S . Ma tculevi ch and S. Repin , Explicit c onstants in Poinc aré-typ e ine qualities for simplicial domains and applic ation to a p osteriori estimates , Comput. Methods Appl. Math., 16 (2016), pp. 277–298. [33] S . Ma tculev ich a nd M. W olfma y r , On the a p osteriori err or analysis for line ar Fokker–Planck mo dels in c onve ction-dominate d diﬀusion pr oblems , Appl. M ath. Comput., 339 (2018), pp. 779–804. [34] S . G. Mikhlin , V ariational metho ds in math ematic al physics , Pergamon Press Oxford, 1964. [35] M . F. Mu rphy, G. H. Golub, and A. J. W a then , A note on pr e co nditioning for indeﬁnite line ar systems , SIAM J. Sci. Comput., 21 (2000) , pp. 1969–1972. GUARANTEED LO WER BOUNDS FOR COST FUNCTIONALS 27 [36] P. Nei tt aan mäki, J. Sprekels, and D. Tiba , O ptimization of El liptic Systems: The ory and Applic ations , Springer, 2006. [37] C. C. P ai ge and M. A. Saunders , Solution of sp arse indeﬁnite systems of line ar e quations , SIAM J. Num er. Anal., 12 (1975), pp. 617–6 29. [38] P. A. Ra viar t and J. M. Thomas , A mixe d ﬁnite element metho d for 2-nd or der el liptic pr oblems , Mathe- matical Asp ects of Finite Element Methods, Lect. Notes Math., 606 (1977), pp. 292–315. [39] S . Repin , Estimates of deviation fr om exact solutions of initial-b oundary v alue pr oblems for the he at e quation , Rend. Mat. A cc. Lincei, 13 (2002), pp. 121–133. [40] , A Posteriori Estimates for Partial Diﬀer ential Equations , Radon Series on Computational and Applied Mathematics 4, W alter de Gruyter, Berlin, 2008. [41] S . R epin and T. R ossi , On quantitative analysis of an il l-p ose d el liptic pr oblem wit h Cauchy b oundary c on- ditions , in Numerical Methods for Diﬀeren tial Equations, Optimization, and T ech nological Problems, Springer, 2013, pp. 133–144. [42] P. Shaky a and R. K. Sinha , A priori and a p osteriori err or estimates of H1-Galerkin mixe d ﬁnite element metho d for p ar ab olic optimal c ontr ol pr oblems , Optim. C ontrol Appl. Methods, 38 (2017), pp. 1056–1070. [43] O. Steinbach , Numeric al appr oximation metho ds for el liptic b oundary value pr oblems: ﬁnit e and b oundary elements , Springer Science & Business Media, 2007. [44] M . Wolfma yr , A note on functional a p osteriori estimates for el liptic optimal c ontr ol pr oblems , Numer. Methods Partial Diﬀer. Equ., 33 (201 7), pp. 403– 424. [45] S . Y a mad a and K. Bessho , Harmon ic ﬁeld c alculation by the c ombination of ﬁnite element analysis and harmonic ba lanc e met ho d , IEEE T rans. Magn., 24 (1988) , pp. 2588–25 90. [46] W. Zulehn er , Nonstandar d norms and r obust estimates for sadd le p oint pr ob lems , SIAM J. Matrix Anal. Appl., 32 (2011), pp. 536–56 0. (M. W olfmayr) F acul ty of Informa tion Techn ology; Un iversity of Jyv ä skylä; P.O. Box 35 (Ago ra); FIN-40014 Jyv äsky lä; Finlan d E-mail addr e ss : monika.k.wolf mayr@jyu.fi

Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment