Optimal prediction for radiative transfer: A new perspective on moment closure

OPTIMAL PREDICTION F OR RADIA TIVE TRANSFER: A NEW PERSPECTIVE ON MOMEN T CLOSURE MAR TIN FRANK A N D BENJAM IN SEIBOLD Abstract. Moment methods are classical approaches th at approx i mate the mesoscopic radia- tiv e transfer equation b y a system of macroscopic moment equat i ons. An expansion in the angular v ariables transforms the original equation into a system of infinitely many momen ts. The truncation of this infinite system is the moment closure problem. Man y t yp es of closures ha ve b een presen ted in the literature. In this note, we demonstrate t hat optimal prediction, an approac h originally developed to appro ximate the mean solution of systems of nonlinear ordi- nary differential equations, ca n be used to deriv e momen t c l osures. T o that end, the formalism is generalized to systems of partial differential equations. Using Gaussian measures, existing linear closures can b e re-derived, suc h as P N , diffusion, and diffusi on correction closures. T hi s prov i des a new p ersp ectiv e on sev eral approx i mations done in the pro cess and give s rise to ideas for m o difications to existing closures. 1. Introduction In man y areas, (systems of ) macrosco pic equations can b e der ived from meso s copic k inetic equa- tions. F or ins ta nce, in the Navier-Stokes and Euler equations, the macr oscopic fluid v aria bles, e.g. densit y and momen tum, a re moments o f the pha se space distribution of the Boltzmann equa- tion. Similarly , in t he equation of radia tive tra nsfer [40], the direction dep endent kinetic equa tions can b e transformed into a coupled system of infinitely many moments. Such moment methods can be deriv ed by p erfor ming an infinite expansion in the kinetic v ar iable, a nd then considering only finitely man y members o f this expa nsion. While moment metho ds are by far not the only way to nu mer ically solve the equations of radiative transfer, they play a n imp or ta nt role, and a deeper understanding of their pro blems is imp ortant. Recen t discussio ns of the role of moment metho ds are provided in ov er view pap ers by Brunner [6], and by McClarren and Hauck [39]. Moment metho ds s tart with an infinite system of moments that is equiv alent to the or iginal equation of radiative transfer. T o a dmit a computation, this infinite system must be approximated by finitely ma ny moments. The challenge to devise approximations that model the influence of the non-consider ed moments on the considered moments as accurately as po ssible is the moment closure problem. Co mmonly known closure strategies are based on truncating and approximating the momen t equations, and observing to which extent the solution of the appr oximate sy s tem is close to the true so lution. The approximations are supp o rted by physical a rguments, such as higher moments b eing small or adjusting instantaneously . Another s trategy is to use asymptotic analysis. In Sect. 2, we outline a commonly used mo men t hiera rch y for r adiative transfer, and present some cla ssical linea r closur e strategies . In this pap er, we prop ose a n alternative strategy: first, an iden tity for the evolution of the low est N moments is formulated, a nd then closures are derived b y approximating this identit y . Sp ecifically , 2000 Mathematics Subje ct Classific ation. 85A25, 78M05, 82Cxx. Key wor ds and phr ases. radiative transfer, m etho d of moments, optimal prediction, measure, di ffusion appro xim ation. The authors thank Martin Grothaus helpf ul suggestions on measures in function spaces. The supp ort by the German Researc h F oundation and the Nat i onal Scienc e F oundation is a ckno wledged. M. F rank wa s supp orted b y DFG gran t KL 1105/14/2. B. Se i bold wa s partially supp orted b y NSF grants DMS–0813648 and DMS–1007899. 1 2 MAR TIN FRANK AND B E NJAMIN SEIBOLD we sho w that the method of optimal prediction [1 5, 12, 14, 10, 11] can b e applied to the equa tion of radiative transfer and yields clo s ed sy stems of finitely many moments. The idea o f optimal prediction, briefly o utlined in Sec t. 3, is to consider the mean solution o f a lar ge system, and approximate it by a smaller sys tem that is der ived b y av era ging the equatio ns with resp ect to an underlying pro bability mea s ure. The a ppr oach can be under sto o d a s a way to r emov e undesired mo des, but in an averaged fashio n, instead of merely neglecting them. W e show that with this formalism, existing linear closur e s ca n be re-derived, such as P N , diffusion, and diffusion cor rection closures. Optimal prediction is an extension of the Mori-Zwanzig formalism [41, 51]. It has b een applied to partial differential equations [17, 2], howev er, only after reducing them to a sys tem of ordinar y differential equations using a F ourier e x pansion or a semi-discretiza tion step. Here, we generalize the formalism to partial differential equations and measures on function spaces. The idea of using the Mori-Zwanzig for malism for momen t closur es in radiative transfer was first pr esented in [46], how ever, the deriv ations done were only formally . In this pap er, we show that the ad-ho c deriv ations done in [46] are in fact justified, by prop erly defining measures on function spaces (see Sect. 3). This genera lization is required so that meas ur es on the space of moments can be considered. W e re s trict ourselves to Gauss ia n measures. This choice is motiv ated by t he fact that it tur ns out repro duce existing linear closure s , rather than by physical consider ations. In Sect. 4 , we presen t linear optimal predictio n in function spaces. Then, in Sect. 5, we apply linear o ptimal prediction to the radiation moment system, and show that existing clo sures ca n be derived with it. In addition, the appr o ach gives rise to new closur es. While the optimal predictio n forma lism do es not re move the a rbitrar ine s s in the clo s ure pro cedure, it introduces it in a ra tional and comprehensible manner through the choice of a mea sure a nd the appr oximation of an integral. Thu s , it pr ovides a new per sp ective o n the err ors incurred due to the truncation of the infinite system. 2. Moment Models for Radia tive Transfer The radiative transfer equation (R TE) is [40] 1 c ∂ t I ( x, Ω , t ) + Ω ∇ I ( x, Ω , t ) + σ t ( x ) I ( x, Ω , t ) = Z 4 π σ s ( x, Ω · Ω ′ ) I ( x, Ω ′ , t ) dΩ ′ + q ( x, t ) . (1) In this equation, the radiative intensit y I ( x, Ω , t ) can be viewed as the the num b er of particles at time t , p osition x , traveling in to direction Ω. Equation (1 ) is a mesos copic phase space equation, mo deling loss due to scatter ing a nd a bsorption ( σ t -term), gain due to anisotr opic sc a ttering ( σ s - term) and containing an emission term q . Due to the large num b er of unknowns, a dire ct n umerica l simulation of (1) is very co stly . Often times only the low est momen ts of the intensit y with respect to the direction Ω are of in ter est. Mo men t mo dels a ttempt to a pproximate (1) by a coupled system of momen ts. F o r the sake of nota tional simplicity , w e co nsider a slab geometry . How ever, all metho ds presented here can b e gener a lized. Consider a plate that is finite alo ng the x -a xis and infinite in the y and z directions. The system is assumed to b e in v ar iant under translations in y and z a nd under rotations around the x -axis. In this case the radiative intensit y can only dep end on the scalar v a riable x and o n the azimuthal angle θ = arccos( µ ) b etw een the x -axis and the direction of fligh t. F ur ther more, w e select units such that c = 1. The system beco mes ∂ t I ( x, µ, t ) + µ∂ x I ( x, µ, t ) + σ t I ( x, µ, t ) = Z 1 − 1 σ s ( x, µ, µ ′ ) I ( x, µ ′ , t ) d µ ′ + q ( x, t ) , (2) with t > 0, x ∈ R , and µ ∈ [ − 1 , 1]. It is supplied with initial conditions I ( x, µ, 0) = ˚ I ( x, µ ). Here, the s cattering kernel is σ s ( x, µ, µ ′ ) = Z 2 π 0 σ s ( x, µµ ′ + p 1 − µ 2 p 1 − µ ′ 2 cos ϕ ) d ϕ . OPTIMAL PREDICTION FOR RADIA TIVE TRANSFER 3 Moment methods start with a n infinite system of momen ts, that is equiv alent to the original equa - tion of r adiative tra nsfer. Macros c o pic appr oximations to (1) can b e de r ived using angular basis functions [43, 5, 7], suc h as spherical harmonics . In 1d slab g eometry (2), Legendre polynomials [19, 9] are c ommonly used, since they form a n orthogonal basis o n [ − 1 , 1]. W e assume that the scattering kernel can b e expanded in to Legendre po lynomials σ s ( x, µ, µ ′ ) = ∞ X n =0 2 n +1 2 σ sn ( x ) P n ( µ ) P n ( µ ′ ) , and intro duce the tra nsp o rt co efficients σ an = σ t − σ sn . T o derive a momen t system, w e define the moments u l ( x, t ) = Z 1 − 1 I ( x, µ, t ) P l ( µ ) d µ . Multiplying (2) with P k and integrating over µ gives ∂ t u k ( x, t ) + ∂ x Z 1 − 1 µP k ( µ ) I ( x, µ, t ) d µ + σ t ( x ) u k ( x, t ) = σ sk ( x ) u k ( x, t ) + 2 q ( x, t ) δ k, 0 . Using the recurs ion relation for the Lege ndr e p olyno mials leads to ∂ t u k + ∂ x ∞ X l =0 b kl u l = ( σ sk ( x ) − σ t ( x )) u k ( x, t ) + 2 q ( x, t ) δ k, 0 . where b kl = k +1 2 k +1 δ k +1 ,l + k 2 k +1 δ k − 1 ,l . This is an infinite tridiagonal sy s tem for k = 0 , 1 , 2 , . . . ∂ t u k + b k,k − 1 ∂ x u k − 1 + b k,k +1 ∂ x u k +1 = − c k u k + q k (3) of first-order partia l differential equations . Using the (infinite) matrix no tation B =         0 1 1 3 0 2 3 2 5 0 3 5 3 7 0 . . . . . . . . .         , C =      σ a 0 σ a 1 σ a 2 . . .      , and q =       2 q 0 . . . . . .       we can write (3) as ∂ t u = − B · ∂ x u − C · u | {z } = R u + q . (4) The infinite momen t sys tem (4) is eq uiv a lent to the tra nsfer equation ( 2 ), since we repr esent a n L 2 function in terms o f its F our ier co mpo nents. In o rder to admit a numerical computatio n, (4) has to be a ppr oximated by a system of finitely man y moments I 0 , . . . , I N , i.e. all modes I l with l > N are not consider e d. In order to obtain a clos ed system, in the equation for I N , the dep endence on I N +1 has to be eliminated. A questio n of fundamen tal interest is ho w t o close the mo ment s ystem, i.e. b y what to replace the dependence on I N +1 . In the follo wing, we provide a brief o verview ov e r commonly used clo sure stra tegies, and wa ys to derive and justify them. 2.1. P N closure. The simplest clos ure, the so-calle d P N closur e [8, 4 3] is based on truncating the sequence I l , i.e. I l = 0 for l > N . The physical arg ument is that if the system is close to equilibrium, then the underlying par ticle distribution is uniquely determined by the low est-order moments. F or example, the P 1 equations are ∂ t u 0 + ∂ x u 1 = − σ a 0 u 0 + 2 q ∂ t u 1 + 1 3 ∂ x u 0 = − σ a 1 u 1 . 4 MAR TIN FRANK AND B E NJAMIN SEIBOLD The P N equations can also be derived by asymptotic analysis in the following w ay [3 5]. Intro duce the dimensionless par ameter ε as the r atio of the tota l scattering mean free path and a characteris- tic macr o scopic length scale. F urthermo r e, a ssume that the co efficients in the R TE asymptotically depe nd on ε as ∂ t I ( x, µ, t ) + µ∂ x I ( x, µ, t ) + σ t ε I ( x, µ, t ) = Z 1 − 1 σ s ( x, µ, µ ′ ) I ( x, µ ′ , t ) d µ ′ + q ( x, t ) . All o ther qua nt ities are O (1 ). If in addition, the sca ttering kernel scale s as σ s ( x, µ, µ ′ ) = N X n =0 2 n +1 2  σ t ( x ) ε − σ an ( x )  P n ( µ ) P n ( µ ′ ) + ∞ X n = N +1 2 n +1 2 σ sn ( x ) ε P n ( µ ) P n ( µ ′ ) , then the P N equations ar e a n asy mptotic limit of the R TE. This mea ns that the solutio n of the scaled R TE conv erges to the solution of the P N equations as ε tends to ze r o. 2.2. Di ffusion clo sure. The classical diffusion closure is defined for N = 1. W e assume I 1 to b e quasi-statio na ry and neg lect I l for l > 1, thus the equa tions r ead ∂ t u 0 + ∂ x u 1 = − σ a 0 u 0 + 2 q 1 3 ∂ x u 0 = − σ a 1 u 1 . Solving the seco nd equation for u 1 and inserting it into the first equatio n yields the diffusion approximation ∂ t u 0 − ∂ x 1 3 σ a 1 ∂ x u 0 = − σ a 0 u 0 + 2 q . (5) Again, an asymptotic sc a ling can be used to justify the diffusion appr oximation [33]. The diffusion- scaled R TE is ε∂ t I ( x, µ, t ) + µ∂ x I ( x, µ, t ) + σ t ε I ( x, µ, t ) = Z 1 − 1 σ s ( x, µ, µ ′ ) I ( x, µ ′ , t ) d µ ′ + εq ( x, t ) , where now the scattering kernel scales as σ s ( x, µ, µ ′ ) = 1 2  σ t ( x ) ε − εσ a 0 ( x )  + ∞ X n =1 2 n +1 2 σ sn ( x ) ε P n ( µ ) P n ( µ ′ ) . 2.3. D N diffusio n correction closure. A new hierarch y o f P N approximations, denoted diffu- sion c orr e ction or D N closure, has b een pr o p osed by Levermore [37, 44]. In sla b geometry , it can be derived in the following wa y: W e a ssume tha t u l = 0 for l > N + 1. Con tra ry to P N , the ( N + 1)-st moment is ass umed to b e qua si-stationar y . Setting ∂ t u N +1 = 0 yields the alg ebraic relation u N +1 = − 1 σ a,N +1 N +1 2 N +3 ∂ x u N , which, substituted into the equation for u N , yields an additional diffusion term for the la st moment: ∂ t u N + b N ,N − 1 ∂ x u N − 1 − θ∂ xx u N = ( 2 q − σ a 0 u 0 if N = 0 − σ aN u N if N > 0 (6) where θ = b N ,N +1 1 σ a,N +1 N +1 2 N +3 = 1 σ a,N +1 ( N +1) 2 (2 N +1)(2 N +3) . F or N = 0 this clo sure becomes the clas sical diffusion closure (5). The orig inal der iv a tion [37, 44] uses as ymptotic scaling a rguments similar to the o nes a bove. The D 1 equations are ∂ t u 0 + ∂ x u 1 = − σ a 0 u 0 + 2 q ∂ t u 1 + 1 3 ∂ x u 0 = − σ a 1 u 1 + 4 15 σ a 2 ∂ xx u 1 . OPTIMAL PREDICTION FOR RADIA TIVE TRANSFER 5 2.4. O ther t yp es of closures. Other higher or der diffusion approximations ex ist, such as the so- called simplifie d P N ( S P N ) equatio ns . These have been derived in an a d ho c fashion [2 5, 26, 27] and hav e subsequently be e n substantiated via asymptotic a nalysis [34] a nd via a v ar iational approa ch [49, 4]. See als o the recent review [38] and the brief deriv ation outlined in [24]. It is demonstrated in [46] that the formalism presen ted in this pap er can be used to derive c e r tain v ers io ns or v aria tions of these types of mo dels . Many nonlinear approximations exist in the literature, most pro minen tly flux-limited diffusion [31], v a riable Eddington facto r s [3 6, 3 2, 4 8], a nd minimum entropy methods [42, 1, 20, 21, 2 3, 5 0, 2 2]. F ur ther more, appr oaches ba sed on par tia l momen ts ex ist [50, 22]. None of these nonlinear closures is cons idered her e. 3. Conditional Expect a tions Optimal prediction, as introduced b y Chorin, Hald, Kast, Kupferman, e t al. [15, 12, 14, 1 0, 1 1] seeks the mean s olution o f a time-dependent system, when only par t of the initial data is known, but a measur e on the pha s e s pa ce is av a ilable. The appr oach has b een dev e lo p ed for o rdinary differential equatio ns and applied in v ar ious proble ms that inv o lve dy na mical systems [45, 18]. The key ideas of the approach ar e as follows. Consider a system of ordinary different ia l equations ˙ x ( t ) = R ( x ( t )) , x (0) = ˚ x . (7) Let the vector of unkno wns b e split x = ( x C , x F ) into the resolved v ariables x C that ar e of in ter est, and the unresolved v ariables x F that should be “av er aged out”. 1 Assume that the initial conditions ˚ x C for the r esolved v ar iables are known, while the initial conditions ˚ x F for the unresolved v ar iables are not known or not of relev ance. In addition, let a pr obability density f ( x ) b e given. Given the known initial co nditions ˚ x C , the measure f induces a co nditio ne d measur e f ˚ x C ( x F ) = ˜ Z − 1 f ( ˚ x C , x F ) for the remaining unknowns, where ˜ Z = R f ( ˚ x C , x F ) d x F . An average of a function u ( x C , x F ) with resp ect to f ˚ x C is the conditional exp ectation P u = E [ u | x C ] = R u ( x C , x F ) f ( x C , x F ) d x F R f ( x C , x F ) d x F . (8) It is an orthog onal pro jection with resp ect to the inner pro duct ( u, v ) = E [ u v ], whic h is defined by the exp ectatio n E [ u ] = R R u ( x C , x F ) f ( x C , x F ) d x C d x F . Let ϕ ( x , t ) deno te the solution of (7), for the initial conditions x = ( x C , x F ). Then o ptimal predictio n seeks for the mean solution P ϕ ( x , t ) = E [ ϕ ( x C , x F , t ) | x C ] . (9) Optimal prediction provides strateg ie s to formulate systems for x C that are faster to compute than solving the o riginal system (7). One appro ach, named firs t order optimal pr ediction, is based on applying the conditional exp e ctation (8) to the original equation’s (7) right hand side. A related approach is based on the Mori-Zwanzig formalism [41, 51] in a v ersio n for conditiona l exp ectatio ns [12]. Applying the forma lism to the Liouville equation for (7) approximates the mean solution by an in tegro -differential equatio n, that inv olves the first order r ight hand side, plus a memor y kernel. In this pa per , we formulate optimal prediction for s ystems of par tial differential equations. In particular, in Sect. 4, we apply the Mori- Zwanzig forma lis m to equations o f the type of the radiative moment system (4). In order to extend the pro jection (8) to these types of sy s tems, we hav e to construct a measure on a suitable infinite-dimens io nal function space, and an e x pression for its conditional e xp e ctation. Both can usually b e ac hieved by considering a suitable s e quence of finite-dimensiona l measur es [29, 3]. In addition, a mea sure ca n b e dir ectly defined b y its 1 In this paper, we borrow a subscript nota tion from the area of multigrid m ethods, in which C stands f or “coarse”, and F stands for “fi ne”. In the following s ections, this notation is al so used f or blo c k matrices and blo ck operators. 6 MAR TIN FRANK AND B E NJAMIN SEIBOLD characteristic functional. F ormally , the character istic fun c tio nal is given by the measure’s F ourier transform. Here w e fo cus on Gaus sian measures, b ecause they are one of the few class es where the constr uction ab ove is possible, and where explicit formulas can be der ived. In the following, we present the formalism for co nditional ex p ecta tions fir st in finitely many dimensions , and then in function space s. 3.1. Gauss ians in Finite Di m ensions . Due to Bo chner’s theorem [3 ], a measur e on R n is uniquely determined by its F ourier tr ansform, or characteristic functional θ ( y ) = Z R n f ( x ) exp( i y T x ) d x . (10) Indeed, if (1) θ ( 0 ) = 1, (2) θ is contin uous on R n , and (3) θ is pos itive definite in the sense that the matr ix ( θ ( y i − y j )) i,j =1 ,...,N is po sitive se mi- definite fo r all N and all y i ∈ R n , then there is unique proba bilit y mea sure λ with density f on the σ -alg e bra of Borel sets of R n such that (10) holds. Let A ∈ R n × n be a symmetric p o sitive definite ma trix, a nd m ∈ R n . Then f ( x ) = det( 2 π A ) − 1 2 exp  − 1 2 ( x − m ) T A − 1 ( x − m )  (11) is a pro ba bility densit y on R n . In tr o ducing the inner pro duct genera ted by A as h x , y i A := x T Ay , the Ga ussian measure with density (11) ha s the characteristic functional θ ( y ) = exp  − 1 2 h y , y i A + i y T m  . This functional satisfies the three conditions ab ove. The conditional exp ecta tion of the Gaussian, given par ts of the vector x , is given b y Lemma 1. De c omp ose the ve ctors x , m and the matrix A into x =  x C x F  , m =  m C m F  and A =  A C C A C F A F C A F F  . Then the c onditional exp e ctation is E [ x | x C ] =  E [ x C | x C ] E [ x F | x C ]  =   R x C f ( x C , x F ) d x F R f ( x C , x F ) d x F R x F f ( x C , x F ) d x F R f ( x C , x F ) d x F   =  x C m F + A F C A − 1 C C ( x C − m C )  . (12) Pr o of. The ident ity E [ x C | x C ] = x C is trivial. T o calculate E [ x F | x C ], consider M = A − 1 , with the s ame blo ck matrix notation for M as for A . One can easily verify that ( x − m ) T A − 1 ( x − m ) = k x F − b F k 2 M F F + ( x C − m C ) T  M C C − M C F M − 1 F F M F C  ( x C − m C ) , where b F = m F − M − 1 F F M F C ( x C − m C ), and the norm is defined as k x F k 2 M F F = x T F M F F x F . This yields for the conditional exp ecta tion E [ x F | x C ] = R ( x F − b F + b F ) exp  − 1 2 k x F − b F k 2 M F F  d x F R exp  − 1 2 k x F − b F k 2 M F F  d x F = b F . Since M = A − 1 , the block matrices satisfy M F C A C C + M F F A F C = 0 , whic h implies − M − 1 F F M F C = A F C A − 1 C C , and thus proves the claim.  OPTIMAL PREDICTION FOR RADIA TIVE TRANSFER 7 Expressio n (12) c o incides with the one given in [1 6]. The co nditional exp ectation is a pro jection, that can b e written in the form P x = E [ x | x C ] = Ex + Fm , using the pro jection matrices E =  I 0 A F C A − 1 C C 0  and F =  0 0 − A F C A − 1 C C I  . (13) The orthogonal complement is then Q x = ( I − P ) x = Fx − Fm . One can easily verify tha t P 2 = P , Q 2 = Q , and P Q = QP = 0. If the measure is cen tered ar ound the o rigin, i.e. m = 0, the pro jections b eco me simple matrix m ultiplica tions P = E and Q = F . 3.2. Gauss ians i n F unction Spaces. The co ns truction of measur es on spaces o f functions uses the character istic functional. F orma lly , all ex pr essions from the finite-dimensio nal case generalize to the infinite-dimensional ca se. Ther e are some mathematical s ubtleties related to this co nstruc- tion. F or the in ter ested r eader, we co llect these in this sectio n. F o llowing [29, 3], we co nstruct measures on the dual S ′ of a Hilber t spa c e S of functions. This construction is based on the Bo chner-Minlos theorem. Its key assumption is that S is nuclear, i.e. the iden tity in S is of Hilb er t- Schmidt type. In tha t case, the three conditions from ab ov e on a characteristic functional ar e necessar y and sufficient f o r the existence of a cor r esp onding measure. The pro of uses a sequence of finite-dimensional measures and Bo chner’s theorem. The following construction of a nuclear Hilber t space ser ves our latter purp oses. Definition 1. L et A b e a symmetric p ositive definite infinite matrix ( i.e. al l finite submatric es ar e symmetric p ositive definite) such t hat ∞ X i,j =1 | A ij | 2 < ∞ , and let V b e a Hilb ert sp ac e. We define the Hilb ert sp ac e l 2 A ( V ) , c onsisting of an infinite ve ct or of elements of V , by the inner pr o duct h f , g i l 2 A ( V ) := ∞ X i,j =1 h A ij f j , g i i V . In our case, we consider X = l 2 A ( L 2 ( R )). In order to o btain a Gelfand triple S ⊂ X ⊂ S ′ with the nuclearity prop erty , w e have to construct a space of smoo th test functions and its dual space of distributions. There ar e several w ays to do this. The following c o nstruction is standard and frequently used [3]. Definition 2. L et H = − d 2 dx 2 + x 2 + 1 , and define the Hilb ert sp ac e H ( R ) as the c ompletion of C ∞ 0 ( R ) in L 2 ( R ) with r esp e ct t o the inner pr o duct h f , g i H ( R ) := h H f , g i L 2 ( R ) . Let H − 1 ( R ) denote the dual of H ( R ). Then with S = l 2 A ( H ( R )) and S ′ = l 2 A ( H − 1 ( R )), we hav e a Hilb ert-Schmidt em b edding S ⊂ X ⊂ S ′ . Thus b y the Bo chner-Minlos theorem [3], the characteristic functional θ ( f ) = Z S ′ exp( i h f , g i S S ′ ) d λ ( g ) = exp  − 1 2 h f , f i X  8 MAR TIN FRANK AND B E NJAMIN SEIBOLD defines a unique probability measure on S ′ which satisfies Z S ′ exp( i h f , g i S S ′ ) d λ ( g ) = θ ( f ) . In addition, a nonzer o e xp e ctation v a lue can b e taken into acc ount b y noting that the modified functional can b e written in terms of a n in tegral whic h easily allo ws c hecking the prop erties in the Bo chner-Minlos theorem: Z S ′ exp( i h f , g + m i S S ′ ) d λ ( g ) = Z S ′ exp( i h f , g i S S ′ ) d λ ( g ) exp( i h f , m i S S ′ ) = exp  − 1 2 h f , f i X + i h f , m i S S ′  . Thu s we hav e Lemma 2. Given m ∈ S ′ , the char acteristic functional θ ( f ) = ex p  − 1 2 h f , f i X + i h f , m i S S ′  (14) defines a unique pr ob ability me asur e on S ′ such that Z S ′ exp( i h f , g i S S ′ ) d λ ( g ) = θ ( f ) . In the same wa y as for the measure, certain moments or conditio na l expectations can be inherited from the finite-dimensional case. Lemma 3 . De c omp ose the ve ctor-value d distribution u ∈ S ′ , the ve ctor-value d exp e ctation value m ∈ S ′ and the matrix A into u =  u C u F  , m =  m C m F  and A =  A C C A C F A F C A F F  . We denote by λ ( u F ) the c onditione d Gaussian m e asu r e with r esp e ct to u F , given u C . Then for al l ve ctor-value d test functions f =  f C , f F  T , we have the c onditional exp e ctations Z h f C , u C i S S ′ d λ ( u F ) = h f C , u C i S S ′ Z h f F , u F i S S ′ d λ ( u F ) = h f F , m F i S S ′ +  f , A F C A − 1 C C ( u C − m C )  S S ′ . Her e, R h f C , u C i S S ′ d λ ( u F ) and R h f F , u F i S S ′ d λ ( u F ) ar e the we ak formulations of the inte gr als R u C d λ ( u F ) and R u F d λ ( u F ) , which c an b e interpr ete d as c onditional exp e ctations of an S ′ -value d r andom variable with pr ob ability distribution λ ( u F ) . Pr o of. The pro of follows the pro o f of the Bo chner-Minlos theorem. W e a ppr oximate the infinite- dimensional Gaussian measur e by a sequence of finite-dimensiona l Gaussian measures. F or each of these, we ha ve a formula for the conditional exp e c tation, given by Lemma 1. By showing that the limit of conditiona l measures exis ts and by showing tha t the mono mials a re measurable, w e obtain the conditional exp ectations ab ove.  As in the finite dimensional ca se, the pro jection ca n b e written in matr ix for m h f , P u i S S ′ = Z h f , u i S S ′ d λ ( u F ) = h f , Eu i S S ′ + h f , Fm i S S ′ , (15) using the sa me pro jection matrices (13) as in the finite dimensiona l case. As a short notation, we write (15) as P u = Eu + Fm , or P = E in the case m = 0 . In the follo wing, whenever we use this short nota tio n, we alwa ys mean in the w ea k sense (15). OPTIMAL PREDICTION FOR RADIA TIVE TRANSFER 9 4. Linear Optimal Prediction W e now apply optimal prediction to a linear ev olution equa tio n under a Gaussian measure. As derived in Sec t. 3, the c o nditional exp ectation is an affine linear pro jectio n. Here, we consider a Ga ussian centered ar ound the orig in, thus P = E . While this choice is reasonable in many cases, its main purpose is to simplify notation. The results transfer to the case m 6 = 0 , with affine linear trans formations instead of matrix multiplications. W e present the Mori-Zwanzig formalism [41, 51] for a linea r e volution equatio n ∂ t u = R u , u (0) = ˚ u , (16) where u is a vector-v alued distr ibutio n and R is a linear differential o pe r ator (or u is a vector and R is a matrix, for a n ordinary differen tial equation) that is independent of space a nd time. Consider a Ga ussian meas ure, defined by a symmetric p ositive definite ma trix A (see Sect. 3). Let the unknowns and the co rresp onding oper ators/ matrices be split u =  u C u F  , R =  R C C R C F R F C R F F  , and A =  A C C A C F A F C A F F  . The conditional exp ecta tion of the co ordinate vector u is P u = E [ u | u C ] = E · u , where we have the pro jection ma trix (1 3 ) E =  I 0 A F C A − 1 C C 0  . Also, define F = I − E a s the orthogonal pro jection matrix . Due to linearity , for any matr ix vector pro duct Bu , the pro jection a lwa ys applies to the vector itself P Bu = E [ Bu | u C ] = B · E [ u | u C ] = B · E · u = B · u , (17) where the pro jected matrix takes the form B =  B C C + B C F A F C A − 1 C C 0 B F C + B F F A F C A − 1 C C 0  . (18) Let the solution oper ator of (16) b e denoted b y e tR . In additio n, we consider the solution oper ator e tR F to the orthog onal dynamics equation ∂ t u = R F u , u (0) = ˚ u . (19) W e assume that b oth the or iginal system (16) and the orthogonal dy na mics eq ua tion (19) are w ell po sed. The existence of solutions to the orthogona l dynamics has b een prov ed for R the Lio uville op erator to a nonlinear differen tial equation [28]. If R is a matrix , then b o th e tR and e tR F are in fact matrix exp o ne ntials. F or R a differential op erator , they stand as a notatio n for the solutio n op erator s genera ted by R and R F . Theorem 4 (Dyson’s formula) . L et R b e a differ ential op er ator and E + F = I a p air of ortho gonal pr oje ction matric es. L et e tR and e tR F denote the evo lu tion op er ators gener ate d by R r esp e ctively R F . Then e tR = e tR F + Z t 0 e ( t − s ) R F R E e sR d s . Pr o of. Define the evolution op e rator M ( t ) = e tR − e tR F − Z t 0 e ( t − s ) R F R E e sR d s . (20) Its time deriv ative equals ∂ t M ( t ) = Re tR − R F e tR F − R E e tR − R F Z t 0 e ( t − s ) R F R E e sR d s = R F M ( t ) . With the initial conditions M (0) = 0, we obtain that M ( t ) = 0 ∀ t ≥ 0.  10 MAR TIN FRANK AND B E NJAMIN SEIBOLD Different ia ting (20) with res pe ct to time yields an identit y for the solution op erator ∂ t  e tR  = R F e tR F + R E e tR + R F Z t 0 e ( t − s ) R F R E e sR d s = R e tR + e tR F R F + Z t 0 K ( t − s ) e sR d s , (21) where R = R E is the pro jected differential o per ator, a nd K ( t ) = e tR F R F R E is a memory k er nel for the dynamics. Matrix E has zeros in the right column, thus R a nd K have the same structure R =  R C C 0 R F C 0  and K =  K C C 0 K F C 0  . As defined by (9), t he mean solution o f (16) with resp ect to the meas ur e defined b y (14) is o bta ined by applying the pro jectio n op era tor to the solution o pe rator. Since the solution op erato r is linear, prop erty (17) yields u m ( t ) = P e tR ˚ u = e tR E ˚ u , i.e. the mean s olution is a pa rticular solution, obtained b y ev olving the pro jected (averaged) initial conditions. Thus, the mean solution op er ator is e tR E . Multiplying th e iden tity (21) from the righ t by E yie lds an ident ity for the mean solution o p e r ator ∂ t  e tR E  = R e tR E + e tR F R FE | {z } =0 + Z t 0 K ( t − s ) e sR E d s , in which the middle term cancels out, since FE = 0 . This yields a new evolution equation for the mean so lution ( ∂ t u m ( t ) = R u m ( t ) + R t 0 K ( t − s ) u m ( s ) d s u m (0) = E ˚ u , which reads in blo ck-comp onents ∂ t u m C = R C C u m C + K C C ∗ u m C , u m C (0) = ˚ u C (22) ∂ t u m F = R F C u m C + K F C ∗ u m C , u m F (0) = A F C A − 1 C C ˚ u C . (23) W e hav e derived an equation (2 2) in which the dynamics for the v ariables o f in teres t is independent of the evolution of the av era ged v ariables . The latter are typically not o f interest, but if desired, they can be obtained b y in tegrating (23). F or nonlinear sys tems of ordinary differential eq uations, an ana logous in teg r o-differential equation can be derived, which approximates the true mea n solution. In that context [13], it is denoted se c ond or der optimal pr e diction . F or the linear problem considered here, equation (22) yields th e true mean solution, hence we call it ful l optimal pr e diction . The simplest approximation to (22 ), called fi rst or der optimal pr e diction , is obtained by neglecting the c o nv olution term, i.e. b y solving the sy stem ∂ t u fo op C = R C C u fo op C , u fo op C (0) = ˚ u C . (24) A b etter approximation can b e obtained if a time scale τ exis ts, after which the kernel b ecomes negligible: K ( t ) ≪ K (0) ∀ t > τ . Assuming that u m = O (1) o ver the time scale of integration, a piecewise-consta nt q uadrature r ule yields the approximation Z t 0 K ( t − s ) u m ( s ) d s = Z t 0 K ( s ) u m ( t − s ) d s ≈ Z τ 0 K ( s ) u m ( t − s ) d s ≈ Z τ 0 K (0) u m ( t ) d s = τ R F R Eu m ( t ) , (25) which leads to the se c ond or der op t imal pr e diction sys tem ∂ t u soop C = R C C u soop C + τ ( R F R E ) C C u soop C , u soop C (0) = ˚ u C . (26) OPTIMAL PREDICTION FOR RADIA TIVE TRANSFER 11 Here ( R F R E ) C C = R C F R F C + R C F R F F A F C A − 1 C C − R C F A F C A − 1 C C R C C − R C F A F C A − 1 C C R C F A F C A − 1 C C is a new linear differential operato r, which is second order if R is a first order op erator . While this piecewise c o nstant approximation to the memory ter m is more a c curate than not considering the memory term a t all, it is not very a ccurate as a quadrature rule. Its use her e is justified for t wo reasons : • Firs t, we wish to restrict our analysis to systems that in volve only the curren t state of the system (i.e. no delay); • Seco nd, this a pproximation tur ns out to yield the diffusion-cor rection closure (6), when applied to ra dia tive transfer (see Sect. 5 ). 5. Applica tion to the Radia tion Moment S ystem W e now tur n our attention to the infinite moment system (4). F o r consistency with the notation de- veloped in Sect. 4, w e denote the (infinite) v ector of momen ts b y u ( x, t ) = ( u 0 ( x, t ) , u 1 ( x, t ) , . . . ) T . In addition, we neglect the source ter m, since it is unaffected by trunca tion. The radiation s ystem (4) can b e written as ∂ t u = R u , where the differential op e rator R = − B ∂ x − C inv olves the (infinite) matrices (2). As intro duced in Sect. 3, we consider a Gaussian measur e o n the s pace of unknowns, defined by an (infinite) matrix A . In Sect. 5.1, w e consider first order optimal pre diction, and in Sect. 5.2, we consider second o rder o ptimal predictio n. 5.1. Fi rst Order Optimal Prediction. W e wish to truncate the system after th e N -th compo - nent . The system and the meas ure are split in to blo cks u =  u C u F  , B =  B C C B C F B F C B F F  , C =  C C C C C F C F C C F F  , and A =  A C C A C F A F C A F F  . F o r the radiation sy stem, w e hav e B C F =    0 . . . 0 . . . . . . . . . N +1 2 N +1 . . . 0    , C C F = 0 , and C F C = 0 . Due to (18), the pro jected differential op erato r ’s upp er left blo c k is R C C = −  B C C + B C F A F C A − 1 C C  ∂ x − C C C . (27) The mo dification term B C F A F C A − 1 C C has nonzero ent r ies only in its las t row. Hence, first order optimal prediction yields a true clos ure relation, since only the last equa tion is mo dified. The mo dification is N +1 2 N +1 times the first r ow of A F C A − 1 C C , i.e. the closure dep ends solely o n the correla tions between the moments, given by the measure. W e ca n see that, depending o n the choice of A , first-or der optimal prediction can ge ne r ate all pos sible linear hyperb olic clos ures. The measure enables us to enco de a co rrelatio n b e t ween the resolved moments u C and the av er - aged moments u F , that co uld come from a dditional knowledge a bo ut the par ticular s etup of the problem. If no suc h additional knowledge is at hand, it is r easonable to prescrib e no correlation betw e en u C and u F , b y consider ing a decoupled measure, i.e. A F C = 0 . In this case, the system is plainly cut off, and thus the classica l P N closure is obtained. 12 MAR TIN FRANK AND B E NJAMIN SEIBOLD 5.2. Se cond Order Optimal Prediction. In Sect. 5.1, we ha ve seen that firs t order o ptimal prediction with a decoupled Gaussian measure yields the clas sical P N closure. F or the same measure, w e now consider the memory term. Since A F C = 0 , we hav e R E =  R C C 0 R F C 0  , R F =  0 R C F 0 R F F  , and R F R E =  R C F R F C 0 R F F R F C 0  . F o r the radiation sy stem (5) we get R F R E =  B C F B F C ∂ xx 0 B F F B F C ∂ xx + C F F B F C ∂ x 0  , where B C F B F C =     0 . . . 0 . . . . . . . . . 0 . . . ( N +1) 2 (2 N +1)(2 N +3)     . F o r mally , the memory ter m uses solution v alues at all previous times. How ever, the differe nt solution comp onents deca y at the rates σ ai . This yields time scales τ i = 1 σ ai , o ver whic h information from the past can be seen in the solution ( τ i are a time scales since we hav e set c = 1). If w e single out one time scale τ , the seco nd order optimal prediction a pproximation (26) b ecomes Z t 0 K C C ( s ) u C ( t − s ) d s ≈ τ ( R F R E ) C C u C =      0 . . . 0 τ ( N +1) 2 (2 N +1)(2 N +3) ∂ xx u N      . Compared to the P N closure, a diffusion ter m is added into the la st comp onent of the truncated system. If we iden tify τ with τ N +1 , i.e. the time scale given b y the rate of decay of the first tr un- cated moment , then this sy stem is ex a ctly the diffusion corre c tion closure (6) by Levermore [37], as outlined in Sect. 2.2. In th e case N = 0, it is equiv alent to the c lassical diffusion appro xima tion. Although here we ha ve assumed s patially homog eneous co efficients, we expect that the equations can b e adapted to the spa ce-dep endent ca se in analogy to diffusion theor y . Sp ecifically , if σ ai ( x ) are space dependent, w e define τ i ( x ) = 1 σ ai ( x ) , and replace τ N +1 ∂ xx u N by ∂ x ( τ N +1 ( x ) ∂ x u N ). The v a lidit y of this a pproximation will b e addresse d in future r e s earch. 6. Discussion W e ha ve formulated the approach of optimal prediction for a sys tem of linear par tia l differen tial equations with an underlying Gaussian mea sure. An identit y for the evolution of a finite num b er of momen ts is obtained. Appro x imations to this identit y yield differen t closures for a truncated version o f the full system. The application o f the formalism to the equation o f radiative tr ansfer allows the re -deriv ation of classical linear closures, such as P N , diffusion, and diffusion cor rection. While traditionally these clos ures a re deriv ed using physical argumen ts or by a symptotic analysis, the optimal prediction formalism generates the closures by choo sing a measure and approximat- ing a mathematical identit y . This connection yields a new int er pretation of diffusion as be ing connected to the memory of the system (or lo ss thereof ). If we follow this interpretation, we observe a p o ssible mo dification of the second order o ptimal prediction approximation (26) (and th us of the diffusion approximation), that may be more accu- rate for s hort times : for t < τ , the in tegr al in (25) cannot stretch ov er the whole length τ . Thus, one should repla ce the co efficient τ by a time dep e nden t function f ( t ) that v anishes for t → 0, increases for 0 < t < τ , and approaches τ for t ≫ τ . F or ex ample, a b etter approximation than OPTIMAL PREDICTION FOR RADIA TIVE TRANSFER 13 (25) is given by Z t 0 K ( t − s ) u m ( s ) d s ≈ min { t, τ } R F R Eu m ( t ) , (28) which leads to repla c ing τ b y min { t, τ } in the seco nd or der system (26). While (26) yields a classical diffusio n cor rection approximation, expressio n (28) leads to a new a ppr oximation, whic h we hav e called cr esc endo-diffusion c orr e ction a ppr oximation in a previous paper [46]. The f unctio n min { t, τ } is just one of ma ny p ossible time dep endent c o efficients for the second order term. A smo other evolution is derived in [46] by approximating the o rthogona l dynamics more accurately . It y ields the diffusion function f ( t ) = τ  1 − e − t/τ  . The in terpr etation tha t diffusion is connected to the memor y of the system lea ds us to a gradua l ramp-up of the diffusion co efficient. This suggests that there is an initia l lay er ov er which the particle system bec o mes diffusiv e. O n the one hand, standard tec hniques do not predict an initial lay er. The ad- ho c condition is to just tak e the moments of the initial condition. This condition also comes out of asymptotic a nalysis (for diffusion in [3 3], for P N in [35]). Computational results [4 6], on the other hand, show that there can be a non-negligible initial layer, and that the c r escendo diffusion idea improv es the quality of the solution. The cr escendo mo dification in tr o duces a n explicit time-dependence. The physical rationale is that at t = 0, the state of the res olved momen ts is known exa c tly . Informa tion is lost a s time evolv es , due to the approximation. In this pap er , optimal pr ediction with r esp ect to Gaus s ian measur es is considere d. How ever, many other measures are co nceiv able, for instance a construction in the following s pirit. Ga us sian measures are supp orted everywhere, and thus do not resp ect the conditio ns o n moments to be realizable a s moments of a non-negative radiative in tensity function [31]. As shown in [30, 47], in the reduced moments N k = u k u 0 ∀ k ≥ 1, this rea lizability domain is contained in a box of sq uare- summable edge lengths, i.e. N k ∈ [ N − k , N + k ] with N + k − N − k ≤ 2 − 2 k +2 . Due to this prop erty it is pla usible that measur es can be defined (e.g ., a uniform distribution in N k ∀ k ≥ 1) on the spa ce o f infinitely many moments, that v a nish o utside the r ealizability doma in. Closures derived as conditional exp ectatio ns with re s pe c t to such measures w ould—in con tr ast to Gaussian measures—b e alwa ys realiza ble and, as a consequence , nonlinear. Hence, it is an important question whether existing nonlinear closures, suc h as flux-limited diffusion [36] and minimum entrop y closures [4 2 , 1, 20] can b e derived by optimal prediction with a non- Gaussian measure. If p o ssible, the appr oach could b e us ed to systematically derive diffusion correctio n terms for these nonlinear clo sures. Another fundamental question is to which extent bo undary conditions ca n be included in to the formalism. Many other t yp es of closure s can p oss ibly be deriv ed from the presented for malism, by appr oximating the memory term with higher order . 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