ADAPTIVE GALERKIN APPROXIMATION ALGORITHMS FOR PARTIAL DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

Transcription

1 ADAPTIVE GALERKIN APPROXIMATION ALGORITHMS FOR PARTIAL DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS CHRISTOPH SCHWAB AND ENDRE SÜLI Abstract. Space-time variational formulations of infinite-dimensional Fokker Planck (FP) and Ornstein Uhlenbeck (OU) equations for functions on a separable Hilbert space H are developed. The well-posedness of these equations in the Hilbert space L 2 (H, µ) of functions on H, which are square-integrable with respect to a Gaussian measure µ on H, is proved. Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin discretizations, based on a tensorized Riesz basis, built from biorthogonal piecewise polynomial wavelet bases in time and the Hermite polynomial chaos in the Wiener Itô decomposition of L 2 (H, µ), are introduced and are shown to converge quasioptimally with respect to the nonlinear, best N-term approximation benchmark. As a consequence, the proposed adaptive Galerkin solution algorithms perform quasioptimally with respect to the best N-term approximation in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of active coordinates identified by the proposed adaptive Galerkin approximation algorithms. 1. Introduction Partial differential equations in infinite dimensions arise in a number of relevant applications; most notably as forward and backward Kolmogorov equations for stochastic partial differential equations (see, e.g., [7, 5] and the references therein). Their numerical solution has, to date, received scant attention, however. Approximations to such equations are typically attempted by path simulations in the corresponding stochastic partial differential equation, whose path-wise solutions belong to function spaces over finite-dimensional domains and can be, therefore, approximated by standard discretization techniques, combined with Monte Carlo path sampling. In the present paper, we propose a novel, adaptive approach to the construction of finite-dimensional numerical approximations to the deterministic forward equation in infinite-dimensional spaces, which exhibit certain optimality properties. The proposed approach is based on space-time variational formulations of these equations, which are posed in Gel fand-triples of Sobolev spaces over a separable Hilbert space H with respect to a Gaussian measure, and on Riesz bases of these spaces, which have been developed for linear and nonlinear parabolic PDEs in finite dimensions in [24, 8, 17]. The structure of the paper is as follows: in the next section we present two space-time variational formulations of linear parabolic equations set in a domain of finite dimension, which we then apply in Section 3 to Fokker Planck equations that arise in bead-spring chain models for d- dimensional polymeric flow, d {2, 3}, with chains consisting of K +1 beads whose kinematics are statistically described by a configuration vector q R Kd, K 1. The probability density function ψ = ψ(q, t) that is sought as the solution of the associated Fokker Planck equation is therefore a function of Kd spatial variables with K 1 and the time variable t. Our aim is to embed this finite-dimensional problem of potentially very high dimension into an infinite-dimensional problem. We therefore turn to Fokker Planck equations in infinite-dimensional domains by recapitulating The research reported in this paper was carried out at the Hausdorff Institute for Mathematics (HIM), Bonn, during the HIM-Trimester on High Dimensional Approximation, May August CS was partially supported by the European Research Council (ERC) under the FP7 programme ERC AdG ES was partially supported by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1). 1

2 2 CHRISTOPH SCHWAB AND ENDRE SÜLI basic facts about Gaussian measures, before proving in Section 5 well-posedness of the second of our two space-time variational formulations in the infinite-dimensional space L 2 (H, µ), where H denotes a separable Hilbert space and µ is a Gaussian measure on H. We show in particular that the solution operator of the variational formulation is an isomorphism between suitable solution and data spaces. We then turn our attention to adaptive Galerkin approximations. We outline the general principle in Section 6, where we introduce the idea of conversion of abstract, well-posed operator equations on separable Hilbert spaces to equivalent, infinite matrix-vector problems in the sequence space l 2 (N). After reviewing some relevant facts concerning N-term approximations in l 2 (N), we introduce abstract adaptive Galerkin approximation algorithms that construct sequences of N-term approximations, which, while not being best N-term approximations, are optimal in the sense that they converge asymptotically at the rate afforded by the best N-term approximation, provided that certain conditions are met by the operators and the Riesz bases used to discretize them. In Section 9 the abstract concepts are applied to infinite-dimensional Fokker Planck equations. Notably, a Wiener polynomial chaos type Riesz basis in L 2 (H, µ) and a wavelet basis in time are used for the Galerkin discretization. In Section 10 we verify the abstract assumptions for the specific equations of interest, and in Section 11 we consider the more general setting of nonsymmetric equations with drift, leading to our main result concerning the optimality of adaptive Galerkin discretizations with dimension-independent bounds, in Section Space-time variational formulation of abstract parabolic problems 2.1. Abstract elliptic equations. Suppose that D R d is either a bounded open domain with Lipschitz continuous boundary D or D = R d. In D, we consider the elliptic partial differential equation Au = f in D (2.1) with suitable homogeneous boundary conditions on D when D is bounded, or subject to a decay condition at infinity when D = R d. In typical weak formulations of elliptic boundary-value problems the solution u is sought in a certain separable Hilbert space V of functions defined on D (usually a hilbertian Sobolev space), so that A L(V, V ) and f V, where V is the dual space of V, resulting in the following weak formulation of problem (2.1): Given f V, find u V such that: a(u, v) = V f, v V v V. Here, a(u, v) = V Au, v V and V, V denotes the V V duality pairing. It is also assumed that there exists a separable Hilbert space H, identified with its dual H, such that V H = H V, (2.2) where the continuous embeddings signified by the symbol are dense and compact. All Hilbert spaces considered here are, for the sake of simplicity, assumed to be over the field R of real numbers. By (, ) H we denote the inner product in H, which extends to the duality pairing V, V on V V by continuity. Example 2.1. The prototypical example of a problem of the above form is Poisson s equation on a bounded open Lipschitz domain D R d, subject to a homogeneous Dirichlet boundary condition on D, in which case: A =, V = H 1 0(D), H = L 2 (D), V = H 1 (D) := H 1 0(D), a(u, v) = u v dx. We shall assume that A is selfadjoint, i.e., A = A (which implies that the bilinear form a(, ) is symmetric on V V) and coercive on V, i.e., there exists a real number γ 0 > 0 such that u V : a(u, u) = V Au, u V γ 0 u 2 V. (2.3) Our assumption A L(V, V ) implies the existence of a positive real number γ 1 = A L(V,V ) γ 0 such that u, v V : a(u, v) γ 1 u V v V ; (2.4) i.e., the bilinear form a is bounded. D

3 ADAPTIVE ALGORITHMS IN INFINITE DIMENSIONS 3 Hence, (2.3) and (2.4) imply that the energy norm a, defined on V by v a := [a(v, v)] 1/2, v V, is equivalent on V with the norm V ; viz., v V : γ 0 v 2 V v 2 a γ 1 v 2 V. We recall the following version of the Hilbert Schmidt theorem (cf. Lemma 15 in [14]). Theorem 2.2. Suppose that H and V are separable Hilbert spaces, with V densely and compactly embedded in H. Let a : V V R be a nonzero, symmetric, coercive and bounded bilinear form. Then, there exists a sequence of real numbers (λ n ) n=1 and a sequence of unit H-norm elements (ϕ n ) n=1 in V, which solve the following eigenvalue problem: find λ R and ϕ V \ {0} such that a(ϕ, v) = λ(ϕ, v) H v V. The real numbers λ n, n N, which can be assumed to be in increasing order with respect to the index n N, are positive, bounded away from 0, and lim n λ n = +. In addition, the ϕ n, n N, form an orthonormal system in H whose closed span in H is equal to H, and the scaled elements ϕ n / λ n, n N, form an orthonormal system with respect to the inner product defined by the bilinear form a(, ), whose closed span with respect to the norm a induced by a(, ) is equal to V. Furthermore, h = (h, ϕ n ) H ϕ n and h 2 H = (h, ϕ n ) 2 H h H (2.5) and v = n=1 and, in addition, n=1 ( ) ϕ n ϕn a v, and h 2 a = λn λn h H and n=1 n=1 ( ) 2 ϕ n a v, v V; λn λ n (h, ϕ n ) 2 H < h V. n=1 By virtue of Theorem 2.2, there exists a countable family σ R + of eigenvalues λ of A, bounded away from zero, whose accumulation point is at +, and an H-orthonormal sequence {ϕ λ : λ σ} of eigenfunctions ϕ λ V, i.e., Aϕ λ = λϕ λ, λ σ, and (ϕ λ, ϕ λ ) H = δ λλ, λ, λ σ. We then have by (2.5), for each v H, the Fourier series expansion v = λ σ v λ ϕ λ, v λ := (v, ϕ λ ) H and Parseval s identity v 2 H = λ σ v λ 2. In particular, therefore, the system {ϕ λ : λ σ} forms a normalized Riesz basis of H. The next two lemmas whose proofs are elementary show that renormalized versions of {ϕ λ : λ σ} constitute Riesz bases in V and V as well. Lemma 2.3. The following two-sided bound holds for each v V: γ 0 v 2 V λ σ λ v λ 2 γ 1 v 2 V. For f V, we have that f = λ σ f λ ϕ λ, f λ := V f, ϕ λ V. Lemma 2.4. The following two-sided bound holds for each f V : 1 f 2 V γ λ 1 f λ 2 1 f 2 V 1 γ. 0 λ σ

4 4 CHRISTOPH SCHWAB AND ENDRE SÜLI As u = A 1 f = λ σ u λ ϕ λ, with u λ = λ 1 f λ for λ σ, it follows that λ u λ 2 = λ 1 f λ 2, (2.6) λ σ λ σ where, by Lemma 2.3, the left-hand side of the equality belongs [ to the interval u 2 V [γ 0, γ 1 ] and, by Lemma 2.3, the right-hand side belongs to the interval f 2 V γ 1 1, ] γ 1 0. Here, for a nonnegative real number α and a nonempty bounded closed interval [s, t] R, the notation x α[s, t] and x [s, t]α are abbreviations for the two-sided inequality: αs x αt. We deduce from (2.6) in particular that γ 0 u V f V γ 1 u V. Thus, the linear operator A 1 is a (bi-lipschitz) quasi-isometric isomorphism between V and V, when these spaces are equipped with the norms V and V, respectively Abstract parabolic equations. Given T > 0 and the triple (2.2), we consider the abstract parabolic differential equation t u + Au = f in Q := (0, T ) D, (2.7) where A L(V, V ), with A L(V,V ) = γ 1 > 0, satisfies A = A, and γ 0 > 0 v V : a(v, v) := V Av, v V γ 0 v 2 V. As our aim is to develop the numerical analysis of space-time adaptive Galerkin discretizations of infinite-dimensional parabolic problems, we begin, following [24], by presenting weak formulations that are amenable to adaptive Galerkin discretizations. To this end, we consider the Bochner spaces L 2 (0, T ; V), L 2 (0, T ; H) and L 2 (0, T ; V ) and we define as well as H 1 (0, T ; V) := {u L 2 (0, T ; V) : u L 2 (0, T ; V)}, H 1 0,{0} (0, T ; V) := {u H1 (0, T ; V) : u(0) = 0 in V}, H 1 0,{T } (0, T ; V) := {u H1 (0, T ; V) : u(t ) = 0 in V}. (2.8a) (2.8b) Here and throughout the rest of the paper u will signify du/dt or u/ t, depending on the context. In the variational formulation of the parabolic problem, an important role is played by the space X defined by X := L 2 (0, T ; V) H 1 (0, T ; V ), which we equip with the norm X defined by v X := ( v 2 L 2 (0,T ;V) + v 2 L 2 (0,T ;V ) With V, H and V as in the triple (2.2) the following continuous embedding holds: ) 1 2. X C([0, T ]; H) (2.9) (in the sense that any v X is equal almost everywhere to a function that is uniformly continuous as a mapping from the nonempty closed interval [0, T ] of the real line into H). Therefore, for u X and 0 t T, the values u(t) are well-defined in H and there exists a constant C = C(T ) > 0 such that u X t [0, T ] : u(t) H C u X. In particular, for u X the values u(0) and u(t ) are well-defined in H and X 0,{0} := {u X : u(0) = 0 in H}, X 0,{T } := {u X : u(t ) = 0 in H} are closed linear subspaces of X. Henceforth we shall write Y = L 2 (0, T ; V) and we denote by Y the dual space L 2 (0, T ; V ) of Y, identifying L 2 (0, T ; H) with its own dual.

5 ADAPTIVE ALGORITHMS IN INFINITE DIMENSIONS First space-time variational formulation. We consider (2.7) in the special case when Lu := t u + Au = f Y in Q := (0, T ) D, u(0) = u 0 H, (2.10) in conjunction with a suitable homogeneous boundary condition incorporated in the (domain of) definition of the linear operator A. We begin by considering the case when the initial condition is also homogeneous, i.e., u 0 = 0 H. The first space-time variational formulation of the parabolic problem (2.10) is based on the bilinear form T ( ) (u, v) X Y B(u, v) := V u, v V + a(u, v) dt R. (2.11) 0 Then, the space-time weak formulation of the parabolic problem (2.10) with homogeneous initial condition u 0 = 0 in H reads as follows: given f Y, find u X 0,{0} such that B(u, v) = f(v) v Y. (2.12) 2.4. Second space-time variational formulation. In (2.12), the initial condition u(0) = 0 is incorporated as essential condition in the function space X 0,{0} in which the solution to the problem was sought. To accommodate a nonhomogeneous initial condition, one may proceed in (at least) two different ways. If u(0) = u 0 0 in H, (2.13) then one can for example first subtract from u a function u p X such that u p t=0 = u 0 ; we may, in particular, choose for this purpose u p = e t u 0, for u 0 V H. The disadvantage of this approach is that u 0 V is needed (instead of u 0 H), which can be viewed as an unnecessarily restrictive demand on the regularity of the initial datum u 0. Alternatively, in order to relax the regularity requirement u 0 V to u 0 H, one may impose (2.13) weakly, either by enforcing it via a multiplier as in [24] or through a space-time variational formulation, which incorporates it as a natural boundary condition, as follows: we integrate the time derivative by parts using that T 0 V u, v V dt = T If u(0) = u 0 0 in H, we require that 0 V v, u V dt + (u, v) T 0, u, v L 2 (0, T ; V) H 1 (0, T ; V ). (2.14) v(t ) = 0. (2.15) Thus, (2.14) and (2.15) lead to the weak formulation (2.18) below. To state it, we recall the spaces Y = L 2 (0, T ; V) and X = L 2 (0, T ; V) H 1 (0, T ; V ) (2.16) and the subspaces (cf. also (2.8)) X 0,{0} := {u X : u(0) = 0 in H}, X 0,{T } := {u X : u(t ) = 0 in H}, (2.17a) (2.17b) equipped with the norm of X ; we shall write X0,{0} and X0,{T } to indicate that the norm of X is applied to an element of X 0,{0} and X 0,{T }, respectively. Thanks to the continuous embedding (2.9), the spaces X 0,{0} and X 0,{T } are correctly defined and are closed subspaces of X ; in particular the expression (2.14) is meaningful for u, v X. The variational form of the parabolic problem (2.10) with weak enforcement of the initial condition, which we shall also refer to as the space-time adjoint weak formulation, then reads: given u 0 H and f X 0,{T }, find Here the bilinear form B (, ) is given by u Y : B (u, v) = l (v) v X 0,{T }. (2.18) B (u, v) := T 0 ( V v, u V + a(u, v) ) dt (2.19)

6 6 CHRISTOPH SCHWAB AND ENDRE SÜLI and the linear functional l is defined by l (v) := X f, v X + (u 0, v(0)) H. (2.20) Clearly, for f X 0,{T } L2 (I; V ) + (H 1 0,{T } ) (I; V) L 2 (I) V + (H 1 0,{T } (I)) V and, for any u 0 H, the functional l in (2.20) is linear and continuous on X 0,{T } ; i.e., there exists a constant C > 0 such that v X 0,{T } : l (v) C ( f X 0,{T } + u 0 H ) v X0,{T } Well-posedness. The well-posedness of the variational problems (2.12) and (2.18) is an immediate consequence of the following result. Theorem 2.5. Suppose that V and H are separable Hilbert spaces over the field R such that V H H V with dense embeddings. Assume further that the bilinear form a(, ) : V V R is continuous, i.e., M a > 0 w, v V : a(w, v) M a v V w V, (2.21) and coercive in the sense that it satisfies a Garding inequality, i.e., there exists m a > 0 and κ 0 such that v V : a(v, v) m a v 2 V κ v 2 H. (2.22) Under these hypotheses, there exists a positive constant C such that the bilinear form B(, ), as defined in (2.11), satisfies the following inf-sup condition: inf u X 0,{0} \{0} sup v Y\{0} B(u, v) u X0,{0} v Y C and v Y \ {0} : sup B(u, v) > 0. u X 0,{0} Analogously, there exists a positive constant C such that the bilinear form B (, ) in (2.19) satisfies the following inf-sup condition: inf u Y\{0} sup v X 0,{T } \{0} B (u, v) u Y v X0,{T } C and v X 0,{T } \ {0} : sup B (u, v) > 0. u Y Furthermore, the bilinear forms B(, ) and B (, ) defined in (2.11) and in (2.18), respectively, induce boundedly invertible, linear operators B L(X 0,{0}, Y ) and B L(Y, (X 0,{T } ) ), where the spaces X 0,{0}, X 0,{T } are defined in (2.16) and (2.17), and Y = L 2 (0, T ; V). The proof is analogous to that of Theorem 5.1 in [24]. The following result is now an immediate consequence of Theorem 2.5. Theorem 2.6. For every u 0 H and every f X0,{T } = L2 (I; V ) + H 1 0,{T }(I; V), there exists a unique weak solution u Y = L 2 (0, T ; V) of (2.10) in the sense that u satisfies (2.18). Moreover, the operator L : Y X0,{T } is an isomorphism. In the next section we apply these abstract results to a class of Fokker Planck equations, posed on high-dimensional domains, that arise in bead-spring chain models in kinetic models of dilute polymers (see, for example, [2] and [3]).

7 ADAPTIVE ALGORITHMS IN INFINITE DIMENSIONS 7 3. Fokker Planck equation Suppose that K is a fixed positive integer and d {2, 3}. Suppose further that D k R d is either a bounded ball in R d centred at the origin of R d and of fixed radius b k, where b k > 0, k = 1,..., K, or that D k = R d for all k = 1,..., K. By identifying R d with a ball of radius +, we can unify the two scenarios by taking b k (0, + ] and by assuming that the b k are either finite for all k = 1,..., K or infinite for all k = 1,..., K. We define D := D 1 D K. On the interval [0, b k 2 ) we consider the function U k C 1 [0, b k 2 ), referred to as a potential, such that U k (0) = 0, U k is strictly monotonic increasing and lim s bk U k = +, k = 1,..., K. We then associate with U k the partial Maxwellian, defined by M k (q k ) := 1 Z k exp ( U k ( 1 2 q k 2)), q k D k, where Z k := exp ( ( U 1 k 2 p k 2)) dp k, D k for k = 1,..., K, and we define the (full) Maxwellian M(q) := M 1 (q 1 ) M K (q K ), q = (q 1,..., q K) D 1 D K = D R Kd. Clearly, M(q) > 0 on D, D M(q) dq = 1 and lim q k b k M k (q k ) = 0, k = 1,..., K. When the domains D k are bounded balls, we shall suppose that there exist positive constants C k1 and C k2, and real numbers α k > 1, k = 1,..., K, such that 0 < C k1 exp ( U k ( 1 2 q k 2)) /(dist(q k, D k )) α k C k2 <, k = 1,..., K. Alternatively, when D k = R d for all k = 1,..., K, we shall assume that U k (s) = s, k = 1,..., K. This section is devoted to the proof of existence and uniqueness of weak solutions to the Fokker Planck equation K ( ( )) ψ t ψ A ij qj M qi = 0, (q, t) D (0, T ], (3.23) M i,j,=1 subject to the initial condition ψ(q, 0) = ψ 0 (q), q D, (3.24) where A R K K is a symmetric positive definite matrix with minimal eigenvalue γ 0 > 0 and maximal eigenvalue γ 1 > 0, and ψ 0 L 1 (D) is a nonnegative function such that D ψ 0 dq = 1. As ψ 0 is a probability density function, and this property needs to be propagated during the course of the evolution over the time interval [0, T ], the boundary condition on D [0, T ] needs to be chosen so that ψ(, t) remains a nonnegative function for all t [0, T ], and ψ(q, t) dq = 1 for all D t [0, T ]. This can be achieved, formally at least, by demanding that K i=1 A ij ( M qi ( ψ M )) q j q j 0 as q j 2 b j, for all j = 1,..., K, (3.25) where either b j (0, ) for all j = 1,..., K when D j is a bounded ball of radius b j ; or b j := + for all j = 1,..., K when D j = R d. By writing ψ := ψ M and ψ0 := ψ 0 M, the initial-value problem (3.23), (3.24) can be restated as follows: M t ψ K i,j,=1 subject to the initial condition ( ) A ij qj M qi ψ = 0, (q, t) D (0, T ], (3.26) M(q) ψ(q, 0) = M(q) ψ 0 (q), q D, (3.27)

8 8 CHRISTOPH SCHWAB AND ENDRE SÜLI together with the (formal) boundary condition K ) q j A ij (M qi ψ q j 0 as q j 2 b j, for all j = 1,..., K, (3.28) i=1 and the adopted convention that b j := + when D j = R d. We consider the Maxwellian-weighted L 2 space L 2 M (D) := { ϕ L 1 loc(d) : M ϕ L 2 (D)}, equipped with the inner product (, ) L 2 M (D) and norm L 2 M (D), defined, respectively, by ( ψ, ϕ) L 2 M (D) := M(q) ψ(q) ϕ(q) dq and ϕ L 2 M (D) := ( ϕ, ϕ) 1 2 L 2 (D), M and the associated Maxwellian-weighted H 1 space D H 1 M (D) := { ϕ L 2 M (D) : qk ϕ L 2 M (D), k = 1,..., K} equipped with the inner product (, ) H 1 M (D) and norm H 1 M (D) defined, respectively, by ( ψ, ϕ) H 1 M (D) = ( ψ, ϕ) L 2 M (D) + K ( qk ψ, qk ϕ) [L 2 M (D)] and ϕ d H 1 M (D) := ( ϕ, ϕ) 1 2 H 1 M (D). We note that when D k are bounded open balls, the open set D, as a Cartesian product of bounded Lipschitz domains, is also a bounded Lipschitz domain (cf. the footnote on p.10 of [14]). Under our assumptions on the potentials U k, k = 1,..., K, L 2 M (D) and H1 M (D) are separable Hilbert spaces over the field of real numbers (cf. Theorems 2.7.1, and in [19]). Clearly, H 1 M (D) is continuously embedded in L2 M (D). In fact the embedding H1 M (D) L2 M (D) is dense and compact (in the case when of each D k, k = 1,..., K, is a bounded ball we refer for the proofs of these statements to the Appendix in Barrett & Süli [1], and Appendices B, C, D in the extended version of Barrett & Süli [2]; in the case when D k = R d for each k = 1,..., K, we refer to the Appendices A and D in Barrett & Süli [3]). Adopting the notations introduced in the previous sections, we take H := L 2 M (D), V := H1 M (D), and consider the linear differential operator K A ϕ := A ij qj (M qi ϕ), ϕ V, i,j=1 that maps V into its dual space V (with respect to the pivot space H). We strengthen our original assumption ψ 0 L 1 (D) by demanding that ψ 0 L 2 M (D) (note that ψ 0 L 1 (D) ψ 0 L 2 M (D) = ψ 0 H for all ψ 0 H = L 2 M (D)), and we consider the following weak formulation of the Fokker Planck initial-boundary-value problem: given ψ 0 H := L 2 M (D), find ψ Y := L 2 (0, T ; V) = L 2 (0, T ; H 1 M (D)) such that B ( ψ, ϕ) := T 0 ( V ϕ, ψ V + a( ψ, ϕ)) dt = ( ψ 0, ϕ(0)) H ϕ X 0,{T }. (3.29) Here, as in the previous sections, ϕ signifies the derivative of ϕ with respect to the variable t. On recalling the Brascamp Lieb inequality (cf. [18], Corollary 2.2) in the case of b k <, k = 1,..., K; and the Poincaré inequality for Gaussian measures (cf. the work of Beckner [4], Theorem 1, inequality (3) with p = 1) when b k =, k = 1,..., K, we deduce from Theorem 2.6 the existence of a unique weak solution ψ L 2 (0, T ; H 1 M (D)) to the Fokker Planck equation (3.26), (3.27). By choosing in the weak formulation (3.29) the test function ϕ from the dense linear subspace { ϕ C ([0, T ]; H 1 M (D)) : ϕ(, T ) = 0} of X 0,{T }, it then follows from (3.29) that ψ H 1 (0, T ; H 1 M (D) ), and therefore ψ L 2 (0, T ; H 1 M (D)) H1 (0, T ; H 1 M (D) ). Thus, returning to our original variable ψ, we deduce the existence of a unique weak solution ψ to (3.23), (3.24), with ψ/m L 2 (0, T ; H 1 M (D)) H1 (0, T ; H 1 M (D) ). The boundary condition (3.25) (or, equivalently, (3.28)) is imposed weakly, by demanding that the function ψ (or, equivalently, ψ) belongs to the appropriate function space for weak solutions.

9 ADAPTIVE ALGORITHMS IN INFINITE DIMENSIONS 9 Henceforth we shall concentrate on the simplest case from the technical point of view, when D k = R d and U k (s) = s for all k = 1,..., K; then, M k (q k ) = 1 exp ( 1 Z 2 q k 2), q k R d, where Z k := exp ( 1 2 p k 2) dp k, k = 1,..., K, k R d and M(q) = M 1 (q 1 ) M K (q K ) = 1 Z exp ( 1 2 q 2), q = (q 1,..., q K) R Kd, where q 2 = q q K 2 and Z = K i=1 Z k, k = 1,..., K. We are particularly interested in the case when K 1; specifically, our objective is to develop adaptive numerical algorithms with quantitative optimality properties, which are dimensionally robust, i.e., whose computational complexity does not exhibit the curse of dimensionality. To this end, we shall embed the weak formulation of the Fokker Planck initial-boundary-value problem (3.29), with a finite, but arbitrarily large number of independent variables q k, k = 1,..., K, into a problem with countably many variables, q k, k = 1, 2,..., corresponding to formally setting K = +. Thus, instead of working on D = R Kd, we shall be interested in the case when D = R (cf. the next section for the definition of R ), which we shall equip with a finite measure, the Gaussian measure on R. We shall employ for this purpose the notion of Gaussian measure on a separable Hilbert space. Our exposition in the next section closely follows Sections 1.1, 1.2, 9.1 and 9.2 in the monograph of Da Prato and Zabczyk [12]. 4. Gaussian measures Suppose that H is a separable Hilbert space with norm H and inner product (, ) H over the field of real numbers. We denote by L(H) the space of all bounded linear operators from H into H equipped with the associated operator norm L(H), and we denote by L + (H) the subset of H consisting of all nonnegative symmetric linear operators. Finally, B(H) will signify the σ-algebra of all Borel subsets of H. A bounded linear operator R L(H) is said to be of trace class if there exist two sequences (a k ) and (b k) in H such that Rh = (h, a k ) H b k, h H, (4.1) and a k H b k H <. The set of all elements of L(H) that are trace class will be denoted by L 1 (H). We note that if R L 1 (H), then R is a compact linear operator on H. The set L 1 (H), endowed with the usual operations of addition and scalar multiplication, is a Banach space with the norm { } R L1(H) := inf a k H b k H : Rh = (h, a k ) H b k, h H, (a k ), (b k ) H. Assuming that R L 1 (H), the trace Tr R of R is defined by the formula Tr R = (Re k, e k ) H, where at this stage (e k ) is any complete orthonormal basis in H. In particular if R L 1(H) is expressed by (4.1), then Tr R = (a k, b k ) H. The definition of trace being independent of the choice of the basis, we have that TrR R L1(H). The next proposition (cf. Pietsch [22], Theorem 8.3.3) sharpens these statements further.

10 10 CHRISTOPH SCHWAB AND ENDRE SÜLI Proposition 4.1. Suppose that R is a compact selfadjoint linear operator on a separable Hilbert space H, and that (λ k ) are its eigenvalues (repeated according to their multiplicity). Then, R L 1 (H) if, and only if, λ k < +. Furthermore, R L1(H) = λ k and Tr R = λ k. In particular if λ k 0 for all k N, then Tr R = R L1(H). Let R denote linear space of all sequences x = (x k ) of real numbers, equipped with the metric d(x, y) := 2 k x k y k 1 + x k y k, x, y R. Let further l 2 (N) denote the Hilbert space of all sequences x = (x k ) R such that ( ) 1/2 x l 2 (N) := x k 2 < + ; the inner product on l 2 (N) is defined by (x, y) l2 (N) := x ky k, for x, y l 2 (N). Let us further consider, for a R and for λ > 0, the Gaussian measure on R with mean a and standard deviation λ defined by N a,λ (dx) := 1 e (x a)2 2λ dx. 2πλ The following result is stated as Theorem in [12]. Theorem 4.2. Suppose that a H and Q L + 1 (H). Then, there exists a unique probability measure µ on (H, B(H)) such that e ı(h,x) H µ(dx) = e ı(a,h) H e 1 2 (Qh,h) H, h H. H Moreover, µ is the restriction to H (identified with the Hilbert space l 2 (N)) of the product measure µ k = N ak,λ k, defined on (R, B(R )). We shall write µ = N a,q, and we call a the mean and the trace class operator Q the covariance operator of µ. The measure µ will be referred to as a Gaussian measure on H with mean a and covariance operator Q. If the law of a random variable is a Gaussian measure, then the random variable is said to be Gaussian. In particular, Theorem 4.2 implies that a random variable X with values in H is Gaussian if, and only if, for any h H the real-valued random variable (h, X) H is Gaussian. For µ = N a,q, we denote by L 2 (H, µ) the Hilbert space of all square-integrable (equivalence classes of) functions from H into R with inner product (u, v) L2 (H,µ) = u(x) v(x) µ(dx), u, v L 2 (H, µ) and norm H u L2 (H,µ) = (u, u) 1 2 L2 (H,µ), u L2 (H, µ). Analogously, we shall denote by L 2 (H, µ; H) the Hilbert space of all square-integrable (equivalence classes of) functions from H into H with inner product (u, v) L 2 (H,µ;H) = (u(x), v(x)) H µ(dx), u, v L 2 (H, µ; H) H and norm u L2 (H,µ;H) = (u, u) 1 2 L2 (H,µ;H) u L 2 (H, µ; H). Throughout the rest of the paper, µ = N Q := N 0,Q, where Q L + 1 (H). Moreover, we shall assume that Ker(Q) = {0}. We shall also suppose that there exists a complete orthonormal system (e k ) in H and a sequence (λ k) of positive real numbers, the eigenvalues of Q, such that

11 ADAPTIVE ALGORITHMS IN INFINITE DIMENSIONS 11 Qe k = λ k e k, k N. The eigenvalues λ k are assumed to be enumerated in decreasing order (and repeated according to their multiplicity), and accumulating only at zero. For x H, we shall then write x k = (x, e k ) H, k N. The subspace Q 1/2 (H) = {Q 1/2 h : h H} is called the reproducing kernel of the Gaussian measure N Q := N 0,Q. If Ker(Q) = {0} as has been assumed, then Q 1/2 (H) is dense in H. In fact, if x 0 H is such that (Q 1/2 h, x 0 ) H = 0 for all h H, then Q 1/2 x 0 = 0, and therefore Qx 0 = 0, which implies that x 0 = 0. For Q L + 1 (H) such that Ker(Q) = {0}, we introduce the isomorphism W from H into L 2 (H, N Q ) as follows: for f Q 1/2 (H) let W f L 2 (H; N Q ) be defined by W f (x) = (Q 1/2 f, x) H, x H. Let µ = N Q. We define Hermite polynomials in L 2 (H, µ). Let us consider to this end the set Γ of all mappings γ : n N γ n {0} N, such that γ := γ k < +. Clearly γ Γ if, and only if, γ n = 0 for all, except possibly finitely many, n N. For any γ Γ we define the Hermite polynomial H γ (x) := H γk (W ek (x)), x H, (4.2) where the factors in the product on the right-hand side are defined by ( e ξ2 2 H n (ξ) = ( 1)n n! e ξ2 2 d n dξ n ), ξ R, n {0} N; (4.3) H n is the classical univariate Hermite polynomial of degree n on R. Note that H 0 1, so that for each γ Γ, the countable product H γ (x) contains only finitely many nontrivial factors and is, therefore, well-defined. Moreover, with the Gaussian measure µ = N Q being a countable product measure, we have that γ, γ Γ : (H γ, H γ ) L 2 (H,µ) = δ γ,γ. (4.4) We shall denote by E(H) the linear space spanned by all exponential functions, that is all functions ϕ : x H ϕ(x) R of the form ϕ(x) = e (h,x) H, h H. It follows from Proposition in Da Prato and Zabczyk [12] that E(H) is dense in L 2 (H, µ). On account of the separability of H, L 2 (H, µ) is separable. For any k N we consider the partial derivative in the direction e k (with e k as above), defined by 1 D k ϕ(x) = lim ε 0 ε (ϕ(x + εe k) ϕ(x)), x H, ϕ E(H). Thus, if ϕ(x) = e (h,x) H with h H, then clearly D k ϕ(x) = e (h,x) H h k, where h k = (h, e k ) H. Let Λ 0 denote the linear span of {H γ e k : γ Γ, k N}, with H γ and e k as above and µ = N Q, and define the linear operator D : E(H) L 2 (H, µ) L 2 (H, µ; H) by Dϕ(x) := D k ϕ(x) e k, x H. Thanks to Proposition in Da Prato and Zabczyk [12], D k is a closable linear operator for all k N. If ϕ belongs to the domain of the closure of D k, which we shall still denote by D k, we shall say that D k ϕ belongs to L 2 (H, µ). Analogously, by Proposition in Da Prato and Zabczyk [12], D is a closable linear operator. If ϕ belongs to the domain of the closure of D, which we shall still denote by D, we shall say that Dϕ belongs to L 2 (H, µ; H).

12 12 CHRISTOPH SCHWAB AND ENDRE SÜLI For µ = N Q, let us denote by W 1,2 (H, µ) the linear space of all functions ϕ L 2 (H, µ) such that Dϕ L 2 (H, µ; H), equipped with the inner product (ϕ, ψ) W 1,2 (H,µ) := (ϕ, ψ) L2 (H,µ) + (Dϕ(x), Dψ(x)) H µ(dx) and norm ϕ W 1,2 (H,µ) = (ϕ, ϕ) 1 2 W 1,2 (H,µ). Then, the Sobolev space W1,2 (H, µ) is complete, and is therefore a separable Hilbert space. In particular, for any ϕ L 2 (H, µ), we have that ϕ = γ Γ ϕ γ H γ, where ϕ γ := (ϕ, H γ ) L2 (H,µ). H The next theorem, proved independently by Da Prato, Malliavin and Nualart [11] and Peszat [21] provides an analogous characterization of functions belonging to W 1,2 (H, µ) in terms of the complete orthonormal basis (H γ ) γ Γ. Theorem 4.3. A function ϕ L 2 (H, µ) belongs to W 1,2 (H, µ) if, and only if, γ, λ 1 ϕ γ 2 < +, (4.5) γ Γ where ϕ γ := (ϕ, H γ ) L2 (H,µ), γ, λ 1 := γ kλ 1 k, and (λ k) is the sequence of (positive) eigenvalues (repeated according to their multiplicity) of the covariance operator Q L + 1 (H), Ker(Q) = {0}. Moreover, if (4.5) holds, then ϕ 2 W 1,2 (H,µ) = ϕ 2 L 2 (H,µ) + γ Γ γ, λ 1 ϕ γ 2. Identifying L 2 (H, µ) with its own dual L 2 (H, µ), we obtain ϕ (W 1,2 (H, µ)) γ Γ γ, λ ϕ γ 2 < +. Furthermore, the embedding of W 1,2 (H, µ) into L 2 (H, µ) is compact. For a proof of these statements we refer, for example, to [12, Theorem ]. As E(H) is dense in both L 2 (H, µ) and W 1,2 (H, µ), the embedding of W 1,2 (H, µ) into L 2 (H, µ) is also dense. 5. Fokker Planck equation in countably many dimensions For k N, let us denote by D k the set R d equipped with the Gaussian measure 1 µ k (dq k ) = N ak,σ k (dq k ) = (2π) d/2 [det(σ k )] exp ( 1 1/2 2 (q k a k ) Σ 1 k (q k a k ) ) dq k with mean a k R d and positive definite covariance matrix Σ k R d d. We shall assume henceforth that a k = 0 for all k N and that the covariance operator Q, represented by the bi-infinite blockdiagonal matrix Σ = diag(σ 1, Σ 2,... ), with d d diagonal blocks Σ k, k = 1, 2,..., is trace class. We define so that D := D k q := (q 1, q 2,... ) D, q k D k, k = 1, 2,.... We equip the domain D with the product measure µ := µ k = N 0,Σk.

13 ADAPTIVE ALGORITHMS IN INFINITE DIMENSIONS 13 Let A = (A ij ) i,j=1 R be a symmetric infinite matrix, i.e., A ij = A ji for all i, j N. Suppose further that there exists a real number γ 0 > 0 such that A ij ξ i ξ j γ 0 ξ 2 l 2 (N) for all ξ = (ξ i ) i=1 l 2 (N) (5.1) i,j=1 and a real number γ 1 > 0 such that A ij ξ i η j γ 1 ξ l2 (N) η l2 (N) for all ξ = (ξ i ) i=1 l 2 (N) and η = (η i ) i=1 l 2 (N). (5.2) i,j=1 Example 5.1. As an example of an infinite matrix A that satisfies (5.1) and (5.2), we mention A[ɛ] = tridiag{(ɛ j, 1, ɛ j ) : j = 1, 2,... }, (5.3) where the sequence ɛ = (ɛ j ) j=1 is assumed to satisfy ɛ l (N) < 1/2. Then, the matrix A[ɛ] in (5.3) satisfies (5.1) with γ 0 = 1 2 ɛ l (N) and (5.2) with γ 1 = ɛ l (N). Example 5.2. More generally, we may consider A of block-diagonal form A = diag{a 11, A 22 }, where A 11 R d d is symmetric, positive definite, and A 22 is as in Example 5.1. Then, (5.1) and (5.2) hold with constants 0 < γ 0 γ 1 < depending on the spectrum of A 11 and on d, however. In order to state the space-time variational formulation of the Fokker Planck equation using the notation introduced in Section 2, we select H = L 2 (D; µ) and V = W 1,2 (D, µ), and we write Y = L 2 (0, T ; V). Guided by the abstract framework in Sections 2.3, 2.4, we consider the space X = L 2 (0, T ; V) H 1 (0, T ; V ) (5.4) and its subspaces X 0,{0}, X 0,{T } as in (2.17), equipped with the norm of X, which are closed due to (2.9). With these spaces, the space-time adjoint weak formulation of the infinite-dimensional Fokker Planck equation reads as follows: given ψ 0 H and g X 0,{T }, find ψ Y : B ( ψ, ϕ) = l ( ϕ) ϕ X 0,{T }, (5.5) where the bilinear form B (, ) : Y X 0,{T } R is defined by with B ( ψ, ϕ) := a( ψ, ϕ) := T 0 ( V ϕ, ψ V + a( ψ, ϕ) ) dt, (5.6) A ij ( qi ψ, qj ϕ) [L2 (D,µ)] d, i,j=1 and the linear functional l is defined by l ( ϕ) := ( ψ 0, ϕ(0)) L2 (D,µ). (5.7) We note that ϕ := e t ϕ X 0,{T } if, and only if, ϕ X 0,{T } ; analogously, ψ := e t ψ Y if, and only if, ψ Y. Thus, by selecting ϕ = e t ϕ as test function in (5.5), we deduce that ψ Y is a solution to (5.5) if, and only if, ψ := e t ψ Y solves ψ Y : B ( ψ, ϕ) = l ( ϕ) ϕ X 0,{T }, where the linear functional l is defined as in (5.7) (note that ψ(0) = ψ(0)), and the bilinear form B (, ) : Y X 0,{T } R is defined by (5.6), except that now a( ψ, ϕ) := A ij ( qi ψ, qj ϕ) [L 2 (D,µ)] + ( ψ, ϕ) d L 2 (D,µ). i,j=1

14 14 CHRISTOPH SCHWAB AND ENDRE SÜLI The solution ψ Y to (5.5) is then directly recovered from ψ by setting ψ := e t ψ. We shall therefore assume (without loss of generality) in what follows that, whenever the problem (5.5) is referred to, it is the bilinear form a defined by a( ψ, ϕ) := A ij ( qi ψ, qj ϕ) [L 2 (D,µ)] + ( ψ, ϕ) d L2 (D,µ) (5.8) i,j=1 that is used in the definition of B, rather than the bilinear form a defined above following (5.6). By adopting this simple convention, the well-posedness of the infinite-dimensional Fokker Planck equation is now an immediate consequence of Theorem 2.5; in particular, the following result holds. Theorem 5.3. For the bilinear form B (, ) in (5.6), where a(, ) is defined by (5.8), there exists a positive constant C such that and inf ψ Y\{0} sup ϕ X 0,{T } \{0} ϕ X 0,{T } \ {0} : B ( ψ, ϕ) ψ Y ϕ X0,{T } C (5.9) sup B ( ψ, ϕ) > 0. (5.10) ψ Y Furthermore, for each ψ 0 H and each g X 0,{T }, there exists a unique ψ Y = L 2 (0, T ; V) that satisfies (5.5); and the linear operator B L(Y, X 0,{T } ) induced by B (, ) is an isomorphism. Following [24], we shall next develop a class of adaptive Galerkin discretization algorithms for (5.5), along the lines of adaptive wavelet discretizations of boundedly invertible operator equations considered in [9, 10]. These algorithms exhibit, in particular, stability by adaptivity, i.e., their stability follows directly from the stability (5.9), (5.10) of the continuous, infinite-dimensional problem through suitable Riesz bases of the spaces Y and X 0,{T }, which we shall construct. Notably, in the present context, these algorithms are dimensionally robust by design: as we shall prove, they deliver a sequence of approximate solutions with finitely supported coefficient vectors, i.e., with only finitely many variables q k being active in each iteration. We will establish certain optimality properties for these finitely supported Galerkin solutions, with all constants in the error and complexity bounds being absolute, i.e., independent of the number of active variables. We first develop the necessary concepts in an abstract setting, before applying them to the Fokker Planck equation (5.5) in space-time variational form. 6. Well-posed operator equations as infinite matrix problems Let us denote for a moment by X, Y two generic separable Hilbert spaces over R, and let us assume that we have available a Riesz basis Ψ X = {ψλ X : λ X } for X, meaning that the synthesis operator : l 2 ( X ) X : c c Ψ X := s Ψ X λ X c λ ψ X λ is boundedly invertible. By identifying l 2 ( X ) with its dual, the adjoint of s Ψ X, known as the analysis operator, is s Ψ X : X l 2 ( X ) : g [g(ψ X λ )] λ X. Similarly, let Ψ Y = {ψ Y λ : λ Y} be a Riesz basis for Y, with synthesis operator s Ψ Y and its adjoint s Ψ Y. Here, X and Y are countable sets of (multi-)indices λ. Now let B L(X, Y ) be boundedly invertible; then, also its adjoint B L(Y, X ) is boundedly invertible and, for every f Y, f X the operator equations Bu = f, B u = f admit unique solutions u X, u Y. Writing u = s Ψ X u and u = s Ψ Y u, these operator equations are equivalent to infinite matrix-vector problems Bu = f, B u = f, (6.1)

15 ADAPTIVE ALGORITHMS IN INFINITE DIMENSIONS 15 where the load vectors f = s Ψ f = [f(ψ Y Y λ )] λ Y l 2 ( Y ), f = s Ψ f = [f(ψ X X λ )] λ X l 2 ( X ); and the system matrix B given by B = s Ψ Y Bs Ψ X = [(BψX µ )(ψ Y λ )] λ Y,µ X L(l 2 ( X ), l 2 ( Y )) and with B defined analogously, are boundedly invertible. We consider the associated bilinear forms B(, ) : X Y R : (w, v) (Bw)(v), and introduce the notations B (, ) : Y X R : (v, w) (B v)(w), B = B(Ψ X, Ψ Y ), f = f(ψ Y ), B = B (Ψ Y, Ψ X ), f = f (Ψ X ). With the Riesz constants Λ X Ψ := s X Ψ X c Ψ X X l 2 ( X ) X = sup, 0 c l 2 ( X ) c l2 ( X ) λ X Ψ X := s 1 1 Ψ X X l 2 ( X ) = inf c Ψ X X, 0 c l 2 ( X ) c l2 ( X ) and Λ Y Ψ Y and λ Y Ψ Y defined analogously, we obviously have that B l 2 ( X ) l 2 ( Y ) B X Y Λ X Ψ X ΛY Ψ Y, (6.2) B 1 l2 ( Y ) l 2 ( X ) B 1 Y X λ X Ψ X λ Y Ψ Y. (6.3) 7. Best N-term approximations and approximation classes We say that u N l 2 ( X ) is a best N-term approximation of u l 2 ( X ), if it is the best possible approximation to u in the norm of l 2 ( X ), given a budget of N coefficients. Determining the best N-term approximation u N of u requires searching the infinite vector u, which is not feasible, i.e., actually locating u N may not be practically feasible. Nevertheless, rates of convergence afforded by best N-term approximations serve as a benchmark for concrete numerical schemes. To this end, we collect all u l 2 ( X ) admitting a best N-term approximation converging with rate s > 0 in the approximation class A s (l 2 ( X )) := { v l 2 ( X ) : v A s (l 2 ( X )) < }, where v A s (l 2 ( X )) := sup ε [min{n N 0 : v v N l2 ( X ) ε}] s, ε>0 which consists of all v l 2 ( X ) whose best N-term approximations converge to v with rate s. Generally, best N-term approximations cannot be realized in practice, in particular not in situations where the vector u to be approximated is only defined implicitly as the solution of an infinite matrix-vector problem, such as (6.1). We will now design, following [24], a spacetime adaptive Galerkin discretization method for the infinite-dimensional Fokker Planck equation (5.5) that produces a sequence of approximations to u, which, whenever u A s (l 2 ( X )) for some s > 0, converge to u with this rate s > 0 in computational complexity that is linear with respect to N, the cardinality of the support of the active coefficients in the computed, finitely supported approximation to u. 8. Adaptive Galerkin methods Let s > 0 be such that u A s (l 2 ( X )). In [9] and [10], adaptive wavelet Galerkin methods for solving elliptic operator equations (6.1) were introduced; the methods considered in both papers are iterative methods. To be able to bound their complexity, one needs a suitable bound on the complexity of an approximate matrix-vector product in terms of the prescribed tolerance. We formalize this idea through the notion of s -admissibility. Definition 8.1. The infinite matrices B L(l 2 ( X ), l 2 ( Y )), B L(l 2 ( Y ), l 2 ( X )) are s -admissible if there exist routines APPLY B [w, ε] z, APPLY B [ w, ε] z

16 16 CHRISTOPH SCHWAB AND ENDRE SÜLI that yield, for any prescribed tolerance ε > 0 and any finitely supported w l 2 ( X ) and w l 2 ( Y ), finitely supported vectors z l 2 ( Y ) and z l 2 ( X ) such that Bw z l 2 ( Y ) + B w z l 2 ( X ) ε, and for which, for any s (0, s ), there exists an admissibility constant a B, s such that #supp z a B, s ε 1/ s w 1/ s A s (l2 ( X )), and likewise for z, w. The number of arithmetic operations and storage locations used by the call APPLY B [w, ε] is bounded by some absolute multiple of a B, s ε 1/ s w 1/ s A s (l2 ( X )) + #supp w + 1. The design of APPLY B [w, ε] and APPLY B [ w, ε] for the system matrices B and B arising from the infinite-dimensional Fokker Planck equation (5.5) is the major technical building block in the adaptive Galerkin discretization of (5.5). In order to approximate u one should be able to approximate f. To this end, we shall assume in what follows the availability of the following routine. RHS f [ε] f ε : For a given ε > 0, the routine yields a finitely supported f ε l 2 ( X ) with f f ε l2 ( X ) ε and #supp f ε min{n : f f N ε}, with the number of arithmetic operations and storage locations used by the call RHS f [ε] being bounded by some absolute multiple of #supp f ε + 1. The availability of the routines APPLY B and RHS f has the following implications. Proposition 8.2. Let B in (6.1) be s -admissible. Then, for any s (0, s ), we have that For z ε := APPLY B [w, ε], we have that Analogous statements hold for B in (6.1). B A s (l 2 ( X )) A s (l2 ( Y )) a s B, s. z ε A s (l 2 ( Y )) a s B, s w A s (l 2 ( X )). A proof of Proposition 8.2 can be given along the lines of the arguments presented in [9, 10]. Using the definition of A s (l 2 ( Y )) and the properties of RHS f, we have the following corollary. Corollary 8.3. Suppose that, in (6.1), B is s -admissible and u A s (l 2 ( Y )) for s < s ; then, for f ε = RHS f [ε], #supp f ε a B,s ε 1/s u 1/s A s (l2 ( Y )), with the number of arithmetic operations and storage locations used by the call RHS f [ε] being bounded by some absolute multiple of a B,s ε 1/s u 1/s A s (l2 ( Y )) + 1. Remark 8.4. Besides f fε l 2 ( Y ) ε, the complexity bounds in Corollary 8.3, with a B,s signifying a constant that depends only on B and s, are essential for the use of RHS f in the adaptive Galerkin methods. The following corollary of Proposition 8.2 from [24] can be used for example for the construction of valid APPLY and RHS routines in case the adaptive Galerkin algorithms are applied to a preconditioned system. Corollary 8.5. Suppose that B L(l 2 ( Y ), l 2 ( X )), C L(l 2 ( X ), l 2 ( Z )) are both s - admissible; then, so is CB L(l 2 ( Y ), l 2 ( Z )). A valid routine APPLY CB is [w, ε] APPLY C [ APPLYB [w, ε/(2 C )], ε/2 ], (8.1) with admissibility constant a CB, s a B, s( C 1/ s + a C, s ) for s (0, s ).

17 ADAPTIVE ALGORITHMS IN INFINITE DIMENSIONS 17 For some s > s, let C L(l 2 ( Y ), l 2 ( Z )) be s -admissible. Then, for we have that RHS Cf [ε] := APPLY C [RHS f [ε/(2 C )], ε/2], (8.2) #supp RHS Cf [ε] a B,s( C 1/s + a C,s ) ε 1/s u 1/s A s (l2 ( Y )) and Cf RHS Cf [ε] l 2 ( Z ) ε, with the number of arithmetic operations and storage locations used by the call RHS Cf [ε] being bounded by some absolute multiple of a B,s( C 1/s + a C,s ) ε 1/s u 1/s A s (l2 ( X )) + 1. Remark 8.6. The properties of RHS Cf stated in Corollary 8.5 show that RHS Cf is a valid routine for approximating Cf in the sense of Remark 8.4. In the special case when B is symmetric positive definite, i.e., X = Y and B = B > 0, the two Galerkin methods considered in the papers [9, 10] were shown to be quasi-optimal in the following sense. Theorem 8.7. Suppose that, in (6.1), B is s -admissible; then, for any ε > 0, the two adaptive Galerkin methods from [9, 10] produce approximations u ε to u with u u ε l 2 ( Y ) ε. Suppose that, in (6.1), for some s > 0 we have u A s (l 2 ( Y )); then #supp u ε ε 1/s u 1/s A s (l2 ( Y )) and if, moreover, s < s, then the number of arithmetic operations and storage locations required by a call of either of these adaptive solvers with tolerance ε > 0 is bounded by some multiple of ε 1/s (1 + a B,s ) u 1/s A s (l2 ( Y )) + 1. The multiples depend on s only when s tends to 0 or, and on B and B 1 when they tend to infinity. The method from [10] consists of the application of a damped Richardson iteration to Bu = f, where the required residual computations are approximated using calls of APPLY B and RHS f within tolerances that decrease linearly with the iteration counter. The method from [9] produces a sequence Ξ 0 Ξ 1 X, together with a corresponding sequence of (approximate) finitely supported Galerkin solutions u i l 2 (Ξ i ). The coefficients of approximate residuals f Bu i are used as indicators how to expand Ξ i to Ξ i+1 so that the latter gives rise to an improved Galerkin approximation. Both methods rely on a recurrent coarsening of the approximation vectors, where small coefficients are removed in order to keep an optimal balance between accuracy and support length; in [15], a modification of the algorithm introduced in [9] was proposed, which does not require the coarsening step. The key to why s -admissibility of B can be expected is the observation that for a large class of operators the stiffness matrix with respect to suitable wavelet bases is close to a computable sparse matrix, in the sense of the following definition. Definition 8.8. B L(l 2 ( X ), l 2 ( Y )) is s -computable if, for each N N, there exists a B [N] L(l 2 ( X ), l 2 ( Y )) having in each column at most N nonzero entries whose joint computation takes an absolute multiple of N operations, such that the computability constants c B, s := sup N B B [N] 1/ s l 2 ( X ) l 2 ( Y ) (8.3) N N are finite for any s (0, s ). The notion of s -computability of B is defined analogously. Theorem 8.9. An s -computable B is s -admissible. Moreover, for s < s, a B, s c B, s where the constant in this estimate depends only on s 0, s s, and on B. This theorem is proved by constructing a suitable APPLY B routine as in [9, 6.4] (a log factor in the complexity estimate there due to sorting was removed later through the use of approximate sorting, see [13] and the references there).

18 18 CHRISTOPH SCHWAB AND ENDRE SÜLI Remark Theorem 8.7 requires that B is s -admissible for an s > s when u A s (l 2 ( X )). Generally this value of s is unknown, and so the condition on s should be interpreted in the sense that s has to be larger than any s for which membership of the solution u in A s (l 2 ( X )) can be expected. The approach from [10] applies also to the saddle point variational principle (5.5) whenever one has a linearly convergent stationary iterative scheme available for the matrix-vector problem B u = f. There is, unfortunately, no such scheme available for a general boundedly invertible B. In particular, for the stiffness matrices B resulting from the space-time saddle-point formulation (5.5) of the infinite-dimensional Fokker Planck equation no directly applicable scheme is available. A remedy proposed in [10] is to apply adaptive Galerkin discretizations to the normal equation BB u = Bf. (8.4) By Theorem 5.3, the operator BB L(l 2 ( Y ), l 2 ( Y )) is boundedly invertible, symmetric positive definite, with BB l2 ( Y ) l 2 ( Y ) B 2 l 2 ( Y ) l 2 ( X ), (BB ) 1 l2 ( Y ) l 2 ( Y ) (B ) 1 2 l 2 ( X ) l 2 ( Y ). Now let u A s (l 2 ( X )), and for some s > s, let B and B be s -admissible. By Corollary 8.5, with B in place of C, a valid RHS Bf routine is given by (8.2), and BB is s -admissible with a valid APPLY BB routine given by (8.1). In the context of (5.5), one execution of APPLY BB corresponds to one (approximate) sweep over the primal problem, followed by one (approximate) sweep over the dual problem, respectively. A combination of Theorem 8.7 and Corollary 8.5 yields the following result obtained in [24]. Theorem For any ε > 0, the adaptive wavelet methods from [10] or from [9] and [15] applied to the normal equations (8.4) using the above APPLY BB and RHS Bf routines produce an approximation u ε to u with u u ε l 2 ( Y ) ε. Suppose that for some s > 0, u A s (l 2 ( Y )); then #supp u ε ε 1/s u 1/s A s (l2 ( Y )), with the constant in this bound only being dependent on s when it tends to 0 or, and on B and (B ) 1 when they tend to infinity. Suppose that s < s ; then, the number of arithmetic operations and storage locations required by a call of either of these adaptive wavelet methods with tolerance ε is bounded by some multiple of 1 + ε 1/s (1 + a B,s(1 + a B,s )) u 1/s A s (l2 ( Y )), where this multiple only depends on s when it tends to 0 or, and on B and (B ) 1 when they tend to infinity. 9. Infinite-dimensional Fokker Planck equation as an infinite matrix vector equation We apply the foregoing abstract concepts to the space-time variational formulation (5.5) of the infinite-dimensional Fokker Planck equation. To construct Riesz bases for X 0,{T } and Y, we use that X 0,{T } = (L 2 (0, T ) V) (H 1 0,{T } (0, T ) V ) and Y = L 2 (0, T ) V with the spaces X 0,{T } as defined in (5.4), and Y = L 2 (0, T ; V); recall that H = L 2 (D, µ), V = W 1,2 (D, µ). Let ( ) qk Υ = {H γ : γ Γ} V, with H γ (q) = H γk, λk denote the polynomial chaos basis (4.2). By Theorem 4.3, Υ is a normalized Riesz basis for [L 2 (H, µ)] d, which, when renormalized by the scaling sequence ( γ, λ 1 ) γ Γ in V and by the scaling sequence ( γ, λ ) γ Γ in V, is a Riesz basis for V and V, respectively. Let Θ = {θ λ : λ t } H 1 0,{T } (0, T )