ACCELERATED SPATIAL APPROXIMATIONS FOR TIME ...

c 2012 Society for Industrial and Applied Mathematics 

SIAM J. MATH. ANAL. Vol. 44, No. 5, pp. 3162–3185

ACCELERATED SPATIAL APPROXIMATIONS FOR TIME DISCRETIZED STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS∗ ERIC JOSEPH HALL† Abstract. The present article investigates the convergence of a class of space-time discretization schemes for the Cauchy problem for linear parabolic stochastic partial differential equations defined on the whole space. Sufficient conditions are given for accelerating the convergence of the scheme with respect to the spatial approximation to higher order accuracy by an application of Richardson’s method. This work extends the results of Gy¨ ongy and Krylov [SIAM J. Math. Anal., 42 (2010), pp. 2275–2296] to schemes that discretize in time as well as space. Key words. Richardson’s method, finite differences, linear stochastic partial differential equations of parabolic type, Cauchy problem AMS subject classifications. 65M06, 60H15, 65B05 DOI. 10.1137/12086412X

1. Introduction. For integers d, d1 ≥ 1 and a constant T ∈ (0, ∞), given a 1 of independent Wiener processes carried by a complete stochassequence (wρ )dρ=1 tic basis (Ω, F , F (t), P ) we consider the Cauchy problem for the stochastic partial differential equation (SPDE) (1.1)

du(t, x) = (Lu(t, x) + f (t, x)) dt +

d1 

(Mρ u(t, x) + g ρ (t, x)) dwρ (t)

ρ=1

on HT := [0, T ] × Rd with given initial condition u(0, x) = u0 (x), where L and Mρ are second order and first order differential operators, respectively, satisfying a strong stochastic parabolicity condition (c.f. Assumption 2.2) and f and g ρ are given random fields on HT . Second order linear SPDEs of parabolic type such as (1.1) arise in the nonlinear filtering of partially observable diffusion processes as the Zakai equation [13, 16, 23, 1]. In such applications, solutions are desired in real-time. Thus there is a keen interest in providing accurate numerical schemes for approximating the solutions to (1.1). The focus of the current paper lies in accelerating the rate of convergence for a class of space-time finite difference approximations to the Cauchy problem for (1.1). Therefore, we introduce the grids for our discretizations and then state our discretization schemes. For fixed τ ∈ (0, 1) we define the time grid Tτ := {ti = iτ ; i ∈ {0, 1, . . . , n}, τ n = T } partitioning [0, T ] with mesh size τ . For functions φ(t) defined for t ∈ [0, T ] we will often write φi := φ(ti ) for ti ∈ Tτ . In particular, let ξiρ = Δwρ (ti−1 ) := wρ (ti ) − wρ (ti−1 ) ∗ Received by the editors January 30, 2012; accepted for publication (in revised form) June 18, 2012; published electronically September 11, 2012. http://www.siam.org/journals/sima/44-5/86412.html † School of Mathematics, University of Edinburgh, King’s Buildings, Edinburgh, EH9 3JZ, Scotland, UK ([email protected]).

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for i ∈ {1, . . . , n}. For h ∈ R \ {0} we define the space grid Gh := {λ1 h + · · · + λp h; p ∈ {1, 2, . . . }, λi ∈ Λ ∪ (−Λ)} with mesh size |h| for a finite subset Λ ⊂ Rd containing the origin. We define the first order spatial difference by δh,λ φ(x) :=

φ(x + hλ) − φ(x) h

for λ ∈ Rd \ {0} and let δh,0 be the identity. We note that from this definition one can obtain both the so-called forward and backward differences as h can be positive or negative. We consider the space-time discretization scheme (1.2)

d1      h,ρ h ρ h Mi−1 (x) + Lhi vih (x) + fi (x) τ + vi−1 (x) + gi−1 (x) ξiρ vih (x) = vi−1 ρ=1

for i ∈ {1, . . . , n} and x ∈ Gh with initial condition v0h = u0 (x). For each i ∈ {0, . . . , n}, the Lhi and Mih,ρ are difference operators given by Lhi φ := aλμ i δh,λ δ−h,μ φ and Mih,ρ φ := bλρ δ φ, for ρ ∈ {1, . . . , d }, where the summation convention is used h,λ 1 i λμ = a (x) are realwith respect to the repeated indices λ, μ ∈ Λ. We assume that aλμ i i λρ d1 λ d1 valued and bi = (bi (x))ρ=1 are R -valued Fi ⊗ B-measurable functions on Ω × Rd for every i ∈ {0, . . . , n} for all λ, μ ∈ Λ, where B = B(Rd ) denotes the σ-algebra of Borel subsets of Rd . We also assume that aλμ = aμλ for all i ∈ {0, . . . , n} and i i λ, μ ∈ Λ. Under certain compatibility assumptions, (1.2) represents an implicit spacetime finite difference method for approximating the solution to the Cauchy problem for (1.1) by replacing the differential operators with finite differences and by carrying out an implicit Euler method in time. Together with (1.2), we consider the time discretization scheme (1.3)

vi (x) = vi−1 (x) + (Li vi (x) + fi (x)) τ +

d1  

 ρ (x) ξiρ Mρi−1 vi−1 (x) + gi−1

ρ=1

for i ∈ {1, . . . , n} and x ∈ R with initial condition v0 = u0 (x). The Li = L(ti ) and Mρi = Mρ (ti ) are second order and first order differential operators given by L(t) := aαβ (t)Dα Dβ and Mρ (t) := bαρ (t)Dα , respectively, where the summation convention is used with respect to the repeated indices α, β ∈ {0, 1, . . . , d}. Here we use the notation Dα = ∂/∂xα for α ∈ {1, . . . , d} and let D0 be the identity. For each α, β ∈ {0, 1, . . . , d} we assume that aαβ (t) = aαβ (t, x) are real-valued and 1 bα (t) = (bαρ (t, x))dρ=1 are Rd1 -valued P ⊗ B-measurable functions on Ω × [0, T ] × Rd , where P denotes the σ-algebra of predictable subsets of Ω × [0, ∞) generated by F (t). Further, we assume that aαβ (t) = aβα (t) for all t ∈ [0, T ] and α, β ∈ {0, . . . , d}. Scheme (1.3) represents an implicit Euler method for approximating the solution to the Cauchy problem for (1.1) in time. Our aim is to show that the strong convergence of the spatial discretization for the solution to the space-time scheme (1.2) to the solution of the Cauchy problem for (1.1) can be accelerated to any order of accuracy with respect to the computational effort. In general, the error of finite difference approximations in the space variable for such equations is proportional to the mesh size |h|; for example, see [21, 22]. We show the strong convergence of the solution of the space-time scheme (1.2) to the solution of d

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the time scheme (1.3) can be accelerated to higher order accuracy by taking suitable mixtures of approximations using different mesh sizes. This technique for obtaining higher order convergence, often referred to as Richardson’s method after L. F. Richardson, who used the idea to accelerate the convergence of finite difference schemes to deterministic partial differential equations (PDEs) (see [17, 18]), falls under a broadly applicable category of extrapolation techniques; for instance, see the survey articles [2, 9]. In particular, in [20, 14, 10] Richardson’s method is implemented to accelerate the weak convergence of Euler approximations for stochastic differential equations. Recently, in [4] Gy¨ ongy and Krylov considered a semidiscrete scheme for solving (1.1) which discretized via finite differences in the space variable, while allowing the scheme to vary continuously in time, and showed that the strong convergence of the spatial approximation can be accelerated by Richardson’s method. The current paper extends these results to the implicit space-time scheme (1.2). We must mention that for the present scheme one cannot also accelerate in time unless certain commutators of the differential operator Mρ in (1.1) vanish; see [3]. For deterministic PDEs we plan to address the simultaneous acceleration of the convergence of approximations with respect to space and time in a future paper. Results concerning acceleration for monotone finite difference schemes for degenerate parabolic and elliptic PDEs are given in [5]; however, our scheme is not necessarily monotone. In the next section, we present our assumptions as well as some preliminaries. Then in section 3 we record the main results, namely, Theorems 3.1, 3.2, and 3.3, the last of which says that the convergence of the spatial approximation can be accelerated to any order of accuracy. In section 4 we provide results which will be needed for the proofs of Theorems 3.1 and 3.2. In particular, we recall the solvability of the spacetime scheme (1.2), for the convenience of the reader and present a new contribution— an estimate for the supremum of the solution to the scheme in appropriate spaces that is independent of h. In section 5 we give the proof of a more general result and show that it implies Theorem 3.2 and hence Theorem 3.1. We end with some notation that will be used throughout this work. For  2 2 |φ(x)| |φ|2 (Gh ) := |h|d x∈Gh

let 2 (Gh ) := {φ ∈ Gh ; |φ|22 (Gh ) < ∞}. We will use the notation Dl φ for the collection of all lth order spatial derivatives of φ. For a nonnegative integer m, let W2m = W2m (Rd ) be the usual Hilbert–Sobolev space of functions on Rd with norm  · m . We note that for L2 = L2 (Rd ) = W20 we will use (·, ·) to denote the usual inner product in that space and we will denote the norm by  · 0 . Let W2m (T ) := L2 (Ω × [0, T ], P, W2m) denote the space of W2m -valued square integrable predictable processes on Ω × [0, T ]. These are the natural spaces in which to seek solutions to (1.1). For basic results and notation concerning the theory of linear SPDEs, see [19]. 2. Preliminaries and assumptions. We begin by setting some assumptions on our operators and recalling well-known results concerning the solvability and rates of convergence for our schemes. In particular, we will discuss an 2 (Gh ) notion of solution and an L2 notion of solution and recall an important lemma relating these function spaces.

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An L2 -valued continuous process u = (u(t))t∈[0,T ] is called a generalized solution to (1.1) if u ∈ W21 for almost every (ω, t) ∈ Ω × [0, T ],  T 2 u(t)1 dt < ∞ 0

almost surely, and



(u(t), φ) = (u0 , φ) +

t

0

 00    a u(s) + aβ Dβ u(s) + f (s), φ − aαβ Dβ u(s), Dα φ ds

d1  t   0ρ  b u(s) + bβρ Dβ u(s) + g ρ (s), φ dwρ (s) + ρ=1

0

holds for all t ∈ [0, T ] and φ ∈ C0∞ (Rd ), where aβ := a0β +aβ0 −Dα aαβ and where the summation convention is used with respect to the repeated indices α, β ∈ {1, . . . , d}. Assumption 2.1. For each (ω, t) ∈ Ω × [0, T ] the functions aαβ are m times and the functions bα are m + 1 times continuously differentiable in x. Moreover there exist constants K0 , . . . , Km+1 such that for l ≤ m  l αβ  D a  ≤ K l and for l ≤ m + 1

 l α D b  ≤ Kl

for all values of α, β ∈ {0, . . . , d} and (ω, t, x) ∈ Ω × HT . Assumption 2.2. There exists a positive constant κ such that d   αβ  2a − bαρ bβρ zα zβ ≥ κ|z|2 α,β=1

for all (ω, t, x) ∈ Ω × HT , z ∈ Rd , and ρ ∈ {1, . . . , d1 }. Assumption 2.3. The initial condition u0 ∈ L2 (Ω, F0 , W2m+1 ), the space of m+1 W2 -valued square integrable F0 -measurable functions on Ω. The f and g ρ , for each ρ ∈ {1, . . . , d1 }, are predictable processes on Ω × [0, T ] taking values in W2m and W2m+1 , respectively. Moreover  T  2 2 2 f (t)m + g(t)m+1 dt < ∞, E u0 m+1 + E 2 g(t)l

d1

0

2 (t)l .

where := ρ=1 g Under Assumptions 2.1, 2.2, and 2.3, the existence of a unique generalized solution u ∈ W2m+2 (T ) to (1.1) is a classical result (see, for example, [15, 12] or Theorem 5.1 from [11]). Remark 2.4. We note that by Sobolev’s embedding of W2m ⊂ Cb , the space of bounded continuous functions, for m > d/2 we can find a continuous function of x which is equal to u0 almost everywhere for almost all ω ∈ Ω. Likewise, for each (ω, t) ∈ Ω × [0, T ] there exists continuous functions of x which coincide with f (t) and g ρ (t) for almost every x ∈ Rd . Thus, if Assumption 2.3 holds with m > d/2 we assume that u0 , f (t), and g ρ (t) are continuous in x for all t ∈ [0, T ]. ¯ := m ∨ 1. We place the Let Λ0 := Λ \ {0} and for a nonnegative integer m let m following additional requirements on our space-time scheme. Assumption 2.5. For all ω ∈ Ω, for i ∈ {0, . . . , n}, for λ, μ ∈ Λ0 , and for ν ∈ Λ: ¯ times continuously differentiable in x; the a0ν and aν0 are m times the aλμ are m ρ

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ERIC JOSEPH HALL

continuously differentiable in x; and the bν are m times continuously differentiable in ¯ we x. Moreover there exist constants A0 , . . . , Am ¯ such that for λ, μ ∈ Λ0 , and l ≤ m have  l λμ  D a  ≤ Al and for λ ∈ Λ and l ≤ m we have  l λ0      D a  ≤ Al , Dl a0λ  ≤ Al , and Dl bλ  ≤ Al for all (ω, x) ∈ Ω × Rd for i ∈ {0, . . . , n}. Assumption 2.6. There exists a positive constant κ such that     2aλμ − bλρ bμρ zλ zμ ≥ κ zλ2 λ,μ∈Λ0

λ∈Λ0

for all (ω, x) ∈ Ω × Rd , i ∈ {0, . . . , n}, ρ ∈ {1, . . . , d1 }, and numbers zλ , for λ ∈ Λ0 . For (1.2) to be consistent with (1.1) we also require the following. Assumption 2.7. For i ∈ {0, . . . , n}  λ∈Λ0



α aλ0 i λ +

00 a00 i = ai , α α0 0α a0μ i μ = ai + ai ,

μ∈Λ0



αβ α β aλμ i λ μ = ai ,

λ,μ∈Λ0 0ρ b0ρ i = bi ,

and 

αρ α bλρ i λ = bi

λ∈Λ0

for all α, β ∈ {1, . . . , d} and ρ ∈ {1, . . . , d1 }. Remark 2.8. If Λ0 is a basis for Rd and Assumption 2.7 holds, then Assumption 2.1 implies 2.5 and 2.2 implies 2.6 with m = m. A solution v h = (vih )ni=1 to (1.2) with an F0 -measurable 2 (Gh )-valued initial condition v0h is understood as a sequence of 2 (Gh )-valued random variables satisfying (1.2) on the grid Gh . The following result is well known and we provide it for completeness. Theorem 2.9. Let f and g ρ be Fi -adapted 2 (Gh )-valued processes and let v0h be an F0 -measurable 2 (Gh )-valued initial condition. If Assumption 2.5 holds, then (1.2) admits a unique 2 (Gh )-valued solution for sufficiently small τ . Proof. By Assumption 2.5, for each i ∈ {1, . . . , n}, (1.2) is a recursion with bounded linear operators on 2 (Gh ). In particular, for each h the operator norm of τ Lh is smaller than a constant less than 1 for sufficiently small τ independently of ω ∈ Ω. Hence (I − τ Lh ) is invertible in 2 (Gh ) for sufficiently small τ by the invertibility of operators in a neighborhood of the (invertible) identity operator I. Therefore, for i ∈ {1, . . . , n} we are guaranteed an 2 (Gh )-valued φ satisfying (I − τ Lhi )φ = ψ for all ψ ∈ 2 (Gh ) and moreover this solution is easily seen to be unique. Thus we can construct a unique solution to the scheme iteratively.

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The rate of convergence of the solution v h of (1.2) (and v of (1.3)) to the solution u of (1.1) with initial condition u0 is known. In [6, 7, 8], Gy¨ongy and Millet obtained the rate of convergence for a class of equations in the nonlinear setting of which our schemes are a special case. Namely, in the situation of Remark 2.8, if Assumptions 2.1, 2.2, and 2.3 hold with aαβ , bα , f , and g ρ all H¨older continuous in time with exponent 1/2, then      δh,λ vih (x) − ui (x) 2 hd E max i≤n

|λ|≤m+1 x∈Gh

+ Eτ

n 



      δh,λ vih (x) − ui (x) 2 hd ≤ N h2 + τ

i=1 |λ|≤m+2 x∈Gh

for sufficiently small τ , h ∈ (0, 1) and for a constant N that is independent of h and τ . The principal interest of this paper is to investigate higher order convergence with respect to the spatial discretization, that is, to obtain an estimate, similar to the above, with a higher power of h by applying Richardson’s method. While it is natural to seek solutions to (1.2) on the grid, carrying out our analysis on the whole space will have certain advantages when it comes to providing estimates for solutions to our schemes. Indeed, we observe that (1.2) is well defined not only on Gh but for all x ∈ Rd . Therefore, we introduce an alternate notion of solution. A solution to (1.2) on Ω × Tτ × Rd with an F0 -measurable L2 -valued initial condition v0h is a sequence v h = (vih )ni=1 of L2 -valued random variables satisfying (1.2). In a similar spirit, solutions to (1.3) with the appropriate initial condition are understood as sequences of W21 -valued random variables satisfying (1.3) in W2−1 . The next result follows immediately from the considerations in the proof of Theorem 2.9. Theorem 2.10. Let f and g ρ be Fi -adapted L2 -valued processes and let v0h be an F0 -measurable L2 -valued initial condition. If Assumption 2.5 holds, then (1.2) admits a unique L2 -valued solution for sufficiently small τ . By Sobolev’s embedding theorem, for l > d/2 there exists a linear operator I : W2l → Cb such that φ(x) = Iφ(x) for almost every x ∈ Rd and supx∈Rd |Iφ(x)| ≤ N φl for all φ ∈ W2l , where N is a constant. We recall the following useful embedding of W2l ⊆ 2 (Gh ) from [4]. Lemma 2.11. Let l > d/2 and |h| ∈ (0, 1). For all φ ∈ W2l the embedding  2 d 2 (2.1) |Iφ(x)| |h| ≤ N φl x∈Gh

holds for a constant N that depends only on d and l. Proof. For z ∈ Rd let Br (x) := {x ∈ Rd ; |x − z| < r}. By the embedding of W2l into Cb , for φ ∈ Cb we have 2

|φ(z)| ≤

sup φ2 (z + hx) x∈B1 (0)

≤N



|h|2|α|



|α|≤l

≤N



|α|≤l

≤ N |h|−d

|h|2|α|−d  |α|≤l

2

B1 (0)

|(Dα φ)(z + hx)| dx



2

Bh (z)

Bh (z)

|(Dα φ)(x)| dx 2

|(Dα φ)(x)| dx

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ERIC JOSEPH HALL

for a constant N depending only on d and l and thus     2 2 |φ|2 (Gh ) = |φ(z)| |h|d ≤ N |α|≤l z∈Gh

z∈Gh

Bh (z)

2

|(Dα φ)(x)| dx,

which yields the desired embedding. We will show that the restriction of a continuous modification of an L2 -valued solution to (1.2) to the grid Gh is also a solution in the 2 (Gh ) sense. Thus we will carry out our analysis in the whole space and obtain estimates independent of h in appropriate Sobolev spaces for the L2 -valued solutions of (1.2) and (1.3). We provide the aforementioned Sobolev space estimates in section 4. We then use these estimates in section 5 to prove the main results, which are the focus of the next section. 3. Main results. To accelerate the convergence of the spatial approximation by Richardson’s method we must have an expansion for the solution v h to (1.2) with initial data v0h = u0 in powers of the discretization parameter h. This relies on the possibility of proving the existence of sequences of random fields v (0) (x), v (1) (x), . . . , v (k) (x), for x ∈ Rd and integer k ≥ 0, satisfying certain properties. Namely, v (0) , . . . , v (k) are independent of h; v (0) is the solution of (1.3) with initial condition u0 ; and the expansion (3.1)

vih (x) =

k  hj j=0

j!

(j)

vi (x) + Riτ,h (x)

holds almost surely for i ∈ {1, . . . , n} and x ∈ Gh , where Rτ,h is an 2 (Gh )-valued adapted process such that  2   2 (3.2) E max sup Riτ,h (x) ≤ N h2(k+1) Km i≤n x∈Gh

for 2

2 Km := E u0 m+1 + Eτ

n  

 2 2 fi m + gi m+1 < ∞

i=0

and a constant N independent of τ and h. Our first result concerns the existence of such an expansion. Theorem 3.1. If Assumptions 2.1, 2.2, 2.3, 2.5, 2.6, and 2.7 hold with m=m>k+1+

d 2

for an integer k ≥ 0, then expansion (3.1) and estimate (3.2) hold for a constant N depending only on d, d1 , Λ, m, K0 , . . . , Km+1 , A0 , . . . , Am , κ, and T . In the proof of Theorem 3.1, as v h is defined not only on Gh but for all x ∈ Rd , we will see that one can replace Gh in (3.2) with Rd . We also note that in the situation of Remark 2.8, if Assumptions 2.1 and 2.2 hold with m > k + 1 + d/2, then the conditions of Theorem 3.1 are satisfied. Taking differences of expansion (3.1) clearly yields δh,λ vih (x) =

k  hj j=0

j!

(j)

δh,λ vi (x) + δh,λ Riτ,h (x)

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for any λ = (λ1 , . . . , λp ) ∈ Λp , for integer p ≥ 0, where Λ0 := {0} and δh,λ := δh,λ1 × · · · × δh,λp . The bound on δh,λ Rτ,h is not obvious; nevertheless we have the following generalization of the above theorem. Theorem 3.2. If the assumptions of Theorem 3.1 hold with m=m>k+p+1+

d 2

for a nonnegative integer p, then for λ ∈ Λp expansion (3.1) and 2 2       τ,h 2 E max sup δh,λ Riτ,h (x) + E max |h|d δh,λ Ri (x) ≤ N h2(k+1) Km i≤n x∈Gh

i≤n

x∈Gh

hold for a constant N depending only on d, d1 , Λ, m, K0 , . . . , Km+1 , A0 , . . . , Am , κ, and T . The proof of Theorems 3.1 and 3.2 appear in section 5 following the considerations in the next section. Currently we shall discuss how to implement Richardson’s method to obtain higher order convergence in the spatial approximation, extending the result from [4] to the space-time scheme. Fix an integer k ≥ 0 and let v¯h :=

(3.3)

k 

−j

βj v 2

h

,

j=0 −j

where v 2 h solves, with 2−j h in place of h, the space-time scheme (1.2) with initial condition u0 . Here β is given by (β0 , β1 , . . . , βk ) := (1, 0, . . . , 0)V −1 , where V −1 is the inverse of the Vandermonde matrix with entries V ij := 2−(i−1)(j−1) for i, j ∈ {1, . . . , k + 1}. Recall that v (0) is the solution to (1.3) with initial condition u0 . Theorem 3.3. Under the assumptions of Theorem 3.1, 2    h (0) 2 (3.4) E max sup ¯ vi (x) − vi (x) ≤ N h2(k+1) Km i≤n x∈Gh

for a constant N depending only on d, d1 , Λ, m, K0 , . . . , Km+1 , A0 , . . . , Am , κ, and T . Proof. By Theorem 3.1 we have the expansion −j

v2

h

= v (0) +

k  −j hi (i) v + rˆτ,2 h hk+1 ij i!2 i=1

−j

−j

for each j ∈ {0, 1, . . . , k}, where rˆτ,2 h := h−(k+1) Rτ,2 h . Then ⎞ ⎛ k k  k k    −j hi v¯h = ⎝ βj ⎠ v (0) + βj ij v (i) + βj rˆτ,2 h hk+1 i!2 j=0 j=0 i=1 j=0 = v (0) +

k  hi i=1

=v

(0)

+

k  j=0

i!

v (i)

k k   −j βj + βj rˆτ,2 h hk+1 ij 2 j=0 j=0

−j

βj rˆτ,2

h k+1

h

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ERIC JOSEPH HALL

k k since j=0 βj = 1 and j=0 βj 2−ij = 0 for each i ∈ {1, 2, . . . , k} by the definition of (β0 , . . . , βk ). Now using the bound on Rτ,h from Theorem 3.1 together with this last calculation yields the desired result. One can also construct rapidly converging approximations of derivatives of v (0) . That is, if the conditions of Theorem 3.1 hold instead with m=m>k+p+1+

d 2

for nonnegative integers k and p, then Theorem 3.3 holds with δh,λ v¯h and δh,λ v (0) in place of v¯h and v (0) , respectively, for λ ∈ Λp . Therefore, using suitable linear combinations of finite differences of v¯h one can construct rapidly converging approximations for the derivatives of v (0) . In the next section, we present material that will be used to prove the main results in this section. In particular, we provide estimates for the L2 -valued solutions of (1.2) and (1.3) in appropriate Sobolev spaces. 4. Auxiliary results. We include the following bound, which is given for the continuous time case in [4], for the convenience of the reader. Lemma 4.1. If Assumptions 2.5 and 2.6 hold, then for all φ ∈ L2  Qi (φ) :=

Rd

2φ(x)Lhi φ(x) +

d1  2  κ    h,ρ 2 2 δh,λ φ0 Mi φ(x) dx ≤ N φ0 − 2 ρ=1 λ∈Λ0

for all i ∈ {1, . . . , n} and for a constant N depending only on κ, A0 , A1 , and the cardinality of Λ. Proof. First observe that for μ ∈ Λ0 the conjugate operator in L2 to δ−h,μ is −δh,μ . Notice also that δh,μ (φψ) = φδh,μ ψ + (Th,μ ψ)δh,μ φ, where Th,μ ψ(x) = ψ(x+hμ). Thus by simple calculations Q = Q(1) +Q(2) +Q(3) +Q(4) , where   (1) λρ μρ Qi (φ) := − ((2aλμ i − bi bi )(δh,λ φ)δh,μ φ)(x) dx, Rd λ,μ∈Λ 0



(2)

Qi (φ) := −2 (3) Qi (φ)





Rd

:= 2 Rd

λ,μ∈Λ0

2 a00 i φ (x) + φ(x)

and (4)

((Th,μ φ)(δh,λ φ)δh,μ aλμ i )(x) dx,

Qi (φ) :=



0λ (aλ0 i δh,λ φ + ai δ−h,λ φ)(x) dx,

λ∈Λ0

 Rd

2 b00 i φ (x) + 2



0ρ bλρ i bi φδh,λ φ(x) dx.

λ∈Λ0

By Assumption 2.6, (1)

Qi (φ) ≤ −κ

 λ∈Λ0

2

δh,λ φ0 ,

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and by Assumption 2.5, Young’s inequality, and the shift invariance of Lebesgue measure, κ  (j) 2 2 Qi (φ) ≤ δh,λ φ0 + N φ0 6 λ∈Λ0

for each j ∈ {2, 3, 4} with a constant N depending only on the cardinality of Λ, κ, A0 , and, for j = 2, also on A1 . We also recall the following discrete Gronwall lemma. Note that we use the convention that summation over an empty set is zero. Lemma 4.2. For constants K ∈ (0, 1) and C, if (ai )ni=0 is a nonnegative sequence j such that aj ≤ C + K i=1 ai holds for each j ∈ {0, . . . , n}, then aj ≤ C(1 − K)−j for j ∈ {0, . . . , n}. j Proof. Let bj := C + K i=1 bi and note that (1 − K)bj = bj−1 . Then aj ≤ bj for j ≤ n by induction, since a0 ≤ C = b0 and aj (1 − K) ≤ C + K

j−1 

ai ≤ C + K

i=1

j−1 

bi = bj (1 − K),

i=1

assuming ai ≤ bi for i ≤ j − 1. Therefore aj ≤ bj = C(1 − K)−j for each j ≤ n and for K ∈ (0, 1). The following provides a Sobolev space estimate for solutions to the space-time scheme that is independent of h. For an integer m ≥ 0, denote by W2m (τ ) the space of W2m -valued predictable processes φ on Ω × Tτ such that [[φ]]m := Eτ

n 

2

φi m < ∞

i=1

and note that we write [[g]]m = Eτ

n 

gi 2m = Eτ

d1 n  

2

giρ m

i=1 ρ=1

i=1

1 for functions g = (g ρ )dρ=1 . Theorem 4.3. Let f, g ρ ∈ W2m (τ ) for ρ ∈ {1, . . . , d1 }. If Assumption 2.5 holds, then for each nonzero h there exists a unique solution ν ∈ W2m (τ ) of

(4.1)

d1      h,ρ ρ Mi−1 ξiρ νi−1 + gi−1 νi = νi−1 + Lhi νi + fi τ + ρ=1

for any F0 -measurable W2m+1 -valued initial condition ν0 . Further, if Assumption 2.6 is also satisfied, then 2

E max νi m + Eτ i≤n

(4.2)

n  

2

2

δh,λ νi m ≤ N Eτ ν0 m+1

i=1 λ∈Λ

+ N Eτ

n    2 2 fi m + gi m i=0

holds for a constant N that depends only on d, d1 , m, Λ, A0 , . . . , Am ¯ , κ, and T .

3172

ERIC JOSEPH HALL

Proof. By Theorem 2.10, the existence of a unique sequence of L2 -valued random variables solving (1.2) is known. For f, g ρ ∈ W2m (τ ), and an W2m+1 -valued initial condition ν0 , there exists a unique sequence of W2m -valued random variables satisfying (4.1). The estimate (4.2) can be achieved easily with a constant N depending on h, so in particular the solution is in W2m (τ ). Next we prove the nestimate (4.2) for a constant independent of h. For convenience we denote Knm := τ i=0 (fi 2m + gi 2m ). Considering the equality a2 − b2 = 2a(a − b) − |a − b|2 we note that (4.1) implies 2

2

2

νi 0 − νi−1 0 = 2 (νi , νi − νi−1 ) − νi − νi−1 0 d1      h,ρ ρ νi−1 , Mi−1 ξiρ = 2 νi , Lhi νi + fi τ + 2 νi−1 + gi−1 ρ=1

+2

d1  

 h,ρ ρ νi − νi−1 , Mi−1 ξiρ − νi − νi−1 20 νi−1 + gi−1

ρ=1 d1      h,ρ ρ = 2 νi , Lhi νi + fi τ + 2 νi−1 + gi−1 νi−1 , Mi−1 ξiρ ρ=1

d  1    2  2  h,ρ ρ ρ + Mi−1 νi−1 + gi−1 ξi  − Lhi νi + fi 0 τ 2 .   ρ=1

0

Summing up over i from 1 to j, we have 2

2

νj 0 ≤ ν0 0 + Hj + Ij + Jj ,

(4.3) where

Hj := 2

j    νi , Lhi νi + fi τ, i=1

Ij := 2

j  d1  

 h,ρ ρ νi−1 , Mi−1 ξiρ , νi−1 + gi−1

i=1 ρ=1

and

d  j  1   2   h,ρ ρ ρ Mi−1 νi−1 + gi−1 ξi  . Jj :=    i=1

ρ=1

0

By an application of Itˆo’s formula, for each π, ρ ∈ {1, . . . , d1 } one has that ρ πρ π ξi+1 = (Δwπ (ti ))(Δwρ (ti )) = Yi+1 − Yiπρ + τ δπρ ξi+1

for all i ∈ {1, . . . , n}, where  t  t   ρ πρ π π ρ Y (t) := w (s) − wγ(s) dw (s) + wρ (s) − wγ(s) dwπ (s), 0

0

γ(s) is the piecewise defined function taking value γ(s) = i for s ∈ [iτ, (i + 1)τ ), and (1) (2) δπρ = 1 when π = ρ and 0 otherwise. Thus we can write Jj = Jj + Jj , where  d j  1  2    (1) h,ρ ρ Mi−1 Jj := νi−1 + gi−1  τ    i=1

ρ=1

0

3173

ACCELERATED SPATIAL APPROXIMATIONS

and (2)

Jj



d1  

tj

:= 0

 h,π h,ρ ρ π Mγ(s) dY πρ (s). νγ(s) + gγ(s) , Mγ(s) νγ(s) + gγ(s)

π,ρ=1

Then we note that since Lemma 4.1 holds for all t ∈ [0, T ], in particular Hj +

(1) Jj



≤τ

δh,λ ν0 20

λ∈Λ 2

≤ Cτ ν0 1 + N τ

 j   1 2 +τ Qi (νi ) + (νi , fi ) + gi−1 0 ε i=1 j 

κ  2 τ δh,λ νi 0 + N Kj0 , 2 i=1 j

2

νi 0 −

i=1

λ∈Λ

where we have used ν0 20 ≤ Cν0 21 , N is a positive constant depending only on κ, A0 , A1 , and the cardinality of Λ, and ε > 0. Thus inequality (4.3) becomes (4.4)

2

νj 0 + τ

j  

2

2

δh,λ νi 0 ≤ N τ ν0 1 + N τ

i=1 λ∈Λ

j 

(2)

2

νi 0 + N Kj0 + Ij + Jj

i=1

for a constant N that depends only on κ, A0 , A1 , C, and the cardinality of Λ. Next we observe that EIj = 2

j  d1 

 E

  h,ρ νi−1 , Mi−1 νi−1 + g ρ E (ξiρ | Fi−1 ) = 0

i=1 ρ=1 ρ since ξi+1 is independent of Fi and νi , Mih,ρ νi , and giρ are all Fi -measurable for (2)

i ∈ {0, . . . , n}. Similarly, we see that EJj = 0 since the expectation of the stochastic integral is zero. Therefore, taking the expectation of (4.4) we have that (4.5)

E νj 20 + Eτ

n  

δh,λ νi 20 ≤ N Eτ ν0 21 + N EKn0 + N Eτ

j 

i=1 λ∈Λ

νi 20

i=1

for each j ∈ {1, . . . , n}. Excluding for the time being the difference term on the left-hand side of (4.5) and applying Lemma 4.2 we obtain   E νj 20 ≤ N Eτ ν0 21 + EKn0 (1 − N τ )−j and, since (1 − N τ )−j = (1 − N Tn )−j ≤ (1 − N Tn )−n ≤ C  eN T , we have 2

2

max E νi 0 ≤ N Eτ ν0 1 + N EKn0

(4.6)

i≤n

for a constant N that depends only on the parameters κ, A0 , A1 , T , and the cardinality of Λ. Using (4.6) we can eliminate the last term on the right-hand side of (4.5). In particular, we have Eτ

n   i=1 λ∈Λ

2

2

δh,λ νi 0 ≤ N Eτ ν0 1 + N EKn0 .

3174

ERIC JOSEPH HALL

Using the Burkholder–Davis–Gundy inequality we can bound max |J (2) | and max |I|. Namely,    (2)  E max Ji  i≤n

d1 

≤3

 E 

d1 

d1 

≤C

T

E

π,ρ=1

⎛

1/2  2  2  h,ρ   h,π ρ π  πρ Mγ(s) νγ(s) + gγ(s)  Mγ(s) νγ(s) + gγ(s)  d Y (s) 0

0

π,ρ=1

≤C

T

0

 4  2  h,ρ   ρ π  Mγ(s) νγ(s) + gγ(s)  wπ (s) − wγ(s)  ds 0

E ⎝ max

π,ρ=1

0

1/2

i≤n

 √    τ Mih,ρ νi + giρ 

0

  1/2 ⎞ 2  2 1 T  h,ρ   ρ π  ⎠, × Mγ(s) νγ(s) + gγ(s)  wπ (s) − wγ(s)  ds τ 0 0

where C is a constant independent of τ and h that is allowed to change from one instance to the next. Therefore, d1    2   (2)    E max Ji  ≤ d1 C τ E max Mih,ρ νi + giρ  i≤n

ρ=1

(4.7) +

0

i≤n

 T d1 2  2 C   h,ρ   ρ π  E Mγ(s) νγ(s) + gγ(s)  wπ (s) − wγ(s)  ds τ π,ρ=1 0 0

by Young’s inequality. The first term on the right-hand side of (4.7) is bounded from above by the sum over all i ∈ {1, . . . , n}, thus d1  ρ=1

d1 n   2  2     h,ρ ρ τ E max Mih,ρ νi + giρ  ≤ Eτ Mi νi + gi  0

i≤n

ρ=1

≤ CEτ

0

i=0 n  i=0





δh,λ νi 20

+

 gi 20

,

λ∈Λ

and the second term on the right-hand side of (4.7) yields  T d1 2  2 1   h,ρ   ρ π  E Mγ(s) νγ(s) + gγ(s)  wπ (s) − wγ(s)  ds τ π,ρ=1 0 0  d1 2  2 T  1   h,ρ   ρ π  E ≤ Mγ(s) νγ(s) + gγ(s)  E wπ (s) − wγ(s)  τ π,ρ=1 0 0   n   2 2 ≤ Eτ δh,λ νi 0 + gi 0 i=0

     Fγ(s) ds 

λ∈Λ

by the tower property for conditional expectations. Combining these estimates we see that E max |J (2) | is estimated by terms already appearing on the right-hand side of (4.4) for a constant N that depends also on d1 .

3175

ACCELERATED SPATIAL APPROXIMATIONS

Similarly, we note that Ij = 2

j  d1    h,ρ ρ ρ νi−1 + gi−1 νi−1 , Mi−1 Δwi−1 i=1 ρ=1

=2

d1   ρ=1

tj



0

 h,ρ ρ νγ(s) , Mγ(s) dwρ (s). νγ(s) + gγ(s)

Applying the Burkholder–Davis–Gundy inequality to obtain E max |Ii | ≤ 6 i≤n

d1 

 E

ρ=1

≤6

d1  ρ=1

T

⎧ ⎨ E

0

2    h,ρ  ρ νγ(s) 2  M ν + gγ(s)  ds 0  γ(s) γ(s)

1/2

0



max νi 0 ⎩ i≤n

T

0

 2  h,ρ  ρ Mγ(s) νγ(s) + gγ(s)  ds

1/2 ⎫ ⎬ ⎭

0

and then applying Young’s inequality  1 2 E max |Ii | ≤ E max νi 0 + CEτ i≤n 2 i≤n i=0 n





 2 δh,λ νi 0

+

2 gi 0

,

λ∈Λ

we see that E max |I| is also estimated by terms already appearing on the right-hand side of (4.4). Returning to (4.4) and taking the maximum followed by the expectation we have E

2 max νi 0 i≤n

+ Eτ

n  

2

2

δh,λ νi 0 ≤ N Eτ ν0 1 + N EKn0 ,

i=1 λ∈Λ

using (4.6) and the estimates on E max |J (2) | and E max |I|. Thus (4.2) holds when m = 0. If m ≥ 1 we differentiate (4.1) with respect to xl and introduce the notation φ˜ for the derivative of a function φ with respect to xl for the directions l ∈ {1, . . . , d}. Then (4.1) becomes d1      λρ ρ ρ ˆi τ + b ν˜i = ν˜i−1 + aλμ δ δ ν ˜ + f δ ν ˜ + g ˆ h,λ −h,μ i h,λ i−1 i i−1 i−1 ξi

(4.8)

ρ=1

with fˆ := f˜+˜ aλμ δh,λ δ−h,μ ν and gˆρ := g˜ρ + ˜bλρ δh,λ ν, where the summation convention is used with respect to the repeated indices λ, μ ∈ Λ. Recalling that for μ ∈ Λ0 , ∂μ = μl Dl , where we use the summation convention with respect to the repeated index l ∈ {1, . . . , d}, we proceed as before but now using the inequality Eτ

n  i=1

(˜ νi , δh,λ δ−h,μ νi ) ≤ τ

n 

E ˜ νi 0 δh,λ ∂μ νi 0

i=1

≤ εEτ

n  d  i=1 l=1

2

Dl δh,λ νi 0 +

 N 2 Eτ ˜ νi 0 , ε i=1 n

3176

ERIC JOSEPH HALL

which holds for arbitrary ε > 0 and N depending only on |μ|. Then for each direction l ∈ {1, . . . , d}, E ˜ νj 20 + Eτ

j  

δh,λ ν˜i 20 ≤ N Eτ ν0 21 + N EKj1 + N Eτ

j 

i=1 λ∈Λ

(4.9)

˜ νi 20

i=1

+

1 Eτ 2d

j 



2

Dl δh,λ νi 0 .

i=1 λ∈Λ

Summing up over each direction, the term with factor (2d)−1 can be seen to be estimated by other terms already appearing on the right-hand side of (4.9). Then by the same procedure as before we obtain (4.10)

E max Dνi 20 + Eτ i≤n

n  

Dδh,λ νi 20 ≤ N Eτ ν0 21 + N EKn1 ,

i=1 λ∈Λ

which proves the theorem when m = 1. Assuming that m ≥ 2 and that (4.2) holds for each integer p < m in place of m, we can differentiate (4.1) (p + 1) times and, repurposing the notation φ˜ for the (p + 1)th order derivatives of φ with respect to x, we obtain (4.8) with different fˆ and gˆρ . Namely, the fˆ will be the sum of f˜ and linear combinations of certain ith order derivatives of aλμ together with certain (p + 1 − i)th order derivatives of δh,λ δ−h,μ ν, for integer i ≤ (p + 1). As before, the L2 -norms of the (p + 1 − i)th derivatives of δh,λ δ−h,μ ν are dominated by the L2 -norms of the (p + 2 − i)th derivatives of δh,λ ν, which are in turn less than the W2p+1 -norm of δh,λ ν. After similar changes are made in gˆρ we obtain the counterpart of (4.10), which then yields (4.2) with (p + 1) in place of m. We also have the following Sobolev space estimate for solutions to the implicit time scheme. Let m be a nonnegative integer. Theorem 4.4. Let f ∈ W2m (τ ) and g ρ ∈ W2m+1 (τ ). If Assumptions 2.1 and 2.2 hold, then (1.3) admits a unique solution v ∈ W2m+2 (τ ) for a given F0 -measurable W2m+1 -valued initial condition v0 . Moreover 2

E max vi m+1 + Eτ i≤n

n 

2

2

vi m+2 ≤ N E v0 m+1 + N Eτ

i=1

n  

2

2

fi m + gi m+1



i=0

holds for a constant N depending only on d, d1 , m, K0 , . . . , Km+1 , κ, and T . Proof. Proving the solvability of (1.3) reduces to solving the elliptic problem (I − τ Li ) vi = vi−1 + τ fi +

d1 

 ρ  ξiρ Mρi−1 vi−1 + gi−1

ρ=1

for each i ∈ {1, . . . , n}, where I is the identity. That is, we claim that A := (I − τ L) is a W2m -valued operator on W2m+2 such that Ai is (i) bounded, i.e., Ai φ2m ≤ Kφ2m+2 for a constant K, (ii) and coercive, i.e., Ai φ, φ ≥ λφ2m+2 for a constant λ > 0, for every i ∈ {1, . . . , n} and for all φ ∈ W2m+2 , where ·, · denotes the duality pairing between W2m+2 and W2m based on the inner product in W2m+1 . Then by the separability of W2m+2 , there exists a countable dense subset {ej }∞ j=1 such that for fixed

ACCELERATED SPATIAL APPROXIMATIONS

3177

p p ≥ 1, φp = j=1 cj ej for constants cj where φp ∈ W2m+2 is not identically zero. We fix i and for every ψ ∈ W2m consider A acting on φp , that is, Aφp , ej  = ψ, ej  for all j ∈ {1, . . . , p}. Taking linear combinations we obtain Aφp , φp  = ψ, φp  from which we derive 2

λ φp m+2 ≤ Aφp , φp  = ψ, φp  ≤ ψm φp m+2 by the coercivity of A and an application of the Cauchy–Schwarz inequality. Thus φp m+2 ≤ λ1 ψm and hence (by the reflexivity of W2m+2 ), there exists a subsequence pk such that φpk converges weakly to φ and in particular Ai φpk , ej  → Aφ, ej  for every j. Therefore for every ψ ∈ W2m there exists a φ ∈ W2m+2 satisfying Ai φ = ψ for every i ∈ {1, . . . , n}. Moreover, this solution is easily seen to be unique. Using the existence and uniqueness to the elliptic problem in each interval, we note that v0 + τ f1 +

d1 

(Mρ0 v0 + g0ρ ) ξ1ρ ∈ W2m

ρ=1

by Assumption 2.3 and therefore there exists a v1 ∈ W2m+2 satisfying (I − τ L1 ) v1 = v0 + τ f1 +

d1 

(Mρ0 v0 + g0ρ ) ξ1ρ .

ρ=0

Further, assuming that there exists a vi ∈ W2m+2 satisfying (1.3) we have that vi + τ fi+1 +

d1 

ρ (Mρi vi + giρ ) ξi+1 ∈ W2m

ρ=0

by the induction hypothesis and Assumption 2.3, and therefore there exists a vi+1 ∈ n W2m+2 satisfying (1.3). Hence we obtain v = (vi )i=1 such that each vi ∈ W2m+2 satisfies (1.3). It only remains to prove the claim concerning ellipticity of A. By Assumption 2.1, clearly Ai is a bounded linear operator for each i. We see that

Aφ, φ = Iφ, φ − τ Lφ, φ = φ2m+1 − τ Lφ, φ, where, by Assumptions 2.1 and 2.2,     

Lφ, φ = a0β − Dα aαβ Dβ φ, φ − aαβ Dβ φ, Dα φ κ 2 2 ≤ C φm+1 − φm+2 2 in the W2m+1 inner product for α, β ∈ {1, . . . , d} and for a constant C depending on K0 and K1 . Therefore κ κ 2 2 2

Aφ, φ ≥ τ φm+2 + (1 − τ K) φm+1 ≥ τ φm+2 2 2 for sufficiently small τ and hence (ii) is satisfied. To prove the estimate, we use a method similar to that in the proof of Theorem 4.3 to arrive at (4.3) with v in place of ν, L in place of Lh , Mρ in place of M h,ρ , all in the W2m+1 -norm instead of the L2 -norm. Again, we decompose J into J (1) and J (2) using the processes Y πρ (t) that arise by applying the Itˆo formula to the product of increments of the independent Wiener processes. However, this time, instead of using

3178

ERIC JOSEPH HALL

Lemma 4.1, we observe that (1)

Hj + Jj

≤ Nτ

j 

  κ  2 2 2 fi m + gi m+1 τ vi m+2 + N τ 2 i=1 i=0 j

2

vi m+1 −

i=1

j

since 

d1 

2v(x)Lv(x) + Rd

2

2

2

|Mρ v(x)| dx ≤ (ε − κC) vm+2 + C vm+1

ρ=1

for ε > 0 by the considerations above. Therefore we have that vj 2m+1 + τ

j 

vi 2m+2 ≤ N v0 2m+1 + N τ

i=1

j    fi 2m + gi 2m+1 i=0

+ N Ij +

(2) N Jj

and the estimate follows by considering the maximum and then taking the expectation. Moreover, with the estimate, it is clear that the solution v ∈ W2m+2 (τ ). We will use the theorem above to obtain estimates in appropriate Sobolev spaces for a system of time discretized equations. For i ∈ {0, . . . , n}, let  λμ (0) Li := ai ∂λ ∂μ , λ,μ∈Λ (0)ρ Mi

:=



bλρ i ∂λ ,

λ∈Λ

and for an integer p ≥ 1 let (p)

Li

:= p!



aλμ i

p  j=0

λ,μ∈Λ0



−1

+ (p + 1) (p)ρ Mi

−1

:= (p + 1)

Ap,j ∂λj+1 ∂μp−j+1 + (p + 1)−1





p+1 aλ0 i ∂λ

λ∈Λ0 p+1 a0μ i ∂μ ,

μ∈Λ0 p+1 bλρ , i ∂λ

λ∈Λ0 h(p)

Oi

:= Lhi −

p  j=0

hj (j) L , j! i

and h(p)ρ

Ri

:= Mih,ρ −

p  hj j=0

j!

(j)ρ

Mi

,

where Ap,j is defined by (4.11)

Ap,j =

(−1)p−j . (j + 1)!(p − j + 1)!

For p ≥ 1, the values of L(p) φ and M(p)ρ φ are obtain by formally taking the pth derivatives in h of Lh φ and M h,ρ φ at h = 0. For a positive integer k ≤ m, the sequences of random fields v (1) , . . . , v (k) needed in (3.1) will be the embeddings of random variables taking values in certain Sobolev

ACCELERATED SPATIAL APPROXIMATIONS

3179

spaces obtained as solutions to a system of time discretized SPDEs. Namely, as the solutions to   p  (p) (p) (p) l (l) (p−l) νi = νi−1 + Li νi + Cp Li νi τ 

(4.12) +

(p) Mρi−1 νi−1

+

l=1 p 



(l)ρ (p−l) Cpl Mi−1 νi−1

ξiρ ,

l=1

= p(p − 1) · · · (p − l + 1)/l! is the binomial coefficient and for p ∈ {1, . . . , k}, where (0) ν is the solution to (1.3) from Theorem 4.4. Theorem 4.5. Let Assumptions 2.1, 2.2, 2.3, and 2.5 hold with m = m ≥ k ≥ 1 and let ν (0) ∈ W2m+2 (τ ) be the solution to (1.3) with initial condition u0 from Theorem 4.4. Then the system (4.12) with initial condition Cpl

(1)

ν0

(2)

= ν0

(k)

= · · · = ν0

=0

has a unique set of solutions (ν (p) )kp=1 such that each ν (p) ∈ W2m+2−p (τ ). Moreover for each p ∈ {1, . . . , k},    (p) 2 E max νi  i≤n

+ Eτ

m+1−p

n     (p) 2 νi 

m+2−p

i=1

(4.13)

≤ N Eu0 2m+1 n    2 2 fi m + gi m+1 + N Eτ i=0

holds for a constant N depending only on d, d1 , Λ, m, K0 , . . . , Km+1 , A0 , . . . , Am , κ, and T . Proof. For convenience let (p)

Fi

:=

p 

(j) (p−j)

Cpj Li νi

j=1

and (p)ρ

Gi

:=

p 

(j)ρ (p−j) νi ,

Cpj Mi

j=1

d1 where we write G(p) = ρ=1 G(p)ρ . Observe that for each p ∈ {1, . . . , k} the equation for ν (p) in (4.12) depends only on ν (l) for l < p and does not involve any of the unknown processes ν (l) with indices l ≥ p. Therefore we shall prove the solvability of the system and the desired properties on ν (p) recursively using Theorem 4.4. For p = 1, we have (4.14)

(1)

νi

d1      (1) (1) (1) (1) (1)ρ τ+ Mρi−1 νi−1 + Gi−1 ξiρ . = νi−1 + Li νi + Fi ρ=1

Since ν (0) ∈ W2m+2 (τ ), for m = m we have that F (1) ∈ W2m−1 (τ ) and G(1) ∈ W2m (τ ) by Assumption 2.5. Hence by Theorem 4.4, there exists a unique ν (1) ∈ W2m+1 (τ ) (1) satisfying (4.14) with initial condition ν0 = 0. Further, ν (1) is estimated by (4.13) and thus Theorem 4.5 holds with p = 1.

3180

ERIC JOSEPH HALL

Now we assume that for m ≥ k ≥ 2 and p ∈ {2, . . . , k} we have unique ν (1) , . . . , ν (p−1) (1) (p−1) solving (4.12) for ν0 = · · · = ν0 = 0 with the desired properties. In particular, observe that for j ∈ {1, . . . , p} [[L(j) ν (p−j) ]]m−p ≤ N [[ν (p−j) ]]m+2−(p−j)

(4.15)

and for each ρ ∈ {1, . . . , d1 } [[M(j)ρ ν (p−j) ]]m−p+1 ≤ N [[ν (p−j) ]]m+1−(p−j)

(4.16)

for a constant N . Therefore it follows that F (p) ∈ W2m−p (τ ) and G(p) ∈ W2m−p+1 (τ ). Applying Theorem 4.4 yields the existence of a unique solution ν (p) ∈ W2m−p+2 (τ ) (p) that satisfies (4.12) with initial condition ν0 = 0. Together with (4.15) and (4.16) the estimate from Theorem 4.4 implies that (4.13) holds. Further, the uniqueness of each ν (p) follows from Theorem 4.4. For the convenience of the reader we record the following lemma and two remarks from [4] that will be used in proving the error estimates. Lemma 4.6. Let φ ∈ W2p+1 and ψ ∈ W2p+2 for a nonnegative integer p and let λ, μ ∈ Λ0 . Set ∂λ φ = λj Dj φ

and

∂λμ = ∂λ ∂μ .

Then we have (4.17)

∂p δh,λ φ(x) = (∂h)p



1 0

θp ∂λp+1 φ(x + hθλ) dθ

and (4.18)

∂p δh,λ δ−h,μ ψ(x) = (∂h)p

 0

1

 0

1

(θ1 ∂λ − θ2 ∂μ )p ∂λμ ψ(x + h(θ1 λ − θ2 μ)) dθ1 dθ2

for almost all x ∈ Rd for each h ∈ R. Furthermore, for integer l ≥ 0, if φ ∈ W2p+2+l and ψ ∈ W2p+3+l , then     p p+1   j    h |h|  p+2  j+1   ∂ φ − φ ≤ φ (4.19) δ  ∂ h,λ   (j + 1)! λ (p + 2)! λ l   j=0 l

and (4.20)

    p i     p+1 j+1 i i−j+1  δh,λ δ−h,μ ψ − h Ai,j ∂λ ∂μ ψ  ≤ N |h| ψl+p+3 ,    i=0 j=0 l

where Ai,j is defined by (4.11) and N depends on λ, μ, d, and p. l,r For integers l ≥ 0 and r ≥ 1, denote by Wh,2 the Hilbert space of functions φ on Rd such that  2 2 (4.21) φl,r,h := δh,λ1 × · · · × δh,λr φl < ∞ λ1 ,...,λr ∈Λ

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ACCELERATED SPATIAL APPROXIMATIONS l,0 and set Wh,2 = W2l . Then for any φ ∈ W2l+r we have

φl,r,h ≤ N φl+r , where N depends only on |Λ0 |2 := λ∈Λ0 |λ|2 and r. Remark 4.7. Formula (4.17) with p = 0 and Minkowski’s integral inequality imply that δh,λ φ0 ≤ ∂λ φ0 . By applying this inequality to finite differences of φ and using induction we can l,r conclude that W2l+r ⊂ Wh,2 . (0)

Remark 4.8. Owing to Assumption 2.7, for i ∈ {0, . . . , n} we have that Li = Li (0)ρ and Mi = Mρi . Also by Lemma 4.6 and Assumptions 2.5 and 2.6, for φ ∈ W2p+2+l and ψ ∈ W2p+3+l we have    h(p)  ψ  ≤ N |h|p+1 ψl+p+3 O l

and

   h(p)ρ  p+1 φ ≤ N |h| φl+p+2 R l

for a constant N depending only on p, d, l, A0 , . . . , Al , and Λ. For integers k, l ≥ 0, let ν (0) , ν (1) , . . . , ν (k) be the functions from Theorem 4.5. We define (0)

riτ,h := νih − νi

(4.22)

−

k  hj j=1

j!

(j)

νi

for i ∈ {1, . . . , n}, where ν h is the unique L2 -valued solution to (1.2) that exists by Theorem 4.3 with initial condition u0 and free terms f and g ρ , ρ ∈ {1, . . . , d1 }. Lemma 4.9. Let Assumptions 2.1, 2.2, 2.3, and 2.5 hold with m = m = l+k+1 for integers k, l ≥ 0, and let rτ,h be defined as in (4.22). Then r0τ,h = 0, rτ,h ∈ W2m−k (τ ) and riτ,h

=

τ,h ri−1

 +

Lhi riτ,h

+

Fiτ,h



d1    h,ρ τ,h τ+ Mi−1 ξiρ ri−1 + Gτ,h,ρ i−1 ρ=1

for i ∈ {1, . . . , n}, where Fiτ,h :=

k  hj j=0

j!

h(k−j) (j) νi

Oi

and Gτ,h,ρ i−1 :=

k  hj j=0

j!

h(k−j)ρ (j) νi−1

Ri−1

and, moreover, F τ,h ∈ W2l (τ ) and Gτ,h,ρ ∈ W2l+1 (τ ).

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ERIC JOSEPH HALL

Proof. By Theorem 4.3 the solution to the space-time scheme ν h ∈ W2m (τ ) and by Theorem 4.4 the solution to the time scheme ν (0) ∈ W2m+2 (τ ). Therefore rτ,h ∈ W2m (τ ) when k = 0 and, by Theorem 4.5, rτ,h ∈ W2m−k (τ ) when k ≥ 1. Observe that k k−i  hi  hj i=0

i!

j=0

j!

k−1 

L(j) ν (i) =

i=0

k−i hi  hj (j) (i) L ν i! j=1 j!

k  k−i i+j  h

=

i!j!

i=1 j=0 k  k 

=

hj L(i) ν (j−i) i!(j − i)!

i=1 j=i k  i 

=

L(i) ν (j)

hi L(j) ν (i−j) =: I τ,h , j!(i − j)!

i=1 j=1

where summations over empty sets are zero. Therefore, we can rewrite F τ,h as F τ,h = Lh ν (0) − Lν (0) +

k  hj j=1

j!

Lh ν (j) −

k  hj j=1

j!

Lν (j) − I τ,h .

Similarly, observe that k k−i  hi  hj i=0

i!

j=0

j!

M(j)ρ ν (i) =

k  i  i=1 j=1

hi M(j)ρ ν (i−j) =: J τ,h,ρ j!(i − j)!

and therefore Gτ,h,ρ = M h,ρ ν (0) − Mρ ν (0) +

k  hj j=1

j!

M h,ρ ν (j) −

k  hj j=1

j!

Mρ ν (j) − J τ,h,ρ .

Thus, following from Remark 4.8 and Theorem 4.5, F τ,h ∈ W2l (τ ) and Gτ,h,ρ ∈ W2l+1 (τ ). With the previous considerations, we are now prepared to prove the main results. 5. Proof of main results. We prove a slightly more general result which implies Theorem 3.2. Here we suppose that m = m. Theorem 5.1. Let Assumptions 2.1, 2.2, 2.3, 2.5, 2.6, and 2.7 hold with m = l + k + 1 for integers l, k ≥ 0. Then for rτ,h as defined in (4.22) we have (5.1)

n   2 2     τ,h  2 E max riτ,h  + Eτ , δh,λ ri  ≤ N h2(k+1) Km i≤n

l

l

i=1 λ∈Λ

where N depends only on d, d1 , Λ, m, K0 , . . . , Km+1 , A0 , . . . , Am , κ, and T . Proof. Recall that by Lemma 4.9 we have that F τ,h ∈ W2l (τ ) and Gτ,h,ρ ∈ l+1 W2 (τ ). Then the left-hand side of (5.1) is dominated by n        τ,h 2  τ,h,ρ 2 (5.2) N Eτ  Fi  + Gi i=1

l

l

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ACCELERATED SPATIAL APPROXIMATIONS

due to Lemma 4.9 and Theorem 4.3. To estimate (5.2) we observe that for j ≤ k, by Remark 4.8 we have that        h(k−j) (j)  k−j+1  (j)  k−j+1  (j)  νi  ≤ N |h| = N |h| , Oi νi  νi  l

l+k−j+3

m+2−j

and combining this result with Theorem 4.5 yields Eτ

n     τ,h 2 2 . Fi  ≤ N h2(k+1) Km i=1

l

The bound on Gτ,h,ρ can be obtained in a similar fashion. Now set Rτ,h := Irτ,h , where I is the embedding operator from Lemma 2.11. We have the following corollary to Theorem 5.1, which implies Theorem 3.2. Corollary 5.2. If the assumptions of Theorem 5.1 hold with l > p + d/2 for a nonnegative integer p, then for λ ∈ Λp 2    2 E max sup δh,λ Riτ,h (x) ≤ N h2(k+1) Km i≤n x∈Rd

and E max i≤n

2    d τ,h 2 δh,λ Ri (x) |h| ≤ N h2(k+1) Km x∈Gh

hold for a constant N depending only on d, d1 , Λ, m, K0 , . . . , Km+1 , A0 , . . . , Am , κ, and T . Proof. Using Sobolev’s embedding of W2l−p into Cb and Remark 4.7, Theorem 5.1 implies   2 2     E max sup δh,λ Riτ,h (x) ≤ CE max riτ,h  i≤n x∈Rd

i≤n

l−p,p,h

 2   ≤ C  E max riτ,h  i≤n

≤ Nh

2(k+1)

l

2 Km ,

where C and C  are constants depending only on m and d, and N is a constant depending only on m, d, d1 , κ, Λ, K0, . . . , Km+1 , and T . Similarly, by Lemma 2.11 above and Remark 4.7,  2 2     d τ,h τ,h  E max δh,λ Ri (x) |h| ≤ CE max δh,λ Ri  i≤n

x∈Gh

i≤n

l−p

 2   ≤ C  E max riτ,h  i≤n

≤ Nh

2(k+1)

l

2 Km .

For the I : W2l → Cb from Lemma 2.11, Theorem 3.2 follows by considering the embeddings vˆh := Iν h , where ν h is the unique L2 -valued solution to (1.2) with initial condition u0 , and v (j) := Iν (j) for j ∈ {0, . . . , k}, where ν (0) is the unique L2 -valued solution to (1.3) with initial condition u0 and the processes ν (1) , . . . , ν (k) are the solutions to the system of time discretized SPDEs (4.12) as given in Theorem 4.5.

3184

ERIC JOSEPH HALL

By Theorem 4.3, ν h is Fi -adapted and W2l -valued for all i ∈ {1, . . . , n}. For each j ∈ {1, . . . , k} the ν (j) are W p+1+k -valued processes by Theorem 4.5. Since l > d/2 and p + 1 − k > d/2 the processes vˆh and v (j) are well defined and clearly (4.22) implies (3.1) with vˆh in place of v h . That is, we have the expansion for a continuous version of the L2 -valued solution. To see that Theorem 3.2 indeed follows from Corollary 5.2 we must show that the restriction of the L2 -valued solution to the grid Gh , a set of Lebesgue measure zero, is indeed equal almost surely to the unique 2 (Gh )-valued solution that one would naturally obtain from (1.2). That is, we must show that vˆih (x) = vih (x)

(5.3)

almost surely for all i ∈ {1, . . . , n} and for each x ∈ Gh where v h is the unique Fi adapted 2 (Gh )-valued solution of (1.2) from Theorem 2.9. Therefore, for a compactly supported nonnegative smooth function φ on Rd with unit integral and for a fixed x ∈ Gh we define   y−x φε (y) := φ ε for y ∈ Rd and ε > 0. Recall by Remark 2.4 that we can obtain versions of u0 , f , and g ρ that are continuous in x. Since vˆh is a L2 -valued solution of (1.2) for each ε, almost surely    h h vˆi (y)φε (y) dy = vˆi−1 (y)φε (y) dy + τ (Lhi vˆih + fi )(y)φε (y) dy Rd

Rd d1 

+

ρ=1

Rd

 ξiρ

Rd

h,ρ h ρ (Mi−1 vˆi−1 + gi−1 )(y)φε (y) dy

for each i ∈ {1, . . . , n}. Letting ε → 0, we see that both sides converge for all i ∈ {1, . . . , n} and ω ∈ Ω. Therefore almost surely d1      h,ρ h ρ h vˆih (x) = vˆi−1 Mi−1 (x) + Lhi vˆih (x) + fi (x) τ + vˆi−1 (x) + gi−1 (x) ξiρ ρ=1

for all i ∈ {1, . . . , n}. Moreover by Lemma 2.11, the restriction of vˆh , the continuous version of ν h , onto Gh is an 2 (Gh )-valued process. Hence (5.3) holds, due to the uniqueness of the Fi -adapted 2 (Gh )-valued solution of (1.2) for any F0 -measurable 2 (Gh )-valued initial data. This finishes the proof of Theorem 3.2. We end with the following generalization of Theorem 3.3. Theorem 5.3. If the assumptions of Theorem 3.2 hold with p = 0 and v¯h as defined in (3.3), then 2 2       h (0) (0) 2 E max sup ¯ vi (x) − vi (x) + E max vih (x) − vi (x) |h|d ≤ N h2(k+1) Km ¯ i≤n x∈Gh

i≤n

x∈Gh

for a constant N depending only on d, d1 , Λ, m, K0 , . . . , Km+1 , A1 , . . . , Am , κ, and T . This follows from Theorem 5.1 and the definition of v¯h .

ACCELERATED SPATIAL APPROXIMATIONS

3185

Acknowledgments. The author would like to express gratitude toward his supervisor, Professor Istv´an Gy¨ongy, for the encouragement and helpful suggestions offered during the preparation of these results, which will form part of the author’s Ph.D. thesis. The author would also like to thank the anonymous referees whose helpful comments improved the quality of this manuscript. REFERENCES [1] A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Stochastic Model. Appl. Probab. 60, Springer, New York, 2009. [2] C. Brezinski, Convergence acceleration during the 20th century, J. Comput. Appl. Math., 122 (2000), pp. 1–21. [3] A. M. Davie and J. G. Gaines, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math. Comp., 70 (2001), pp. 121–134. ¨ ngy and N. Krylov, Accelerated finite difference schemes for linear stochastic partial [4] I. Gyo differential equations in the whole space, SIAM J. Math. Anal., 42 (2010), pp. 2275–2296. ¨ ngy and N. Krylov, Accelerated finite difference schemes for second order degenerate [5] I. Gyo elliptic and parabolic problems in the whole space, Math. Comp., 80 (2011), pp. 1431–1458. ¨ ngy and A. Millet, On discretization schemes for stochastic evolution equations, Po[6] I. Gyo tential Anal., 23 (2005), pp. 99–134. ¨ ngy and A. Millet, Rate of convergence of implicit approximations for stochastic evolu[7] I. Gyo tion equations, in Stochastic Differential Equations: Theory and Applications, Interdiscip. Math. Sci. 2, World Science Publishers, Hackensack, NJ, 2007. ¨ ngy and A. Millet, Rate of convergence of space time approximations for stochastic [8] I. Gyo evolution equations, Potential Anal., 30 (2009), pp. 29–64. [9] D. C. Joyce, Survey of extrapolation processes in numerical analysis, SIAM Rev., 13 (1971), pp. 435–490. [10] P. E. Kloeden, E. Platen, and N. Hofmann, Extrapolation methods for the weak approximation of Itˆ o diffusions, SIAM J. Numer. Anal., 32 (1995), pp. 1519–1534. [11] N. V. Krylov, An analytic approach to SPDEs, in Stochastic Partial Differential Equations: Six Perspectives, Math. Surveys Monogr. 64, AMS, Providence, RI, 1999. [12] N. V. Krylov and B. L. Rozovski˘ı, The Cauchy problem for linear stochastic partial differential equations, Math. USSR Izv., 11 (1977), pp. 1267–1284. [13] H. Kunita, Cauchy problem for stochastic partial differential equations arising in nonlinear filtering theory, Systems Control Lett., 1 (1981/82), pp. 37–41. [14] P. Malliavin and A. Thalmaier, Numerical error for SDE: Asymptotic expansion and hyperdistributions, C. R. Math. Acad. Sci. Paris, 336 (2003), pp. 851–856. ´ [15] E. Pardoux, Equations aux d´ eriv´ ees partielles stochastiques de type monotone, in S´ eminaire ´ sur les Equations aux D´eriv´ ees Partielles (1974–1975), III, Exp. No. 2, Coll`ege de France, Paris, 1975. [16] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), pp. 127–167. [17] L. F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Engrg. Sci., 210 (1911), pp. 307–357. [18] L. F. Richardson and J. A. Gaunt, The deferred approach to the limit. Part I. Single lattice. Part II. Interpenetrating lattices, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Engrg. Sci., 226 (1927), pp. 299–361. [19] B. L. Rozovski˘ı, Stochastic evolution systems: Linear theory and applications to nonlinear filtering, Math. Appl. (Soviet Series) 35, Kluwer Academic, Dordrecht, The Netherlands, 1990. [20] D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl., 8 (1990), pp. 483–509. [21] H. Yoo, An Analytic Aapproach to Stochastic Partial Differential Equations and its Applications, Ph.D. thesis, Department of Mathematics, University of Minnesota, Minneapolis, 1998. [22] H. Yoo, Semi-discretization of stochastic partial differential equations on R1 by a finitedifference method, Math. Comp., 69 (2000), pp. 653–666. [23] M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11 (1969), pp. 230–243.