Discrete maximal regularity of an implicit Euler--Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

An implicit Euler--Maruyama method with non-uniform step-size applied to a class of stochastic partial differential equations is studied. A spectral method is used for the spatial discretization and the truncation of the Wiener process. A discrete analogue of maximal $L^2$-regularity of the scheme and the discretised stochastic convolution is established, which has the same form as their continuous counterpart.


Introduction
Our interest in this paper lies in a discrete analogue of maximal regularity for a class of stochastic partial differential equations (SPDEs) of parabolic type. In more detail, with a positive self-adjoint generator −A with compact inverse densely defined on a separable Hilbert space H, we consider the equation dX(t) = AX(t)dt + B(t, X(t))dW (t), for t ∈ (0, 1] X(0) = ξ, (1.1) where the mild solution X takes values in H. The assumption on B and the Q-Wiener process W will be discussed later. The aim of this paper is to show a property of a prototypical discretisation to simulate the solution of such equations: we show a discrete analogue of an estimate called maximal regularity (Corollary 4.6). Maximal regularity is a fundamental concept in the theory of deterministic partial differential equations (see, for example [2,18,21] and references therein). Similarly, in the study of stochastic partial differential equations, the maximal regularity is an important analysis tool [9,8] as well as an active research area [27,26,5,28]. In our setting, the above equation (1.1) can be shown to satisfy the maximal regularity estimate of the form where ι ≥ 0 is a suitable parameter depending on the operator B, D(A ι+ 1 2 ) is the domain of the fractional power A ι+ 1 2 of A in H, and L 2 (H 0 , D(A ι )) is the space of Hilbert-Schmidt operator from H 0 , the Cameron-Martin space associated with Q, to D(A ι ). More details will be discussed later.
In recent years, the study of discrete analogues of the maximal regularity has been attracting attention for deterministic partial differential equations [1,4,7,13,14,15,20]; to the best of the author's knowledge, corresponding properties of numerical methods for stochastic PDEs have not been addressed in the literature.
Maximal regularity of stochastic and deterministic equations are different in nature. As we will see in (1.2), given a suitable smoothness of the initial data, the solution is "one-half spatially smoother" than the range of the diffusion operator B(t, x). This estimate is optimal, in that the solution cannot be spatially smoother in general (see [17,Example 5.3]). To put it another way, as described in [9,Chapter 6], the regularity one can obtain is the half of the corresponding regularity for the deterministic case.
We focus on the case where the operator A and the covariance operator Q share the same eigensystems. This prototypical setting is partly motivated by applications in environmental modelling and astrophysics, where covariance operators-of the random fields [6,22], and of the Wiener process for the stochastic heat equations [19,3], for example-the eigenspaces of which are the same as those of the Laplace operators play important roles. In simulations, it is desirable that discretisations users employ inherit properties of the solution of the model considered. Our results show the method we consider preserves a spatial regularity-maximal regularity-in a suitable sense.
As a spatial discretisation we consider the spectral-Galerkin method. The Wiener process, which is assumed to admit a series representation, takes its value in an infinite-dimensional space. In practice, we can simulate only finitely many of them. We approximate the Wiener process by truncation, i.e., we use a type of truncated Karhunen-Loève approximation.
Temporally, we consider the implicit Euler-Maruyama method with a non-uniform time discretisation. The aforementioned approximation of the Wiener process introduces one-dimensional Wiener processes multiplied by varying scalars-the eigenvalues of the covariance operator. Motivated by this observation, we allow the discretisation of each Wiener process to depend on these scalars. The algorithm we consider is first proposed by Müller-Gronbach and Ritter [24,23], for the stochastic heat equation on the unit cube. In [24,23], the resulting non-uniform scheme was shown to achieve an asymptotic optimality under a suitable step size, which in general cannot be achieved by schemes with uniform step-size.
The results we establish show that the non-uniform discretisation still preserves maximal regularity in a suitable sense. The algorithm we consider includes the implicit Euler-Maruyama method with the uniform time discretisation as a special case-the case where one uses the same step size for all one-dimensional Wiener processes-even though we, in general, lose the aforementioned optimality. As a consequence, we obtain a discrete analogue of maximal regularity for the standard implicit Euler-Maruyama method: the discretisation with the uniform step size.
The structure of this paper is as follows. Section 2 recalls some definitions and basic results needed in this paper. Section 3 introduces the discretised scheme we consider. Then, in Section 4 we show a discrete maximal regularity. Then, we conclude this paper in Section 5.

Setting
By H we denote a separable R-Hilbert space (H, ·, · , · ). Let −A : D(A) ⊂ H → H be a self-adjoint, positive definite linear operator that is densely defined on H, with compact inverse. Then, A is the generator of the C 0 -semigroup (S(t)) t≥0 := (e At ) t≥0 acting on H that is analytic. Further, there exists a complete orthonormal system {h j } for H such that −Ah j = λ j h j , each eigenspace is of finite dimensional, and 0 < λ 1 < λ 2 < · · · < λ j < · · · , and λ j → ∞ as j → ∞ unless the compact inverse −A −1 is finite rank. For simplicity, we assume the dimension of each eigenspace is 1. Then, we have the spectral representation For r ∈ R, let us define the domain D(A r ) of the fractional power A r of A by We obtain a separable Hilbert space (D(A r ), ·, · D(A r ) , · D(A r ) ) by setting ·, · D(A r ) := A r ·, A r · . For more details for the set up above, see for example [12,21,25,29]. Let (Ω, F , P) be a probability space equipped with a filtration satisfying the usual conditions. By W : [0, 1] × Ω → H we denote the Q-Wiener process with a covariance operator Q of the trace class. We assume that the Wiener process W is adapted to the filtration. Further, we assume that the eigenfunctions h ℓ of A is also eigenfunctions of Q with It is well-known that W taking values in H can be characterised as where β ℓ are independent one-dimensional standard Brownian motions with the zero initial condition realised on (Ω, F , P) that are adapted to the underlying filtration, and that the series converges in the Bochner space L 2 (Ω; C([0, 1]; H)). The Q-Wiener process takes values in H by construction. Here, since A and Q are assumed to share the same eigenfunctions, we can provide finer characterisations of the regularity.
In the following, a b means that a can be bounded by some constant times b uniformly with respect to any parameters on which a and b may depend. Throughout this paper, we assume the following.
where for a given normed space (X , · X ) the Borel σ-algebra associated with the norm topology is denoted by B(X ). Further, let B satisfy Moreover, let ι ∈ [0, 1/2] be given. We assume for any t Thus, together with (2.1) we see that Assumption 1 implies We recall the following existence result, which can be found in, for example, [9].
Theorem 2.1. Suppose that the mapping B satisfies Assumption 1 with some ι ≥ 0. Then, for ξ ∈ H there exists an H-valued continuous process (X(t)) t∈[0,1] adapted to the underlying filtration satisfying the usual conditions such that Moreover, this process is uniquely determined a.s., and it is called the mild solution of (1.1). Further, for any p ≥ 2 we have For more details, see for example [9,Sec. 7.1]. For the mild solution X, let Then, the processes X j = (X j (t)) t∈[0,1] satisfy the following bi-inifinite system of stochastic differential equations: Each process X j is given as where the series in the second term is convergent in L 2 (Ω), due to (2.5) and Assumption 1.
We have the following spatial regularity result.
Proposition 2.2. Suppose that Assumption 1 is satisfied with some ι ∈ [0, 1/2], and that the initial condition satisfies ξ ∈ D(A ι ). Then, we have the estimate where B * (r, X(r)) denotes the adjoint operator of B(r, X(r)). Therefore, it holds that and thus summing over j ≥ 1 yields the desired result. We note that for ι ∈ [0, 1/2] the right hand side of (2.6) is finite. To see this, we first note that (2.3) together with (2.5) implies Thus, from (2.2) we have for some constant c ι > 0.
Remark 2. We note that the solution is spatially one half smoother than the range of B(t, x). This is in general optimal, in that the solution cannot be spatially smoother in general ([17, Example 5.3]). For more details, see [17,16] and references therein. For recent developments of maximal regularity theory, see [27,26].

Discretisation
This section introduces the scheme proposed by Müller-Gronbach and Ritter [24,23]. In this regard, let us first discretise the interval [0, 1] with a uniform partition, i.e., we partition the interval with t i = i/n, for i = 0, 1, 2, . . . , n. For integers J, L ∈ N, an Itô-Galerkin approximation X(t i ) to (2.4) with the temporal discretisation being the implicit Euler-Maruyama scheme with a uniform time discretisation is given by with coefficients X(t i ), h j defined by X J,L j (0) = ξ, h j , and Müller-Gronbach and Ritter [24,23] noted that the projected Q-Wiener processes √ q ℓ β ℓ = Qh ℓ , h ℓ β ℓ = W (t), h ℓ have varying variances depending on the index ℓ. This observation motivated them to use different step-sizes depending on ℓ. Following them, we evaluate the standard one-dimensional Wiener process β ℓ at each level ℓ = 1, . . . , L at the corresponding n ℓ ∈ N nodes Then, the discretisation of the truncated Q-Wiener process We further introduce the following notation for the product of eigenvalues of the operator (I − 1 τ ν −τ ν−1 A) −1 , which we use for the approximation of the semigroup generated by A. For any τ η 1 ≤ τ η 2 , we let with the convention ∅ = 1. Note that s η,ℓ , t i−1,ℓ ∈ {τ 1 , . . . , τ N }. Then, for η = 1, . . . , N , the drift-implicit Euler-Maruyama scheme in the recursive form is given by, Equivalently, the above can be written in the convolution form Then, we use for η = 1, . . . , N as our approximate solution. We note that this scheme generalises the aforementioned approximation X J,L with the uniform time step as in (3.1): X J,L is nothing but X J,L with n ℓ = N for ℓ = 1, . . . , L.
Using the previous result, we obtain the following estimate.
Then, we have Proof. For any η ∈ {1, . . . , N }, from Lemma 4.1 we have It follows that where in the first equality Itô's isometry, and in the last inequality the condition (4.7) is used. Thus, the statement follows.
We need the following estimate for the process (φ J,(η) ℓ ) ν−1 as in (4.4a) and (4.4b) in terms of the Hilbert-Schmidt norm.
The following lemma is important to show the maximal regularity estimate of the same form as the continuous counterpart (2.6), studied in [9, Proposition 6.18] and [8].
We are ready to state our main result.
As a consequence of the previous result, given a suitable regularity of the initial condition, the approximate solution has the spatial regularity "one-half smoother"-the same as the continuous counterpart [9]-than the range of the operator B(t, x). .
In particular, X J,L defined as in (3.1) satisfies Proof. From Lemma 4.4 we have Then, from (4.3) and Theorem 4.5 the first statement follows. Letting n ℓ = N for ℓ = 1, . . . , L establishes the second statement.

Conclusion
In this paper, we considered an implicit Euler-Maruyama scheme for a class of stochastic partial differential equations with a non-uniform time discretisation. For this scheme, we showed that a discrete analogue of the maximal L 2 -regularity holds, which has the same form as the maximal regularity of the original problem.