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Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space−time noise Arnulf Jentzen and Peter E Kloeden Proc. R. Soc. A 2009 465, 649-667 doi: 10.1098/rspa.2008.0325
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Proc. R. Soc. A (2009) 465, 649–667 doi:10.1098/rspa.2008.0325 Published online 18 November 2008
Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space–time noise B Y A RNULF J ENTZEN *
AND
P ETER E. K LOEDEN
Institut fu ¨r Mathematik, Johann Wolfgang Goethe Universita ¨t, 60054 Frankfurt am Main, Germany We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler scheme, and show that it converges (in the strong sense) faster than the classical numerical schemes, such as the linear-implicit Euler scheme or the Crank–Nicholson scheme, for this equation with the general noise. In particular, we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical schemes using only basic increments of the noise process reported previously. By contrast, our scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier. Keywords: parabolic stochastic partial differential equation; Galerkin approximation; computational order barrier; exponential Euler scheme
1. Introduction The numerical approximation of stochastic partial differential equations (SPDEs) encounters all the difficulties that arise in the numerical solution of both deterministic partial differential equations (PDEs) and finite-dimensional stochastic differential equations (SDEs) plus more due to the infinite dimensional nature of the driving noise processes. See, for example, Gyo ¨ngy & Nualart (1995, 1997), Grecksch & Kloeden (1996), Gyo ¨ngy (1998, 1999), Shardlow (1999), Davie & Gaines (2000), Kloeden & Shott (2001), Hausenblas (2003), Lord & Rougemont (2004) and Mu ¨ller-Gronbach & Ritter (2007). In this paper, we consider the numerical approximation of a parabolic SPDE with additive noise, i.e. of the form dUt Z ½AUt C f ðUt Þdt C dWt ; U0 Z u 0 ; ð1:1Þ where A is in general an unbounded operator (e.g. AZD), f is a nonlinear continuous function and Wt is, for example, a space–time white noise (see §2 for a precise description of equation (1.1)). * Author for correspondence (
[email protected]). Received 7 August 2008 Accepted 20 October 2008
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Various approximation schemes have been introduced in the literature for this problem (see the papers cited above). All have the property that their rate of convergence for the one-dimensional semilinear stochastic heat equation with additive space–time white noise is 1/6. Indeed, Davie & Gaines (2000; see also Mu ¨ller-Gronbach & Ritter 2007) showed that no numerical schemes using only equidistant evaluations of the noise can converge with an overall rate faster than 1/6 (with strong convergence in time). Here, we present a new scheme that overcomes the rate barrier of 1/6 and converges with the rate of 1/3. It uses linear functionals of the noise, which are Gaussian distributed and are thus not difficult to generate. Of course, the choice of a semilinear heat equation is just for illustration and the situation is analogous for SPDE with the more general structure (1.1). The smoothening effect of the semigroup generated by the operator A in the SPDE (1.1) and various semigroup estimates play an important role in the proof. In §2, we give a precise description of the equations under consideration and the assumptions that we need. Then, in §3, we will introduce our numerical scheme and state the convergence theorem, while in §4 we present an example, briefly review the above-cited literature and give some numerical results. Proofs are given in §5. 2. Mathematical setting and assumptions Let TO0 and let (U, F , P) be a probability space with a normal filtration F t , t2[0,T ]. In addition, let (H, h$,$i) be a separable Hilbert space with norm denoted by j$j. We will interpret the SPDE (1.1) in such a space H in the mild sense, i.e. as satisfying the integral equation ðt ðt At AðtKsÞ AðtKsÞ P Ut Z e u 0 C e ð2:1Þ f ðUs Þds C e dWs Z 1; 0
0
for all t2[0,T ]. See, for example, Da Prato & Zabczyk (1992) for details. The objects A, u 0, f and Wt here are specified through the following assumptions. Assumption 2.1: linear operator A. There exist sequences of real eigenvalues 0!l1%l2%/ and eigenfunctions {en}nR1 of A such that the linear operator A : D(A)3H/H is given by N X Av Z Kln hen ; vien ; nZ1
for all v2D(A) with DðAÞZ fv 2 H j
PN
nZ1
jln j2 jhen ; vij2 !Ng.
Furthermore, let D((KA)r ) with r2R denote the interpolation spaces of the operator KA (e.g. Sell & You 2002). Assumption 2.2: cylindrical Brownian motion Wt. There exist a sequence qnR0, nR1, of positive real numbers, a real number g2(0,1) such that N X nZ1
Proc. R. Soc. A (2009)
ðln Þ2gK1 qn !N;
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and independent real valued F t -Brownian motions bnt , tR0, for nR1, i.e. each bnt is F t -adapted and the increments bntCh Kbnt , hO0, are independent of F t . Then, the cylindrical Brownian motion Wt is given by N X pffiffiffiffiffi n Wt Z qn en bt : nZ1
Remark 2.3. The above series may not converge in H, but in some space U1 into which H can be embedded, see Da Prato & Zabczyk (1992) and Pre´vot & Ro ¨ckner (2007). In our example with the Laplace operator in one dimension, we will have lnZKp2n2 and qnh1 for nR1. This is the important case of space– time white noise. Assumption 2.4: nonlinearity f. The nonlinearity f : H/H is two times continuously Fre´chet differentiable, it and its derivatives satisfy jf 0 ðxÞKf 0 ðyÞj% Ljx Kyj;
jðKAÞðKrÞ f 0 ðxÞðKAÞr vj% Ljvj;
for all x, y2H, v2D((KA)r ) and rZ0, 1/2, 1 and they satisfy jA K1 f 00 ðxÞðv; wÞj% LjðKAÞ Kð1=2Þ vjjðKAÞ Kð1=2Þ wj; for all v, w, x2H, where LO0 is a positive constant. Remark 2.5. The function f is usually given as a real-valued function of a real variable, but in the SPDE (1.1) it is considered as a function defined on H and taking values in some function space such as a subspace of H. Assumption 2.6: initial value u 0. The initial value u 0 is a D((KA)g) valued random variable, which satisfies EjðKAÞg u 0 j4 !N; where gO0 is given in assumption 2.2. Later, for the definition of the numerical scheme, we will also use the particular knowledge of the eigen decomposition of the linear operator A given in assumption Ð2.1. Under assumption 2.2, it is well known that the stochastic convolution 0t eAðtKsÞ dWs for t2[0,T ] has a predictable modification with values in D((KA)g). Hence, under assumptions 2.1, 2.2, 2.4 and 2.6 it is well known (see Da Prato & Zabczyk 1992) that the SPDE (1.1) has an up to modifications unique mild solution Ut on the time interval [0,T ], where Ut is the predictable stochastic process with values in D((KA)g), which fulfils (2.1). The following well-known results will be needed later: sup EjðKAÞg Ut j4 !N;
EjðKAÞ Kð1=2Þ ðUt K Us Þj4 % C ðtKsÞ2
0%t%T
and g
ðt
AðtKsÞ
4
9 > > > =
sup EjðKAÞ e dWs j !N; 0 ðt > > Kð1=2Þ eAðtKrÞ dWr j4 % C ðtKsÞ2 ; > EjðKAÞ ; 0%t%T
s
for all t, s 2[0,T ] and a constant CO0 (see Jentzen 2008). Proc. R. Soc. A (2009)
ð2:2Þ
ð2:3Þ
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3. Main theorem We will now apply a Galerkin projection to the SPDE (1.1) and then introduce a numerical scheme, which we will call the exponential Euler scheme, to the finitedimensional Galerkin SDE and, finally, we will state a convergence theorem for this scheme. We define finite-dimensional subspaces HN of H by HN dspanhe1 ; .; eN i; and the projections PN : H/HN by PN v Z
N X hen ; vien ; nZ1
for N2N. Then, we truncate or project the initial random variable u 0, the nonlinearity f and the Brownian motion Wt by uN 0
fN ðvÞ dðPN f ÞðvÞ;
dPN u 0 ;
WN t
N X pffiffiffiffiffi n qn en bt ; Z PN Wt Z nZ1
for all v2H and each NR1. Using this notation, we introduce a finite-dimensional SDE in the space HN (or, equivalently, in RN ) by ð3:1Þ dUtN Z AN UtN C fN ðUtN Þ dt C dW N t ; which is the Galerkin projection of the SPDE (1.1) onto HN, where ANZPNA is the (matrix) operator AN : HN/HH. Since assumption 2.4 also applies to fN, the SDE (3.1) also has a unique solution on [0,T ], which is given (implicitly) by (see Kloeden & Platen 1992) ðt ðt N N AN t N Ut Z e u 0 C expðAN ðt KsÞÞfN Us ds C expðAN ðtKsÞÞdW N ð3:2Þ s : 0
0
Now, we introduce a numerical method to approximate UtN in time on the interval [0,T ], which we will call the exponential Euler scheme: let V0N ;M du N 0 and define ð tkC1 N ;M Z eAN h VkN ;M C ANK1 eAN h KI fN VkN ;M C expðAN ðtkC1 KsÞÞdW N VkC1 s ; tk
ð3:3Þ with time-step hZT/M for some M2N and discretization times tkZkh for kZ0, 1, ., M. Note that AN is a diagonal matrix with diagonal entries l1, ., lN. Therefore, we can easily compute the expressions eAN h and ANK1 directly, which can also be seen below. Of course, we need to know the eigen decomposition of the linear operator A in order for the method to be applicable (see assumption 2.1). Then, this scheme is easier to simulate than may seem on first sight. Proc. R. Soc. A (2009)
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Denoting the components of VkN ;M and fN by D E N ;M Vk;i Z ei ; VkN ;M ; fNi Z hei ; fN i;
653
i Z 1; .; N ;
we can rewrite the numerical scheme (3.3) as !1=2 Kl1 h 1Ke q N ;M N ;M 1 VkC1;1 Z eKl1 h Vk;1 C fN1 VkN ;M C ð1Ke K2l1 h Þ R1k ; l1 2l1 «
N ;M VkC1;N
«
KlN h
Z e
N ;M Vk;N
« !1=2 1KeKlN h N N ;M qN K2lN h C fN Vk 1Ke RN C k ; lN 2lN
where the Rik for iZ1, ., N and kZ0, 1, ., MK1 are independent, standard normally distributed random variables. With this notation, we are able to present our main theorem, which states the strong convergence of the above numerical scheme and also provides a rate for this strong convergence. Theorem 3.1. Suppose that assumptions 2.1, 2.2, 2.4 and 2.6 are satisfied. Then, there is a constant CTO0 such that
2 1=2 logðM Þ N ;M Kg EUtk KVk % CT l N C sup ; ð3:4Þ M kZ0; . ; M holds for all N, MR2, where Ut is the solution of SDE (1.1), VkN ;M is the numerical solution given by (3.3 ), tk Z Tðk=M Þ for kZ0, 1, ., M and gO0 is the constant given in assumption 2.2. It follows immediately that the exponential Euler scheme (3.3) converges in time with strong order 1K3 for an arbitrary small 3O0, since log(M ) can be estimated by M 3. Exponential integrators for SDEs have already been studied intensively in the literature, see, for example, Jimenez et al. (1999) or also Carbonell et al. (2005). The schemes there applied to the finite-dimensional Galerkin SDE (3.1) are similar to the scheme proposed in equation (3.3). However, in Jimenez et al. (1999; see eqns (12) and (13) there), they only used evaluations of the noise process instead of linear functionals and so this would not lead to a higher order than the linear-implicit Euler scheme (see §4c). Furthermore, in Carbonell et al. (2005), they used linear functionals of the noise but they compute the derivative fN0 , which is an N!N matrix, in each computation step, which is very inefficient with respect to the computational cost. They considered a general SDE with additive noise and did not take into account the special structure coming from the SPDE (1.1). In the theorem above, we estimated the mean square error, which is in a way standard for strong convergence (see theorem 10.6.3 in Kloeden & Platen 1992). Alternatively, one could estimate the pth moment for every pR1 and so also obtain pathwise convergence (see Jentzen et al. accepted), which is a task for further research. For it, one needs a Burkholder–Davis–Gundy inequality in Hilbert spaces, which Proc. R. Soc. A (2009)
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is a non-trivial tool. Finally, note that for deterministic parabolic PDEs, exponential integrators, i.e. numerical schemes with exponential terms, have for example been considered in Hochbruck & Ostermann (2005a,b). 4. An example To present more clearly the consequences of theorem 3.1, we consider a much studied example of the SPDE (1.1). After introducing the example and applying theorem 3.1 to it we will review results in the literature concerning the numerical approximation of this example and, more generally, for the SPDE (1.1). Finally, we will illustrate our results with some numerical calculations. (a ) The semilinear stochastic heat equation As a simple example of the SPDE (1.1), we consider a semilinear stochastic heat equation with additive space–time white noise on the one-dimensional domain [0,1] over the time interval [0,T ] with TZ1, i.e. vu v2 u _ t; ð4:1Þ Z 2 C f ðuÞ C W vt vx with the Dirichlet boundary condition and the initial value u(x,0)Zu 0(x) for x2(0,1). Here, HZL2[0,1] and the linear operator A is the one-dimensional Laplace operator with the Dirichlet boundary condition AZDdD(D)/H, where DðDÞ Z H 2 ð0; 1Þh H01 ð0; 1Þ; given by
x 2 ð0; 1Þ; ðDuÞðxÞ du 00 ðxÞ; 00 for all u2D(D), where u denotes the second weak derivative of the function u. The eigenfunctions en in H and eigenvalues Kln of D are given by pffiffiffi ln Z n2 p2 ; en ðxÞ Z 2sinðnpxÞ; for all nR1. Hence ðDuÞðxÞ Z
N X
Kn 2 p2 un en ðxÞ;
x 2 ð0; 1Þ;
nZ1
with
ð1
un Z hen ; ui Z uðxÞen ðxÞdx; P 0 2 4 for all u2D(D) with DðDÞZ fv 2 H j N nZ1 jhen ; vij n !Ng. The noise Wt here is the space–time Wiener process Wt Z
N X
en bnt ;
nZ1
with qnh1 for all nR1 in view of assumption 2.2. (The summation here is just formal. In particular, it does not converge in H.) Therefore, we have gZ ð1=4ÞK3 with an arbitrary small 3O0 in our situation. Proc. R. Soc. A (2009)
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(b ) Consequences of theorem 3.1 applied to the SPDE (4.1) From theorem 3.1, the exponential Euler scheme (3.1) applied to the SPDE (4.1) has the error estimate
2 1=2 logðM Þ N ;M Kð1=2ÞC3 sup EUt k KVk %C N C : ð4:2Þ M kZ0; . ; M Since M$N arithmetical operations, random number and function evaluations are needed to calculate VkN ;M , the computational cost of the scheme (3.3) is KZN$M. In view of the above error bound, it is optimal to choose N Z K 2=3 and M Z K 1=3 and we have the optimal error bound 2 1=2 K 2=3 ;K 1=3 sup EUt k KVk % C $K ðKð1=3ÞC3Þ : ð4:3Þ kZ0; . ; M
Hence, the convergence rate of the scheme with respect to the computational cost is (1/3)K3 for an arbitrary small 3O0. ;M Note that, in general, if a numerical solution W N for some scheme with k computational cost N$M applied to this SPDE converges in the sense
2 1=2 1 1 ;M sup EUt k KW N % C C ; ð4:4Þ k Na Mb kZ0; . ; M for a, bO0, then it converges with a rate of ab=ðaC bÞ with respect to the computational cost. In our situation, we have aZ ð1=2ÞK3 and bZ1K3, so we obtain the overall rate (1/3)K3. (c ) Articles in the literature on the SPDE (4.1) We briefly review some articles in the literature concerning the numerical approximation of the SPDE (4.1) and the more general SPDE (1.1). Gyo¨ngy & Nualart (1995) introduced an implicit numerical scheme for the SPDE (4.1) and showed that it converges strongly to the exact solution without giving a rate (see their theorem 2.1). Grecksch & Kloeden (1996) considered a parabolic SPDE driven by a (possibly multiplicative) scalar Wiener process and applied explicit Itoˆ–Taylor schemes (see Kloeden & Platen 1992) to the Galerkin SDEs, while Kloeden & Shott (2001) applied linear-implicit Itoˆ–Taylor schemes to the Galerkin SDEs. In both cases, a higher order of strong convergence was obtained, but this was due to the special nature of the noise. In the general situation with space–time white noise, these schemes only converge with the order of 1/6. Gyo¨ngy & Nualart (1997) considered an SPDE with multiplicative space–time white noise and applied a temporal numerical scheme, which is based on the simulation of random variables of the form ð t kC1 sðs; Xk ÞdWs ; tk
where s is the diffusion coefficient and Xk is the numerical scheme at time tk. They showed that this scheme converges with a rate of 1/8 in time (see their theorem 3.1). Proc. R. Soc. A (2009)
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Shardlow (1999) applied finite differences to the SPDE (4.1) to obtain a spatial discretization that he then discretized in time with a q-method. This had an overall convergence rate 1/6 with respect to the computational cost (see his theorem 3.7). Gyo¨ngy (1998, 1999) also applied finite differences for an SPDE driven by space–time white noise and then used several temporal implicit and explicit schemes, in particular, the linear-implicit Euler scheme. He showed that these schemes converge with order 1/2 in space and with order 1/4 in time (assuming a smooth initial value). Hence, he obtained an overall convergence rate of 1/6 with respect to the computational cost in space and time (see theorem 3.1(iii) in Gyo¨ngy 1999). In a seminal paper, Davie & Gaines (2000) showed that any numerical scheme applied to the SPDE (4.1) with fZ0 which uses only equidistant values of the noise Wt cannot converge faster than the rate of 1/6 with respect to the computational cost (see §2.1 in Davie & Gaines 2000). Moreover, for the special situation with fZ0, they simulated the solution exactly (see also Mu ¨llerGronbach et al. 2007). Mu ¨ller-Gronbach & Ritter (2007) also showed that this is a lower bound for the convergence rate for the SPDE above, but with multiplicative noise. (They even showed that in one dimension (see equation (4.1)), one cannot improve this rate of convergence by choosing non-uniform evaluations of the noise.) However, Davie & Gaines (2000, p. 129) remarked that it may be possible to improve the convergence rate by using suitable linear functionals of the noise, which is essentially what we have done in this paper. Hausenblas (2003) applied the linear-implicit and explicit Euler scheme and the Crank–Nicholson scheme to an SPDE (4.1) driven by an infinite dimensional noise. In the case of a smoother noise, i.e. trace-class noise, she obtained the order 1/4 with respect to the computational cost. However, in the general case of space–time white noise, the convergence rate is no better than 1/6. Finally, Lord & Rougemont (2004) also considered the SPDE (4.1) with a smoother noise. They discretized the Galerkin SDE in time with a numerical scheme that also uses the factor eAN t with ANZPNA for AZD. The Lord– Rougemont scheme is given by N ;M N XkC1 Z eAN h XkN ;M C hfN ðXkN ;M Þ C W N ð4:5Þ t kC1 KW t k ; where XkN ;M is the numerical solution at time tk. They showed that this scheme is useful when the noise is very smooth in space, in particular, with Gevrey regularity. However, in the general case of space–time white noise, the scheme (4.5) converges also only with the rate of 1/6 with respect to the computational cost, which is a consequence of the work of Davie & Gaines (2000) and can also be seen in the numerical simulations below. (d ) Numerical results We illustrate theorem 3.1 with a numerical example. In particular, we consider the semilinear stochastic heat equation (4.1) on the one-dimensional domain (0,1) with f ðuÞZ ð1=2Þu, i.e. vu v2 u 1 _ t; Z 2 C u CW vt 2 vx Proc. R. Soc. A (2009)
ð4:6Þ
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10–1
error
10–2
10–3
10– 4
10–5 101
102 time steps M
103
Figure 1. Mean square error versus time steps to approximate the SDE (3.2) with NZ100. Thin solid line, linear-implicit Euler; dot-dashed line, Lord–Rougemont; thick solid line, exponential Euler; dotted line, orderlines 0.25 and 1.
with the Dirichlet boundary condition and the initial value u 0 ðxÞZ P N nZ1 ð1=nÞen ðxÞ, TZ1. Of course, in general, f will be nonlinear, but we choose f to be linear here since we have an exact solution for comparison with the numerical solution. We applied the linear-implicit Euler scheme (with just the Laplacian part implicit) to the SPDE (4.6), which is given by ;M N ;M N ;M K1 N N EN C W ; Z ðI K hA Þ E C hf E KW t t N kC1 k k kC1 k and the Lord–Rougemont scheme (4.5) as well as the exponential Euler scheme (3.3). The spatial error of the Galerkin projection, i.e. the difference of equation (2.1) and (3.2), is of the order of 1/2, so we first focus on the convergence rate in the time of the numerical schemes above. We fix NZ100 (space discretization) and then apply the above schemes with different MZ10, 20, 25, 40, 50, 80, 100, 200, 500 and 1000 (time discretization). Figure 1 shows the error as log–log plot, where the expectation in the error is approximated by the mean of 100 independent realizations. The linear-implicit Euler and Lord–Rougemont schemes clearly converge with the rate 1/4 in time, while the exponential Euler scheme converges with the temporal rate 1. Now, we focus on their convergence rate with respect to their computational cost. Here, figure 2 shows the error as log–log plot, where the expectation in the error is approximated again by the mean of 100 independent realizations. Here, the linear-implicit Euler and Lord–Rougemont schemes clearly converge with the rate 1/6, while the exponential Euler scheme converges with the rate 1/3. All the three schemes thus converge with their theoretically predicted order. Proc. R. Soc. A (2009)
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error
1
10–1
10–2 102
103 computational effort
104
Figure 2. Mean square error versus computational effort to approximate the SPDE (1.1). Thin solid line, linear implicit Euler; dot-dashed line, Lord–Rougemont; thick solid line, exponential Euler; dotted line, orderlines 1/6 and 1/3.
Finally in figure 3, we present the approximation of the solution of equation (4.6) computed by the numerical scheme introduced in that paper at the four different times tZ0, 1/1000, 1/100 and tZTZ1. 5. Proof of theorem 3.1 In what follows, C is a constant which changes from line to line. Let N, M2N be arbitrary with MR2. We divide the proof into two parts. Our main goal is to show
logðM Þ N ;M Kg sup Ut k KVk 2 % C lN C ; ð5:1Þ L ðUÞ M kZ0; .; M where j$jL 2 ðUÞ denotes the L2(U)-norm, given by jXjL2 ðUÞ Z ðEjXj2 Þ1=2 ; for an H-valued random variable X. First, we will show in §5a that the spatial error satisfies C ð5:2Þ sup Ut k KUtNk L2 ðUÞ % g ; lN kZ0; .; M and then, in §5b, that the temporal error satisfies C logðM Þ sup UtNk KVkN ;M 2 % : L ðUÞ M kZ0; .; M Combining both results will then give the desired estimate (5.1). Proc. R. Soc. A (2009)
ð5:3Þ
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Numerical approximation of SPDEs (a)
(b)
solution
2 1 0 –1 (c)
(d )
solution
2 1 0 –1
0
0.5 space
0
1.0
0.5 space
1.0
Figure 3. A single realization of the exponential Euler scheme (3.3) at times (a) tZ0, (b) tZ1/1000, (c) tZ1/100, (d ) tZ1 and with MZ10.000 time steps in NZ500 dimensions.
(a ) Spatial error Before we begin, we note the inequality
Kg Kg jvjg Z jðI K PN Þvj% jðI K PN ÞðKAÞ j$jvjg Z sup ðli Þ iRNC1
g
1 jvj ; ðlNC1 Þg g
g
for all v2D((KA) ), where jvjgZj(KA) vj. We have ðt ðt AN t N PN Ut Z e u 0 C expðAN ðtKsÞÞfN ðUs Þds C expðAN ðtKsÞÞdW N s ; 0
0
from which we obtain PN Ut KUtN
ðt Z
0
expðAN ðtKsÞÞ fN ðUs ÞK fN UsN ds;
which implies Ut KUtN % jUt K PN Ut j C PN Ut KUtN %ðlNC1 Þ Kg jUt jg C
ðt 0
%ðlNC1 Þ Kg jUt jg C L Proc. R. Soc. A (2009)
jeAðtKsÞ jfN ðUs ÞK fN UsN ds
ðt 0
Us KUsN ds;
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due to assumption 2.4. From this, it follows that: ð t
2 N 2 K2g 2 2 N E Ut KUt % 2ðlNC1 Þ EjUt jg C 2L E Us KUs ds 0
%2ðlNC1 Þ K2g EjUt j2g C 2L2 T
ðt 0
2 EUs KUsN ds;
where we have used the Ho ¨lder inequality. The Gronwall lemma then yields
1 N 2 2L2 T 2 2 sup EjUt jg ; E Ut KUt % 2e ðlNC1 Þ2g 0%t%T and therefore Ut KUtN
2
L T L2 ðUÞ % 2e
2
sup jðKAÞg Ut jL2 ðUÞ
0%t%T
1 C % ; ðlNC1 Þg ðlNC1 Þg
due to equation (2.2). This proves the estimate (5.2). (b ) Time discretization Now let k 2 {0, 1, ., MK1} be arbitrary. We have N ;M VkC1
AN t kC1
Ze
uN 0
ð t kC1 C 0
C
k ð t lC1 X tl
lZ0
expðAN ðtkC1 KsÞÞfN VlN ;M ds
expðAN ðtkC1 KsÞÞdW N s
and UtNkC1
AN t kC1
Ze
ð t kC1 C 0
uN 0
ð t kC1 C 0
expðAN ðtkC1 KsÞÞfN UsN ds
expðAN ðtkC1 KsÞÞdW N s :
Thus, we need to estimate N ;M N ;M VkC1 KUtNkC1 2 % VkC1 KX
L2 ðUÞ
L ðUÞ
C X KUtNkC1 L2 ðUÞ ;
where X de
At kC1
uN 0
ð t kC1 C 0
Proc. R. Soc. A (2009)
C
k ð t lC1 X lZ0
tl
expðAN ðtkC1 KsÞÞfN UtNl ds
expðAN ðtkC1 KsÞÞdW N s :
ð5:4Þ
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Of course, we have N ;M VkC1 KX
L2 ðUÞ
We will now show that
%
k X lZ0
N ;M N LhVl KUtl
X KUtN kC1
L2 ðUÞ
L2 ðUÞ %
:
C logðM Þ ; M
which by inequality (5.4), implies that k C logðM Þ X N ;M LhVlN ;M KUtNl 2 : C VkC1 KUtNkC1 2 % L ðUÞ L ðUÞ M lZ0 Thus, by Gronwall’s Lemma, we have N ;M VkC1 KUtNkC1
L2 ðUÞ
%
C logðM Þ ; M
which implies the estimate (5.3). Hence, it remains to estimate the quantity X KUtNkC1 L2 ðUÞ . For this, we use the Taylor Expansion ð 1
fN UsN Z fN UtNl C fN0 UtNl C r UsN KUtNl dr UsN KUtNl 0 N N Z fN Ut l C Il;s Us KUtNl Z fN UtNl C Il;s ðexpðAN ðs K tl ÞÞKI ÞUtNl ðs ðs N C Il;s expðAN ðs KrÞÞfN ðUr Þdr C Il;s expðAN ðs KrÞÞdWr ; tl
tl
with
ð1 Il;s d
We obtain jX KUtNkC1 j
Zj
0
fN0 UtNl C r UsN KUtNl dr:
k ð t lC1 X lZ0
tl
expðAN ðtkC1 KsÞÞIl;s UsN KUtNl dsj;
%E1 C E2 C E3 C E4 ; with
X k ð t lC1 E1 Z expðAN ðtkC1 KsÞÞIl;s ðexpðAN ðs K t l ÞÞKI ÞUtNl ds; lZ0 t l ðs X k ð t lC1 N E2 Z expðAN ðtkC1 KsÞÞIl;s expðAN ðs K rÞÞfN Ur dr ds; lZ0 t l tl ð X k ð t lC1 N s 0 E3 Z expðAN ðtkC1 KsÞÞf N Utl expðAN ðs K rÞÞdWr ds; lZ0 t l tl ð X k ð t lC1 N s 0 E4 Z expðAN ðtkC1 KsÞÞ Il;s Kf N Ut l expðAN ðs KrÞÞdWr ds: lZ0 t l tl
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In §5b(i)–(iv), we show that jEi jL2 ðUÞ % ðC logðM ÞÞ=M for iZ1, ., 4, from which will finally follow that: X KUtN 2 % C logðM Þ ; kC1 L ðUÞ M and this implies inequality (5.3). (i) First term First, we consider the E1 term. k ð t lC1 X expðAN ðtkC1 KsÞÞIl;s ðexpðAN ðs K t l ÞÞKI ÞUtN ds: E1 % l tl
lZ0
To begin, we have ð t kC1 N t jexpðAN ðtkC1 KsÞÞIl;s ðexpðAN ðsKtl ÞÞKI ÞUt l jds k
%C j L2 ðUÞ
Ð t kC1 tk
jUtNl jdsjL2 ðUÞ ;
%Ch;
due to assumption 2.4. Hence kK1 ð tlC1 X N 2 jE1 jL ðUÞ % Ch C expðAN ðtkC1 KsÞÞIl;s ðexpðAN ðs K tl ÞÞKI ÞUtl ds lZ0
L2 ðUÞ
tl
kK1 ð tlC1 X %Ch C jAN expðAN ðtkC1 KsÞÞj tl lZ0
K1 !AN Il;s ðexpðAN ðs K tl ÞÞKI ÞUtNl ds
L2 ðUÞ
kK1 ð tlC1 X K1 K1 N %Ch C C t ðtk K tl Þ AN ðexpðAN ðs K tl ÞÞKI ÞUtl ds l
lZ0
kK1 ð tlC1 X K1 N %Ch C C t ðk KlÞ Utl ds
!
ð t lC1 N K1 ðk KlÞ Utl ds t
!
%Ch C C
kK1 X lZ0
L2 ðUÞ
l
lZ0
due to assumption 2.4. Finally, we obtain jE1 jL2 ðUÞ % Ch C Ch
kK1 X
ðk KlÞ
lZ0
Z Ch C Ch
k X 1 lZ1
which is the claim for E1. Proc. R. Soc. A (2009)
;
L2 ðUÞ
l
l
! K1
! % Ch
M X 1 lZ1
l
! %
C logðM Þ ; M
! L2 ðUÞ
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663
Remark 5.1. In the literature, the term E1 is usually estimated in the following way: k ð tlC1 X expðAN ðtkC1 KsÞÞIl;s ðexpðAN ðs K tl ÞÞKI ÞUtN ds E1 % l tl
lZ0
%C
k ð tlC1 X ðexpðAN ðs K tl ÞÞKI ÞUtN ds l lZ0
ZC
tl
k ð tlC1 X lZ0
tl
jðexpðAN ðs K tl ÞÞKI ÞðKAN Þg jðKAN Þ Kg UtNl ds
k ð tlC1 X ðKAN Þ Kg UtN ds; %Ch l g
lZ0
tl
which yields jE1 jL2 ðUÞ % Chg : In this way, one can only obtain a convergence rate of g in time, which for our example, would be gZ ð1=4ÞK3, 3O0 small. To obtain a higher order, we need to use the smoothening effect of the term expðAN ðt kC1 KsÞÞ as we did earlier, which is based on the estimate AN eAN t % C 1 : t The situation is similar for the estimates of the terms E3 and E4. (ii) Second term We need the estimate j f ðvÞj% C ð1 C jvjÞ;
ð5:5Þ
for all v2H, for which this is true, since by assumption 2.4, jf ðvÞj% jf ðvÞKf ð0Þj C jf ð0Þj% Ljvj C C % C ð1 C jv jÞ; for all v2H. Thus for E2, we obtain E2 %
k ð t lC1 X lZ0
tl
jexpðAN ðtkC1 KsÞÞIl;s j
ðs tl
expðAN ðs KrÞÞfN UrN dr ds
! ! k ð t lC1 ð s k ð t lC1 ð s X X N N fN ðUr Þdr ds % C 1 C Ur dr ds : %C lZ0
Proc. R. Soc. A (2009)
tl
tl
lZ0
tl
tl
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A. Jentzen and P. E. Kloeden
Therefore, we have ð t ð s
2 ! k X lC1 N EjE2 j % CM E 1 C Ur dr ds 2
lZ0
tl
tl
! k ð t lC1 ð s X N 2 1 C E Ur %Ch dr ds % Ch2 ; lZ0
tl
tl
which implies jE2 jL2 ðUÞ % Ch: (iii) Third term Here, we have ð t kC1 Ej
tk
expðAN ðtkC1 KsÞÞfN0
N Utl
ðs tk
2
expðAN ðs KrÞÞdWr dsj
ð t
2 ðs kC1 %C $E j expðAN ðs K rÞÞdWr jds tk
tk
ð t kC1 ð s %Ch$ Ej expðAN ðs KrÞÞdWr j2 ds% Ch2 ; tk
tk
due to assumption 2.4 and inequality (2.3). Therefore, we obtain 2 ð kK1 ð tlC1 X N s 2 2 0 EjE3 j % Ch C C $E expðAN ðtkC1 KsÞÞf N Utl expðAN ðs K rÞÞdWr ds lZ0 tl tl 2 ð kK1 ð t lC1 X s Z Ch 2 C C E expðAN ðtkC1 KsÞÞf N0 UtNl expðAN ðs KrÞÞdWr ds tl
lZ0
tl
2 ! ðs k K1 ð t lC1 X 2 K1 Kð1=2Þ %Ch C Ch E ðtkC1 KsÞ ðKAN Þ expðAN ðs K rÞÞ dWr ds lZ0
%Ch2 C C
kK1 X lZ0
%Ch2 C Ch 2
tl
1 ðk KlÞ
k K1 X lZ0
tl
2 ! ð t lC1 ðs EðKAN Þ Kð1=2Þ expðAN ðs K rÞÞdWr ds tl
!
!
tl
M X 1 1 % Ch 2 % Ch 2 logðM Þ; ðk KlÞ l lZ1
due to assumption 2.4 again. Finally, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C logðM Þ C logðM Þ % : jE3 jL2 ðUÞ % M M Proc. R. Soc. A (2009)
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(iv) Fourth term Note that Il;s KfN0 UtNl Z
ð 1 0
fN00 UtNl C r UsN KUtNl ð1KrÞdr UsN KUtNl :
Hence, assumption 2.4 implies that ð
K1 s AN Il;s KfN0 UtN expðAN ðs KrÞÞdWr l tl
ðs N Kð1=2Þ N Kð1=2Þ AN ðsKrÞ %C ðKAN Þ Us KUt l ðKAN Þ exp dWr :
ð5:6Þ
tl
Now, we estimate E4. Since ð t ð kC1 N s 0 expðAN ðs KrÞÞdWr ds t expðAN ðtkC1 KsÞÞ Il;s KfN Ut l t k
L2 ðUÞ
l
ð t ð kC1 N s N %C Us KUt l expðAN ðs K rÞÞdWr ds t t k
l
L2 ðUÞ
2 1=2 ð pffiffiffi ð t kC1 N 2 s N %C h E Us KUt l expðAN ðs KrÞÞdWr ds tl
tk
2 ð s expðAN ðs KrÞÞdWr 2 L ðUÞ t
pffiffiffi ð t kC1 N 2 %C h Us KUtNl tk
!1=2 ds
L2 ðUÞ
l
%Ch; and since 2 ð t ðs lC1 N Kð1=2Þ N Kð1=2Þ Us KUt l ðKAN Þ expðAN ðs K rÞÞdWr ds E ðKAN Þ tl
ð t lC1 2 %Ch E ðKAN Þ Kð1=2Þ UsN KUtNl
tl
tl
2
ðs Kð1=2Þ expðAN ðs KrÞÞdWr !ðKAN Þ tl ð t lC1 N 2 Kð1=2Þ N %Ch Us KUt l ðKAN Þ tl
L2 ðUÞ
2 ðs !ðKAN Þ Kð1=2Þ expðAN ðs K rÞÞdWr tl
L2 ðUÞ
Proc. R. Soc. A (2009)
ds% Ch4 ;
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666
A. Jentzen and P. E. Kloeden
holds for lZ0, 1, . , kK1 due to the inequalities (2.2) and (2.3), we obtain kK1 ð tlC1 X expðAN ðtkC1 KsÞÞ Il;s KfN0 UtN jE4 jL2 ðUÞ % Ch C l tl
lZ0
ðs ! tl
expðAN ðs KrÞÞdWr jds
L2 ðUÞ
kK1 ð tlC1 X K1 %Ch C Il;s KfN0 UtNl t jAN expðAN ðtkC1 KsÞÞj A l
lZ0
ðs ! tl
expðAN ðs KrÞÞdWr jds
L2 ðUÞ
%Ch C C
kK1 X
ðtk K tl Þ
ð t lC1 N Kð1=2Þ N Us KUtl ðKAÞ Kð1=2Þ t ðKAN Þ
K1
l
lZ0
ðs ! tl
expðAN ðs KrÞÞdWr jds
Z Ch C Ch
% Ch C Ch
2
L2 ðUÞ
kK1 X
kK1 X
! ðtk K tl Þ
K1
lZ0
! ðk KlÞ
K1
lZ0
%Ch C Ch
k X 1 lZ1
l
! % Ch logðM Þ Z
in view of inequality (5.6).
C logðM Þ ; M &
We thank the referees for their valuable comments. This work was supported by the DFG project ‘Pathwise numerical analysis of stochastic evolution equations’.
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Proc. R. Soc. A (2009)
