A random regularized approximate solution of the inverse problem for the Burgers' equation

TL;DR

This paper develops a regularized method combining quasi-reversibility, truncated expansion, and nonparametric regression to approximate solutions for the ill-posed inverse Burgers' equation problem, analyzing convergence rates.

Contribution

It introduces a novel regularization approach for the inverse Burgers' equation using random perturbations and nonparametric regression, enhancing solution stability.

Findings

Proposes a regularized solution method for the inverse Burgers' equation.

Analyzes the convergence rate of the regularized solution.

Demonstrates improved stability in the inverse problem solution.

Abstract

In this paper, we find a regularized approximate solution for an inverse problem for the Burgers' equation. The solution of the inverse problem for the Burgers' equation is ill-posed, i.e., the solution does not depend continuously on the data. The approximate solution is the solution of a regularized equation with randomly perturbed coefficients and randomly perturbed final value and source functions. To find the regularized solution, we use the modified quasi-reversibility method associated with the truncated expansion method with nonparametric regression. We also investigate the convergence rate.

Equations

$$\left\{\begin{array}[]{l l l}{\bf u}_{t}-(A(x,t){\bf u}_{x})_{x}&={\bf u}{\bf u}_{x}+G(x,t),&\qquad(x,t)\in\Omega\times(0,T),\\ {\bf u}(x,t)&=0,&\qquad x\in\partial{\Omega},\\ {\bf u}(x,T)&=H(x),&\qquad x\in{\Omega},\end{array}\right.$$

$$\tilde{H}(x_k)=H(x_k)+{\sigma }_k{ϵ}_k,{\tilde{G}}_k(t)=G(x_k,t)+\vartheta {\xi }_k(t),\text{for}k=\overline{1,n},$$

$${\tilde{A}}_k(t)=A(x_k,t)+\overline{\vartheta }{\xi }_k(t),\text{for}k=\overline{1,n}.$$

$$Af(x):=-\Delta f(x)=-\frac{{\partial }^{2}f(x)}{\partial x^2}.$$

$${\lambda }_0=0<{\lambda }_1\le {\lambda }_2\le {\lambda }_3\le \cdots \le {\lambda }_p\le \cdots$$

$$\{\begin{array}{ll}A{\phi }_p(x)=-{\lambda }_p{\varphi }_p(x),& x\in \Omega \\ {\partial }_x{\varphi }_p(x)=0,& x\in \partial \Omega .\end{array}$$

$$H^{\gamma}(\Omega)=\Bigg{\{}v\in L^{2}(\Omega):\sum\limits_{p=0}^{\infty}\lambda_{p}^{2\gamma}|\left\langle{v},{\phi_{p}}\right\rangle|^{2}<+\infty\Bigg{\}},$$

$$\Vert v{\Vert }_{H^{\gamma }(\Omega )}={\left(\sum _{p=1}^{\infty }{\lambda }_p^{2\gamma }|⟨v,{\varphi }_p⟩{|}^2\right)}^{1/2}.$$

$$\mathcal{W}_{\beta_{n}}=\Big{\{}{p}\in\mathbb{N}:|{p}|\leq\sqrt{\beta_{n}}\Big{\}}$$

$$\underset{|n|\to +\infty }{\lim }{\beta }_n=+\infty .$$

$$\widehat{H}_{\beta_{n}}({x})=\sum_{{p}\in\mathcal{W}_{\beta_{n}}}\Bigg{[}\frac{\pi}{n}\sum_{k=1}^{n}\widetilde{D}_{k}\psi_{p}(x_{k})\Bigg{]}\psi_{p}({x}),~{}~{}\widehat{G}_{\beta_{n}}({x},t)=\sum_{{p}\in\mathcal{W}_{\beta_{n}}}\Bigg{[}\frac{\pi}{n}\sum_{k=1}^{n}\widetilde{G}_{k}(t)\psi_{p}(x_{k})\Bigg{]}\psi_{p}({x}).$$

$$\displaystyle\begin{aligned} &{\bf E}\Big{\|}\widehat{H}_{\beta_{n}}-H\Big{\|}^{2}_{L^{2}(\Omega)}\leq\overline{C}(\mu_{0},H)\sqrt{\beta_{n}}n^{-4\mu_{0}}+{4\beta_{n}^{-\mu_{0}}}\Big{\|}H\Big{\|}^{2}_{\mathcal{H}^{\mu_{0}}(\Omega)},\\ &{\bf E}\Big{\|}\widehat{G}_{\beta_{\bf n}}(.,t)-G(.,t)\Big{\|}^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}\leq\overline{C}(\mu_{0},G)\sqrt{\beta_{n}}n^{-4\mu}+{4\beta_{n}^{-\mu_{0}}}\Big{\|}G\Big{\|}^{2}_{L^{\infty}(0,T;\mathcal{H}^{\mu_{0}}(\Omega))},\end{aligned}$$

$$\overline{C}(\mu_{0},H)=8\pi V_{max}^{2}\frac{2\pi^{1/2}}{\Gamma(1/2)}+\frac{16\mathcal{C}^{2}\mu_{0}\pi^{1/2}}{\Gamma(1/2)}\Big{\|}H\Big{\|}^{2}_{\mathcal{H}^{\mu_{0}}(\Omega)}.$$

$$\overline{C}(\mu_{0},G)=8\pi V_{max}^{2}\frac{2\pi^{1/2}}{\Gamma(1/2)}+\frac{16\mathcal{C}^{2}\mu\pi^{1/2}}{\Gamma(1/2)}\Big{\|}G\Big{\|}^{2}_{L^{\infty}(0,T;\mathcal{H}^{\mu_{0}}(\Omega))}.$$

$$\max\Bigg{(}\sqrt{\beta_{n}}n^{-4\mu_{0}},~{}\beta_{n}^{-\mu_{0}}\Bigg{)}.$$

$$\mathcal{Z}_{\gamma,B}(\Omega):=\Bigg{\{}f\in L^{2}(\Omega),\sum_{{p}\in\mathbb{N}}{p}^{2+2\gamma}e^{2B{p}^{2}}\big{\langle}f,\psi_{p}\big{\rangle}_{L^{2}(\Omega)}^{2}<+\infty\Bigg{\}},$$

$${\mathbf{P}}_{{\rho }_n}(v)$$

$$\Vert {\mathbf{P}}_{{\rho }_{\mathbf{n}}}(v){\Vert }_{L^2(\Omega )}\le {\rho }_n\Vert v{\Vert }_{L^2(\Omega )},$$

$$\Vert \mathbf{P}v-{\mathbf{P}}_{{\rho }_n}v{\Vert }_{L^2(\Omega )}\le A_1{\rho }_n^{-\gamma }{e}^{-T{\rho }_n}\Vert v{\Vert }_{{\mathcal{Z}}_{\gamma ,T{A}_1}(\Omega )}.$$

$$\Vert {\mathbf{P}}_{{\rho }_{\mathbf{n}}}(v){\Vert }_{L^2(\Omega )}^2$$

$$\displaystyle\leq\rho_{n}^{2}\sum_{{p}\leq\sqrt{\frac{\rho_{n}}{A_{1}}}}^{\infty}\big{\langle}v(x),\psi_{p}\big{\rangle}^{2}_{L^{2}(\Omega)}=\rho_{n}^{2}\|v\|_{L^{2}(\Omega)}^{2},$$

$$\Vert \mathbf{P}v-{\mathbf{P}}_{{\rho }_n}(v){\Vert }_{L^2(\Omega )}^2$$

$$\displaystyle\leq A_{1}^{2}\rho_{n}^{-2\gamma}e^{-2TA_{1}\rho_{n}}\sum_{{p}>\sqrt{\frac{\rho_{n}}{A_{1}}}}^{\infty}{p}^{4+4\gamma}e^{2TA_{1}{p}^{2}}\big{\langle}v(x),\psi_{p}\big{\rangle}^{2}_{L^{2}(\Omega)}$$

$$=A_1^2{\rho }_n^{-2\gamma }{e}^{-2T{\rho }_n}\Vert v{\Vert }_{{\mathcal{Z}}_{\gamma ,T{A}_1}(\Omega )}^2.$$

$$\{\begin{array}{c}\frac{\partial {\tilde{U}}_{{\rho }_n,{\beta }_n}}{\partial t}-{\left({\hat{A}}_{{\beta }_n}(x,t)\frac{\partial {\tilde{U}}_{{\rho }_n,{\beta }_n}}{\partial x}\right)}_x-\mathbf{P}{\tilde{U}}_{{\rho }_n,{\beta }_n}+{\mathbf{P}}_{{\rho }_n}{\tilde{U}}_{{\rho }_n,{\beta }_n}=\\ ={{\overline{\mathbf{F}}}^0}_{{\hat{Q}}_n}\left({\tilde{U}}_{{\rho }_n,{\beta }_n},\frac{\partial {\tilde{U}}_{{\rho }_n,{\beta }_n}}{\partial x}\right)+{\hat{G}}_{{\rho }_n}(x,t),0<t<T,\text{hfill}\\ {\tilde{U}}_{{\rho }_n,{\beta }_n}(x,t)=0,x\in \partial \Omega ,\text{hfill}\\ {\tilde{U}}_{{\rho }_n,{\beta }_n}(x,T)={\hat{H}}_{{\beta }_n}(x).\text{hfill}\end{array}$$

$$\widehat{A}_{\beta_{n}}(x,t)=\sum_{p\in\mathcal{W}_{\beta_{n}}}\Bigg{[}\frac{\pi}{n}\sum_{k=1}^{n}\widetilde{A}_{k}(t)\psi_{p}(x_{k})\Bigg{]}\psi_{p}(x)$$

$${{\overline{\mathbf{F}}}^0}_{{\hat{Q}}_n}(v,\hat{v}):=\{\begin{array}{ll}{\hat{Q}}_n^2,& \max \{v,\hat{v}\}\in ({\hat{Q}}_n,+\infty ),\\ v\hat{v},& \max \{v,\hat{v}\}\in [-{\hat{Q}}_n,{\hat{Q}}_n],\\ {\hat{Q}}_n^2,& \max \{v,\hat{v}\}\in (-\infty ,-{\hat{Q}}_n).\end{array}$$

$$\widehat{Q}_{n}\geq\max\Big{(}\|\textbf{u}\|_{L^{\infty}((0,T);L^{2}(\Omega))},\|\textbf{u}_{x}\|_{L^{\infty}((0,T);L^{2}(\Omega))}\Big{)}.$$

$$\displaystyle\Big{|}{\bf\overline{F}^{0}}_{\widehat{Q}_{n}}(v,\widehat{v})-{\bf\overline{F}^{0}}_{\widehat{Q}_{n}}(w,\widehat{w})\Big{|}\leq\widehat{Q}_{n}\Big{(}|v-\widehat{v}|+|w-\widehat{w}|\Big{)}.$$

$$\displaystyle\Big{|}{\bf\overline{F}^{0}}_{\widehat{Q}_{n}}(v,\widehat{v})-{\bf\overline{F}^{0}}_{\widehat{Q}_{n}}(w,\widehat{w})\Big{|}$$

$$\le {\hat{Q}}_n|w+{\hat{Q}}_n|+|w||\hat{w}+{\hat{Q}}_n|$$

$$\displaystyle\leq\widehat{Q}_{n}\Big{(}|w+\widehat{Q}_{n}|+|\widehat{w}+\widehat{Q}_{n}|\Big{)}\leq\widehat{Q}_{n}\Big{(}|w-v|+|\widehat{w}-\widehat{v}|\Big{)}.$$

$$\displaystyle\Big{|}{\bf\overline{F}^{0}}_{\widehat{Q}_{n}}(v,\widehat{v})-{\bf\overline{F}^{0}}_{\widehat{Q}_{n}}(w,\widehat{w})\Big{|}$$

Full text

1 Introduction

In this work, we consider the backward in time problem for 1-D Burgers’ equation

[TABLE]

where $\Omega=(0,\pi)$. The Burgers equation is a fundamental partial differential equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow [7].

One can see that the term $(A(x,t){\bf u}_{x})_{x}$ is $\Delta{\bf u}={\bf u}_{xx}$ if $A=1$. However, one can not use spectral methods to study the operator $(A(x,t){\bf u}_{x})_{x}$ So, the problem is more difficult. The second observation is that for the equation ${\bf u}_{t}-(A(x,t){\bf u}_{x})_{x}=f({\bf u},{\bf u}_{x})$ when $A(x,t)$ is deterministic and $f({\bf u},{\bf u}_{x})=f({\bf u})$, the problem is a consequence of Theorem 4.1 in our recent paper [4]. However, if $A(x,t)$ is randomly perturbed and $f({\bf u},{\bf u}_{x})$ depends on ${\bf u}$ and ${\bf u}_{x}$ then the problem is more challenging.

Until now, the deterministic Burgers’ equation with the randomly perturbed case have not been studied. Hence, the paper is the first study of Burgers’ equation backward in time. The inclusion of the gradient term in $uu_{x}$ in the right hand side of the Burgers’ equation makes the Burgers’ equation more difficult to study. We need to find an approximate function for ${\bf u}{\bf u}_{x}$. This task is nontrivial.

This paper is a continuation of our study of backward problems in the two recent papers [4, 5]. In those papers the equations did not have random coefficients in the main equations. The paper [4] does not consider the random operator. The paper [5] considers the simple coefficient $A(x,t)=A(t)$ and the source function is ${\bf u}-{\bf u}^{3}$. Hence, one can see that the Burgers’ equation considered here is more difficult since the gradient term in the right hand side and the coefficient $A(x,t)$ depends on both $x$ and $t$.

It is known that the backward problem mentioned above is ill-posed in general [7], i.e., a solution does not always exist. When the solution exists, the solution does not depend continuously on the given initial data. In fact, from a small noise of a physical measurement, the corresponding solution may have a large error. This makes the numerical computation troublesome. Hence a regularization is required. It is well-known that there are some difficulties to study the nonlocal Burger’s equation. First, by the given form of coefficient $A(x,t)$ in the main equation (1.1), the solution of Problem (1.1) can not be transformed into a nonlinear integral equation. Hence, classical spectral method cannot be applied. The second thing that makes the Burger’s equation more difficult to study is the gradient term ${\bf u}_{x}$ in the right hand side. Until now, although there are limited number of works on the backward problem for Burgers’ equation [1, 3], there are no results for regularizing the problem.

As is well-known, measurements always are given at a discrete set of points and contain errors. These errors may be generated from controllable sources or uncontrollable sources. In the first case, the error is often deterministic. If the errors are generated from uncontrollable sources as wind, rain, humidity, etc., then the model is random. Methods used for the deterministic cases cannot be applied directly to the random case.

In this paper, we consider the following model as follows

[TABLE]

and

[TABLE]

where $x_{k}=\pi\dfrac{2k-1}{2n}$ and $\epsilon_{k}$ are unknown independent random errors. Moreover, $\epsilon_{k}\sim\mathcal{N}(0,1)$, and $\sigma_{k},~{}\vartheta,\overline{\vartheta}$ are unknown positive constants which are bounded by a positive constant $V_{max}$, i.e., $0\leq\sigma_{k}<V_{\text{max}}$ for all $k=1,\cdots,n$. $\xi_{k}(t)$ are Brownian motions. The noises $\epsilon_{k},\xi_{k}(t)$ are mutually independent. Our task is reconstructing the initial data ${u}(x,0)$.

We next want to mention about the organization of the paper and our methods in this paper. We prove some preliminary results in section 2. We state and prove our main result in section 3. The existence and uniqueness of solution of equation (1.1) is an open problem, and we do not investigate this problem here. For inverse problem, we assume that the solution of the Burgers’ equation (1.1) exists. In this case its solution is not stable. In this paper we establish an approximation of the backward in time problem for 1-D Burgers’ equation (1.1) with the solution of a regularized equation with randomly perturbed equation (2.18). The random perturbation in equation (2.18) is explained in equations (1.2), (1.3), (2.18) and (2.19).

2 Some Notation

We first introduce notation, and then state the first set of our main results in this paper. We define fractional powers of the Neummann-Laplacian.

[TABLE]

Since $A$ is a linear densely defined self-adjoint and positive definite elliptic operator on the connected bounded domain $\Omega=(0,\pi)$ with Dirichlet boundary condition, the eigenvalues of $A$ satisfy

[TABLE]

with $\lambda_{p}=p^{2}\to\infty$ as $p\to\infty$. The corresponding eigenfunctions are denoted respectively by $\varphi_{p}(x)=\sqrt{\frac{2}{\pi}}\sin(px)$. Thus the eigenpairs $(\lambda_{p},\phi_{p})$, $p=0,1,2,...$, satisfy

[TABLE]

The functions $\varphi_{p}$ are normalized so that $\{\phi_{p}\}_{p=0}^{\infty}$ is an orthonormal basis of $L^{2}(\Omega)$.

Defining

[TABLE]

where $\left\langle{\cdot},{\cdot}\right\rangle$ is the inner product in $L^{2}(\Omega)$, then $H^{\gamma}(\Omega)$ is a Hilbert space equipped with norm

[TABLE]

First, we state following Lemmas that will be used in this paper

Theorem 2.1** (Theorem 2.1 in [4]).**

Define the set $\mathcal{W}_{\beta_{n}}$ for any $n\in\mathbb{N}$

[TABLE]

where $\beta_{n}$ satisfies

[TABLE]

For a given ${n}$ and $\beta_{n}$ we define functions that are approximating $H,G$ as follows

[TABLE]

Let us choose $\mu_{0}>\frac{1}{2}$. If $H\in\mathcal{H}^{\mu_{0}}(\Omega)$ and $G\in L^{\infty}(0,T;\mathcal{H}^{\mu_{0}}(\Omega))$ then the following estimates hold

[TABLE]

where

[TABLE]

and

[TABLE]

Corollary 2.1** (Corollary 2.1 in [4]).**

Let $H,G$ be as in Theorem (2.1). Then the term {\bf E}\Big{\|}\widehat{H}_{\beta_{n}}-H\Big{\|}_{L^{2}(\Omega)}^{2}+T{\bf E}\Big{\|}\widehat{G}_{\beta_{n}}-G\Big{\|}_{L^{\infty}(0,T;L^{2}(\Omega))}^{2} is of order

[TABLE]

Lemma 2.1**.**

Define the following space of functions

[TABLE]

for any $\gamma\geq 0$ and $B\geq 0$. Define also the operator ${\bf P}=A_{1}\Delta$ and ${\bf P}_{\rho_{n}}$ is defined as follows

[TABLE]

for any function $v\in L^{2}(\Omega)$. Then for any $v\in L^{2}(\Omega)$

[TABLE]

and for $v\in\mathcal{Z}_{\gamma,TA_{1}}(\Omega)$ then

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Proof.

First, for any $v\in L^{2}(\Omega)$, we have

[TABLE]

and

[TABLE]

∎

Now, we can assume that $\widehat{A}_{\rho_{n}}(x,t),A(x,t)\leq A_{0}$ for all $(x,t)\in\Omega\times(0,T)$ and we choose $A_{1}>A_{0}$. We describe our regularized problem by defining the following problem

[TABLE]

Here $\widehat{A}_{\beta_{n}}$ is defined by

[TABLE]

where $\psi_{p}(x)=\sqrt{\frac{2}{\pi}}\sin(px)$. Noting as above, the function $F({\bf u},{\bf u}_{x})={\bf u}{\bf u_{x}}$ in the first equation of Problem (1.1) is locally Lipschitz function and is approximated by the function ${\bf\overline{F}^{0}}_{\widehat{Q}_{n}}\left(\widetilde{U}_{\rho_{n}},\frac{\partial\widetilde{U}_{\rho_{n}}}{\partial x}\right)$ in the first equation of Problem (2.18) where

[TABLE]

Here the function $\widehat{Q}_{n}$ is increasing function and $\lim_{n\rightarrow+\infty}\widehat{Q}_{n}=+\infty$. For a sufficiently large $n>0$ such that

[TABLE]

We show that ${\bf\overline{F}^{0}}_{\widehat{Q}_{n}}$ is a globally Lipschitz function by the following Lemma

Lemma 2.2**.**

For any $(v,\widehat{v})\in\mathbb{R}^{2},(w,\widehat{w})\in\mathbb{R}^{2}$, we obtain

[TABLE]

Proof.

We divide the proof into 5 cases:

Case 1. If $\max\{v,\widehat{v}\}<-\widehat{Q}_{n}$ and $\max\{w,\widehat{w}\}<-\widehat{Q}_{n}$ then it is easy to see that ${\bf\overline{F}^{0}}_{\widehat{Q}_{n}}(v,\widehat{v})-{\bf\overline{F}^{0}}_{\widehat{Q}_{n}}(w,\widehat{w})=0$.

Case 2. If $\max\{v,\widehat{v}\}<-\widehat{Q}_{n}\leq\max\{w,\widehat{w}\}\leq\widehat{Q}_{n}$ then using triangle inequality, we get

[TABLE]

Case 3. If $\max\{v,\widehat{v}\}<-\widehat{Q}_{n}<\widehat{Q}_{n}\leq\max\{w,\widehat{w}\}$ then

[TABLE]

Case 4. If $-\widehat{Q}_{n}<\max\{v,\widehat{v}\},~{}\max\{w,\widehat{w}\}\leq\widehat{Q}_{n}$ then

[TABLE]

Case 5. If $\max\{v,\widehat{v}\}>\widehat{Q}_{n}$ and $\max\{w,\widehat{w}\}>\widehat{Q}_{n}$ then

[TABLE]

By all cases above, we complete the proof of Lemma (2.2). ∎

3 Regularized solutions for backward problem for Burgers’ equation

Our main result in this paper is stated as follows

Theorem 3.1**.**

Let the functions $H\in\mathcal{H}^{\mu_{0}}(\Omega)$ and $A,G\in L^{\infty}(0,T;\mathcal{H}^{\mu_{0}}(\Omega))$, for $\mu_{0}>\frac{1}{2}$. Then problem (2.18) has unique solution ${\widetilde{U}_{\rho_{n}}}\in C([0,T];L^{2}(\Omega))$. Assume that Problem (1.1) has unique solution ${\bf u}\in L^{\infty}\left(0,T;\mathcal{Z}_{\gamma,TA_{1}}(\Omega)\right)$ . Let us choose $\widehat{Q}_{n}$ such that

[TABLE]

Then for $n$ large enough, ${\bf E}\|\widetilde{U}_{\rho_{n},\beta_{n}}(x,t)-\textbf{u}(x,t)\|^{2}_{L^{2}(\Omega)}$ is of order

[TABLE]

Proof.

Denote by

[TABLE]

The first equation of Problem (1.1) can be written as

[TABLE]

and the first equation of Problem (2.18) is rewritten as

[TABLE]

For $\kappa_{n}>0$, we put

[TABLE]

Then the last two equations, and a simple computation gives

[TABLE]

and $\mathbf{Y}_{\rho_{n},\beta_{n}}|_{\partial\Omega}=0,~{}\mathbf{Y}_{\rho_{n},\beta_{n}}(x,T)=\overline{H}_{\beta_{n}}(x)-H(x)$.

By taking the inner product of the two sides of the last equality with $\mathbf{Y}_{\rho_{n},\beta_{n}}$ and noting the equality

[TABLE]

one deduces that

[TABLE]

For $\widetilde{\mathcal{J}}_{12,n}$, we have the following

[TABLE]

where we used inequality (2.9). And for $\widetilde{\mathcal{J}}_{13,n}$, using Cauchy-Schwartz and (2.10), we have the following upper bound

[TABLE]

The Cauchy-Schwartz inequality leads to the following estimation

[TABLE]

For $\widetilde{\mathcal{J}}_{15,n}$ , we note that ${\bf\overline{F}^{0}}_{\widehat{Q}_{n}}({\bf u},{\bf u}_{x})={\bf u}{\bf u}_{x}$ and thanks to (2.21), we obtain

[TABLE]

where we note that $\left\|\mathbf{Y}_{\rho_{n},\beta_{n}}\right\|_{L^{2}(\Omega)}\leq\left\|\frac{\partial\mathbf{Y}_{\rho_{n},\beta_{n}}}{\partial x}\right\|_{L^{2}(\Omega)}$. This implies that

[TABLE]

The term \big{|}\widetilde{\mathcal{J}}_{16,n}\big{|} can be bounded by

[TABLE]

Combining all the previous estimates, we get

[TABLE]

By taking the integral from $t$ to $T$ and by a simple calculation yields

[TABLE]

where we used the fact that

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Let us choose $\kappa_{n}=\rho_{n}$ then we obtain

[TABLE]

Multiplying both sides of the last inequality by $e^{2\kappa_{n}T}$, we obtain

[TABLE]

Applying Gronwall’s inequality, we deduce that

[TABLE]

where

[TABLE]

Thanks to Theorem 2.1, we have that

[TABLE]

is of order \max\Big{(}\beta_{n}^{1/2}n^{-4\mu},\beta_{n}^{-\mu_{0}}\Big{)} for any $\mu>\frac{1}{2}$. This together with (3.29) implies that ${\bf E}\|\widetilde{U}_{\rho_{n},\beta_{n}}(.,t)-\textbf{u}(.,t)\|^{2}_{L^{2}(\Omega)}$ is of order

[TABLE]

∎