Stochastic Boundary Value Problems via Wiener Chaos Expansion ^†

Abstract

In this work, we study stochastic boundary value problems that arise in acoustics and linear elasticity via a Wiener chaos expansion. In particular, for both cases, we provide the appropriate variational formulation for the stochastic-source Helmholtz equation, as well as for the Navier equation with stochastic boundary data. The main idea is to reduce our stochastic problems into an infinite hierarchy of deterministic boundary value problems, for each of which an appropriate variational formulation is considered. Furthermore, we present well-posedness for the above hierarchy of deterministic problems, we give the appropriate linchpin frame with the stochastic problem and we exploit uniqueness and existence arguments for the weighted Wiener chaos solution. Finally, some useful remarks and conclusions are also given.

1. Introduction

In this paper, we study stochastic boundary value problems that arise in acoustics and linear elasticity. Our methodology is based upon the use of an appropriate Wiener chaos expansion for the Helmholtz equation with a stochastic source, as well as for the case of the Navier equation with stochastic boundary data. Although the corresponding deterministic problems have been widely studied, there is relatively little work for the corresponding stochastic problems required to incorporate the effects of randomness and uncertainty, which turn the original partial differential equation (PDE) problem into a stochastic partial differential equation (SPDE) [1].

The aim of this work is to establish the existence and uniqueness of solutions for stochastic boundary value problems due to the Helmholtz and Navier equations. Building on previous work on elliptic and parabolic equations (see, e.g., [1,2,3,4] and references therein), the key idea is to use the Wiener chaos expansion and decompose the SPDE into an infinite hierarchy of deterministic PDE problems whose properties are well studied, and then compose the solution of the SPDE as a generalized random series, thus allowing us to obtain well-posed results for the SPDE. The results of the present paper are motivated by and can be considered as a first step towards our final goal of applying this method to acoustic and elastic scattering problems for obstacles with various boundary conditions.

Our paper is organized as follows. In Section 2, and for the convenience of the reader, we give preliminary mathematical notations, as well as the appropriate functional space setting. In Section 3, we deal with the stochastic boundary value problem for the Helmholtz equation, for which an analogous approach due to [1,5] is applied. In Section 4, we study and give results for a stochastic elastic boundary value problem, where the boundary condition is a random variable [3,6]. Finally, in Section 5, we provide some useful remarks and conclusions.

2. Mathematical Preliminaries

In this section, we present mathematical notations and suitable functional space setting. Initially, we consider the Wiener chaos expansion of elements of the space of square-integrable functions defined in the space of tempered distributions [5].

Let $S\left({\mathbb{R}}^d\right)$ be the Schwartz space of rapidly decreasing $C^{\infty }$ functions on ${\mathbb{R}}^d$, where its dual space $S^{*}\left({\mathbb{R}}^d\right)$ is the space of tempered distributions. We also mention that there exists a unique probability measure P on F, where F is the family of Borel subsets of $S^{*}\left({\mathbb{R}}^d\right)$, such that

$$E\left[e^{i\left(·,\varphi \right)}\right]:=\underset{S^{*}}{\int }{e}^{i〈\omega ,\varphi 〉}dP\left(\omega \right)=exp\left(-\frac{1}{2}{∥\varphi ∥}_{L^2\left({\mathbb{R}}^d\right)}^2\right)\forall \varphi \in S,$$

(1)

where $〈\omega ,\varphi 〉=\omega \left(\varphi \right)$ is the process of $\omega \in S^{*}$ on $\varphi \in S$ (Bochner–Minlos theorem), [5].

The Hermite polynomials are defined as $h_n\left(x\right)={\left(-1\right)}^n{e}^{\frac{x^2}{2}}\frac{d^n}{d{x}^n}\left(e^{-\frac{x^2}{2}}\right),n=0,1,2,\dots$ and thus Hermite functions ${\xi }_n\left(x\right)$ are also defined as

$${\xi }_n\left(x\right)={\pi }^{-\frac{1}{4}}{\left(\left(n-1\right)!\right)}^{-\frac{1}{2}}{e}^{-\frac{x^2}{2}}{h}_{n-1}\left(x\right),n=1,2,3,\dots$$

We can easily see that the Hermite functions ${\xi }_n\left(x\right)n=1,2,3,\dots$ constitute an orthonormal basis in $L^2\left({\mathbb{R}}^d\right)$ with respect to the weight $e^{-\frac{x^2}{2}}$.

Let ${\delta }^j=\left({\delta }_1^j,{\delta }_2^j,\dots ,{\delta }_d^j\right)$, where ${\delta }_i^j\in \mathbb{N}$, and assume the following tensor products:

$${\xi }_{{\delta }^j}:={\xi }_{{\delta }_1^j}\otimes {\xi }_{{\delta }_2^j}\otimes \dots \otimes {\xi }_{{\delta }_d^j},j=1,2,3,\dots$$

where for the $i<j$ inequality, ${\delta }_1^i+{\delta }_2^i+\dots +{\delta }_d^i\le {\delta }_1^j+{\delta }_2^j+\dots +{\delta }_d^j$ holds. The family of tensor products ${\left\{{\xi }_{{\delta }^j}\right\}}_{j=1}^{\infty }$ constitutes an orthogonal basis in $L^2\left({\mathbb{R}}^d\right)$. We also introduce the countable multi-index via $I=\left\{a=\left(a_1,a_2,\dots \right)|a_i\in \mathbb{N}\cup \left\{0\right\}\right\}$ for which there exists a finite number of $a_i\ne 0$. For each $a\in I$, we define stochastic Hermite polynomials $H_a$ given by

$$H_a\left(\omega \right)=\prod _{i=1}^{\infty }{h}_{a_i}\left(〈\omega ,{\xi }_{{\delta }^i}〉\right),\omega \in \mathsf{\Omega }.$$

We can see that $H_a$ forms an orthogonal basis in $L^2\left(\mathsf{\Omega }\right)$ and the norm $∥H_a∥$ satisfies

$${∥H_a∥}_{L^2\left(\mathsf{\Omega }\right)}^2=a!=a_1!a_2!\dots$$

Theorem 1.

Every $f\in L^2\left(\mathsf{\Omega }\right)$ has a unique Wiener–Ito chaos expansion in terms of stochastic Hermite polynomials, given by

$$f\left(\omega \right)=\sum _{a\in I}{c}_a{H}_a\left(\omega \right),c_a\in \mathbb{R}$$

(2)

where

$$c_a=E\left(f\left(\omega \right)H_a\left(\omega \right)\right)=\underset{\mathsf{\Omega }}{\int }f\left(\omega \right)H_a\left(\omega \right)dP\left(\omega \right).$$

In what follows, we define the stochastic Hilbert space ${\left(S\right)}^{\rho ,z,L^2\left(\mathsf{\Omega }\right)}$, for $\rho \in \left[-1,1\right],z\in \mathbb{R}$, as the set of all sums

$$f=\sum _{a\in J}{f}_a{H}_a,f_a\in L^2\left(\mathsf{\Omega }\right),\forall a\in I$$

(3)

with the finite norm

$${∥f∥}_{\rho ,z,L^2\left(\mathsf{\Omega }\right)}={\left(\sum _{a\in I}{∥f_a∥}_{L^2\left(\mathsf{\Omega }\right)}^2{\left(a!\right)}^{1+\rho }{\left(2\mathbb{N}\right)}^{za}\right)}^{1/2}.$$

(4)

The norm given by (4) is induced by the inner product

$${\left(f,g\right)}_{\rho ,z,L^2\left(\mathsf{\Omega }\right)}=\sum _{a\in I}{\left(f_a,g_a\right)}_{L^2\left(\mathsf{\Omega }\right)}{\left(a!\right)}^{1+\rho }{\left(2\mathbb{N}\right)}^{za},f,g\in {\left(S\right)}^{\rho ,z,L^2\left(\mathsf{\Omega }\right)}$$

where

$$f=\sum _{a\in I}{f}_a{H}_a,g=\sum _{a\in I}{g}_a{H}_a$$

(5)

and

$${\left(2\mathbb{N}\right)}^{za}:=\prod _{j=1}^{\infty }{\left(2j\right)}^{z{a}_j}.$$

Finally, we also define the usual Sobolev space $H_0^1\left(D\right)$, given by

$$H_0^1\left(D\right):=\left\{\upsilon \in H^1\left(D\right)and\upsilon =0on\partial D\right\}.$$

3. The Stochastic Helmholtz Boundary Value Problem

In this section, we present the construction of an infinite hierarchy of deterministic equations for the stochastic Helmholtz equation. Furthermore, we study the well-posedness of our stochastic problem through the existence and uniqueness of the hierarchy of deterministic problems for each solution.

We consider the following stochastic boundary value problem

$$\mathsf{\Delta }u+k^{2}u=f\mathrm{in}D$$

(6)

$$u=g,\mathrm{on}\partial D$$

(7)

where f is a generalized stochastic source, g a stochastic boundary condition and $I=\left\{a=\left(a_1,a_2,\dots \right)|a_i\in \mathbb{N}\cup \left\{0\right\}\right\}$, as given above (see page 2). For the stochastic problem (6) and (7), we use the relations given in (5), as well as $u={\displaystyle \sum _a}{u}_a{H}_a$, in order to obtain the infinite hierarchy of deterministic problems

$$\mathsf{\Delta }{u}_a+k^2{u}_a=f_a\mathrm{in}D\mathrm{and}{u}_a=g_a\mathrm{on}\partial D$$

(8)

For the above deterministic problems we can obtain their corresponding variational formulations, and for the sake of brevity, we only give the variational formulation of the problem for $\left|a\right|=n$.

$$\mathsf{\Delta }{u}_n+k^2{u}_n=f_n\mathrm{in}D$$

(9)

$$u_n=g_n\mathrm{on}\partial D$$

(10)

given by

$$\alpha \left(u_n,v\right)=\ell \left(v\right)\forall v\in H^1\left(D\right).$$

(11)

In (11), the bilinear form $\alpha \left(u_n,v\right)$ on $H^1\left(D\right)\times H^1\left(D\right)$ is given by

$$\alpha \left(u_n,v\right)=\underset{D}{\int }\left(-\nabla u_n·\nabla v+k^2{u}_{n}v\right)dx$$

(12)

and the linear functional $\ell \left(vs.\right)$ on $H^1\left(D\right)$ by

$$\ell \left(vs.\right)=\underset{D}{\int }{f}_{n}vdx-\underset{\partial D}{\int }{g}_{n}vdx,$$

(13)

where the function $f_n\in L^2\left(D\right)$ and $g_n\in L^2\left(\partial D\right)$. In what follows, we give the following proposition.

Proposition 1.

Let D be a bounded open subset of ${\mathbb{R}}^d$, $f_n\in L^2\left(D\right)$, $g_n\in L^2\left(\partial D\right)$ and $k^2\in L^{\infty }\left(D\right)$. Then, the problem (11) has a unique solution $u_n\in H^1\left(D\right)$, which satisfies the following inequality:

$${∥u_n∥}_{H^1\left(D\right)}\le c_n({∥f_n∥}_{L^2\left(D\right)}+{∥g_n∥}_{H^{1/2}\left(\partial D\right)})$$

(14)

The proof of the proposition uses the hypothesis of the Lax–Milgram theorem [7], and is omitted here for brevity. We now give the following main result.

Proposition 2.

If we define the weights $w_a={\left(a!\right)}^{1+\rho }{\left(2\mathbb{N}\right)}^{za},\left|a\right|=0,1,\dots ,n$, then the stochastic problem (6) and (7) has a unique weighted Wiener chaos solution $u\in {\left(S\right)}^{\rho ,z,L^2\left(D\right)}$.

Proof. 

From Proposition 1, each of the deterministic problems (8) has a unique solution $u_a\in H^1\left(D\right)$, and via relation $u={\displaystyle \sum _a}{u}_a{H}_a$ our stochastic problem (6) and (7) has a unique solution. In relation (14), $c_n$ depends on $f_n$, $g_n$, and hence there is a positive constant c being the supremum of $c_n,n=0,1,2,\dots$, which satisfies inequality (14). Furthermore, if we raise each one of the inequalities (14) for $n=0,1,2,\dots$ to the square power, multiply both sides by the weights $w_a$, and add them, we can obtain

$$\sum _{a\in I}{w}_a{∥u_a∥}_{H^1\left(D\right)}^2\le c^2\sum _{a\in I}{w}_a\left({∥f_a∥}_{H^1\left(D\right)}^2+{∥g_a∥}_{H^{1/2}\left(\partial D\right)}^2\right)$$

(15)

for a positive constant $c=Sup\left(c_n\right),n=0,1,2,\dots$ Using the fact that

$${∥u∥}_{{\left(S\right)}^{\rho ,z,L^2\left(D\right)}}^2=\sum _{a\in I}{w}_a{∥u_a∥}_{L^2\left(D\right)}^2$$

(16)

and taking into account that ${\parallel u_{\alpha }\parallel }_{L^2\left(D\right)}\le {\parallel u_{\alpha }\parallel }_{H^1\left(D\right)}$ via (15), we can easily obtain

$${∥u∥}_{{\left(S\right)}^{\rho ,z,L^2\left(D\right)}}^2\le c^2\sum _{a\in I}{w}_a\left({∥f_a∥}_{H^1\left(D\right)}^2+{∥g_a∥}_{H^{1/2}\left(\partial D\right)}^2\right)<\infty .$$

(17)

We also remark that an analogous estimation for the solution as in (17) is also valid in the space ${\left(S\right)}^{\rho ,z,H^1\left(D\right)}$. □

4. Stochastic Boundary Data for Navier Equation

In this section, we study the stochastic boundary value problem for the Navier equation. Initially, similar to the acoustic case, we construct an infinite hierarchy of deterministic problems and establish the well-posedness of our stochastic problem via the uniqueness and existence of each deterministic one. Let $D\subset {\mathbb{R}}^{{}^2}$ be an open-bounded domain with the boundary $\partial D\equiv \mathsf{\Gamma }$ being Lipschitz. Throughout this paper, $\widehat{\mathbf{n}}=\widehat{\mathbf{n}}\left(\mathbf{r}\right)$ denotes the outward unit normal vector at the point $\mathbf{r}\in \mathsf{\Gamma }$. The problem is formulated as follows:

Find a vector function $\mathbf{u}\in {\left(S\right)}^{\rho ,z,[L^2\left(D\right){]}^2}$ such that

$${\mathsf{\Delta }}^{*}\mathbf{u}\left(\mathbf{r}\right)+\varrho {\omega }^2\mathbf{u}\left(\mathbf{r}\right)=\mathbf{0},\mathbf{r}\in D,$$

(18)

$$\mathbf{u}\left(\mathbf{r}\right)=\mathbf{g}:=\sum _{\alpha }{\mathbf{g}}_{\alpha }{H}_{\alpha },\mathbf{r}\in \mathsf{\Gamma },$$

(19)

where the explicit expression for ${\mathsf{\Delta }}^{*}$, is given by

$${\mathsf{\Delta }}^{*}\mathbf{u}\left(\mathbf{r}\right):=\mu \mathsf{\Delta }\mathbf{u}\left(\mathbf{r}\right)+\left(\lambda +\mu \right)\nabla \nabla ·\mathbf{u}\left(\mathbf{r}\right)$$

(20)

where $\omega \in \mathbb{R}$ in (18) denotes the so-called angular frequency, $\lambda$ and $\mu$ are the Lamé constants and $\varrho$ is the mass density. Since any element of the space ${\left(S\right)}^{\rho ,z,[L^2\left(D\right){]}^2}$ allows a Wiener chaos expansion [4,5], substituting the projections ${\mathbf{u}}_{\alpha }$ of $\mathbf{u}$ on $H_{\alpha }$ into the relation $\mathbf{u}\left(\mathbf{r}\right)={\sum }_{\alpha }{\mathbf{u}}_{\alpha }{H}_{\alpha }$, we can construct the solution $\mathbf{u}$. We transform our stochastic problem into an infinite hierarchy of deterministic problems and we exploit uniqueness and existence results for each one [1]. Via the projections ${\mathbf{u}}_{\alpha },\alpha \in I$, we obtain the following hierarchy of problems:

$${\mathsf{\Delta }}^{*}{\mathbf{u}}_{\alpha }\left(\mathbf{r}\right)+\varrho {\omega }^2{\mathbf{u}}_{\alpha }\left(\mathbf{r}\right)=\mathbf{0},\mathbf{r}\in D,$$

(21)

$${\mathbf{u}}_{\alpha }\left(\mathbf{r}\right)={\mathbf{g}}_{\alpha },\mathbf{r}\in \mathsf{\Gamma },$$

(22)

For the above deterministic problems, we can obtain their corresponding variational formulations; for the sake of brevity, we only give the variational formulation of the problem for $\left|a\right|=n$ (21) and (22):

Find a solution ${\mathbf{u}}_n\in {\left[H^1\left(\overline{D}\right)\right]}^2$ such that

$$\alpha ({\mathbf{u}}_n,\mathbf{v})=\ell \left(\mathbf{v}\right), for every \mathbf{v}\in {\left[H^1\left(D\right)\right]}^2$$

(23)

where the bilinear form $\alpha ({\mathbf{u}}_n,\mathbf{v})$ on ${\left[H^1\left(D\right)\right]}^2\times {\left[H^1\left(D\right)\right]}^2$ is given by

$$\begin{array}{ccc}\hfill \alpha ({\mathbf{u}}_n,\mathbf{v})& =& -\mu {\int }_D(\nabla {\mathbf{u}}_n):(\nabla \overline{\mathbf{v}})dr-(\lambda +\mu ){\int }_D(\nabla ·{\mathbf{u}}_n)(\nabla ·\overline{\mathbf{v}})dr\hfill \\ & & +{\int }_D\rho {\omega }^2{\mathbf{u}}_n·\overline{\mathbf{v}}dr\hfill \end{array}$$

(24)

and the linear functional $\ell \left(\mathbf{v}\right)$ on ${\left[H^1\left(D\right)\right]}^2$ by

$$\begin{array}{ccc}\hfill \ell \left(\mathbf{v}\right)& =& -\mu {\int }_{\mathsf{\Gamma }}\widehat{\mathbf{n}}·(\nabla {\mathbf{g}}_n)·\overline{\mathbf{v}}ds-(\lambda +\mu ){\int }_{\mathsf{\Gamma }}(\nabla ·{\mathbf{g}}_n)\widehat{\mathbf{n}}·\overline{\mathbf{v}}ds\hfill \end{array}$$

(25)

Proposition 3.

Let D be an open subset of ${\mathbb{R}}^2$ and ${\mathbf{g}}_n\in {\left[L^2\left(D\right)\right]}^2$; then, problem (21) and (22) is uniquely solvable and, furthermore, the solution ${\mathbf{u}}_n\in {\left[H^1\left(\overline{D}\right)\right]}^2$ satisfies

$${\parallel {\mathbf{u}}_n\parallel }_{H^1\left(D\right)}\le c{\parallel {\mathbf{g}}_n\parallel }_{H^{1/2}\left(\mathsf{\Gamma }\right)}forsomepositiveconstantc.$$

(26)

In order to establish the existence and uniqueness of (23), we need the following three lemmas, for which the proofs are omitted here for brevity.

Lemma 1.

The bilinear form $\alpha ({\mathbf{u}}_n,\mathbf{v})$ is bounded, i.e.,

$$|\alpha ({\mathbf{u}}_n,\mathbf{v})|\le c_3\parallel {\mathbf{u}}_n{\parallel }_{H^1\left(D\right)}{\parallel \mathbf{v}\parallel }_{H^1\left(D\right)}.$$

(27)

Lemma 2.

The following coercivity property for $\alpha ({\mathbf{u}}_n,{\mathbf{u}}_n)$ holds:

$$Re\left\{\alpha ({\mathbf{u}}_n,{\mathbf{u}}_n)\right\}\ge c{\parallel {\mathbf{u}}_n\parallel }_{H^1\left(D\right)}^2.$$

(28)

Lemma 3.

The linear functional $\ell \left(\mathbf{v}\right)$ is bounded, i.e., there exists a positive constant $c_1$ such that

$$|\ell \left(\mathbf{v}\right)|\le c_1{\parallel \mathbf{v}\parallel }_{H^1\left(D\right)}.$$

(29)

The above procedure uses the hypothesis of the Lax–Milgram theorem [7] in order to derive the assertion of Proposition 3.

Proposition 4.

The stochastic problem (18) and (19) admits a unique Wiener chaos solution $\mathbf{u}\in {\left(S\right)}^{\rho ,z,{\left[L^2\left(D\right)\right]}^2}$ that satisfies

$${\parallel \mathbf{u}\parallel }_{{\left(S\right)}^{\rho ,z,[L^2\left(D\right){]}^2}}^2\le c^2\sum _{\alpha }{w}_{\alpha }\parallel {\mathbf{g}}_{\alpha }{\parallel }_{H^{1/2}\left(\mathsf{\Gamma }\right)}^{2}where{w}_{\alpha }={\left(a!\right)}^{1+\rho }{\left(2\mathbb{N}\right)}^{za},\left|\alpha \right|=0,1,2,\dots$$

(30)

Proof. 

We state here that each of the deterministic problems (21) and (22) admits a unique solution, and we also mention that $c_n$ depends on ${\mathbf{g}}_n$, and hence there is a positive constant c being the supremum of $c_n$, i.e., $c=sup\{c_n,n=0,1,2,\dots \}$. Thus, the following inequalities hold:

$$\begin{array}{ccc}& & {\parallel {\mathbf{u}}_0\parallel }_{H^1\left(D\right)}\le c{\parallel {\mathbf{g}}_0\parallel }_{H^{1/2}\left(\mathsf{\Gamma }\right)}\hfill \\ & & {\parallel {\mathbf{u}}_1\parallel }_{H^1\left(D\right)}\le c{\parallel {\mathbf{g}}_1\parallel }_{H^{1/2}\left(\mathsf{\Gamma }\right)}\hfill \\ & & \begin{array}{cc}& ⋮\end{array}\hfill \\ & & {\parallel {\mathbf{u}}_n\parallel }_{H^1\left(D\right)}\le c{\parallel {\mathbf{g}}_n\parallel }_{H^{1/2}\left(\mathsf{\Gamma }\right)}\hfill \\ & & \begin{array}{ccc}& ⋮& \end{array}\hfill \end{array}$$

(31)

Raising these inequalities to the second power, multiplying both sides of each inequality by $w_{\alpha }$, adding them, and taking into account ${\parallel u_{\alpha }\parallel }_{{\left[L^2\left(D\right)\right]}^2}\le {\parallel u_{\alpha }\parallel }_{{\left[H^1\left(D\right)\right]}^2}$, we obtain

$$\sum _{\alpha }{w}_{\alpha }\parallel {\mathbf{u}}_a{\parallel }_{L^2\left(D\right)}^2\le c^2\sum _{\alpha }{w}_{\alpha }{\parallel {\mathbf{g}}_{\alpha }\parallel }_{H^{1/2}\left(\mathsf{\Gamma }\right)}^2.$$

(32)

Hence, we easily arrive at

$${\parallel \mathbf{u}\parallel }_{{\left(S\right)}^{\rho ,z,{\left[L^2\left(D\right)\right]}^2}}^2\le c^2\sum _{\alpha }{w}_{\alpha }{\parallel {\mathbf{g}}_{\alpha }\parallel }_{H^{1/2}\left(\mathsf{\Gamma }\right)}^2<\infty .$$

(33)

An analogous estimation for the solution, as in (33), is also valid in the space ${\left(S\right)}^{\rho ,z,{\left[H^1\left(D\right)\right]}^2}$. □

5. Conclusions

In this paper, the well-posedness of solutions for stochastic boundary value problems due to the Helmholtz and Navier equations was established, via the study of the corresponding hierarchies of deterministic problems. Uniqueness, existence and regularity issues were addressed, and we also make the following remarks:

(i)

For the stochastic Helmholtz equation, with a stochastic source and stochastic boundary conditions, we proved that the stochastic problem (6) and (7) has a unique weighted Wiener chaos solution.

(ii)

In the case of stochastic boundary data for the Navier equation, a unique Wiener chaos solution for the stochastic problem (18) and (19) was proved.

(iii)

The proposed method can also be extended to cover the case of a stochastic boundary value problem where the randomness is present in the equation (e.g., in k for the Helmholtz equation, or $\varrho$, $\lambda$, and $\mu$ for the Navier equation), as well as in the boundary condition. The study of such cases is under progress and will be communicated separately.