A note on inverse problem for strongly damped wave equation with Gaussian white noise

In this paper, we study for the first time the inverse initial problem for the one-dimensional strongly damped wave with Gaussian white noise data. Under some a priori assumptions on the true solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. Error estimates are given in both the L2− and Hp−norms.


Introduction
Let T be a positive number and D = (0, π).We are interested in the problem of recovering the initial state u(x, 0), x ∈ D, for the following strongly damped wave equation u tt − αu xxt − u xx = 0, (x, t) ∈ D × (0, T ), (1.1) subject to the final conditions      u(x, T ) = g(x), (x, t) ∈ D × (0, T ), u t (x, T ) = 0, (x, t) ∈ D × (0, T ), (1.2) and the boundary condition u(0, t) = u(π, t) = 0, t ∈ (0, T ), (1.3) where α > 2 is a fixed positive constant.In physics, the homogeneous Dirichlet boundary condition (1.3) expresses that two ends of the string are fixed.We can consider other types of ITM Web of Conferences 20, 02003 (2018) https://doi.org/10.1051/itmconf/20182002003ICM 2018 boundary condition instead.In practice, we cannot measure g exactly, but we observe with the presence of a Gaussian white noise process ξ g obs (x) = g(x) + ξ(x), (1.4) where > 0 is the amplitude of the noise.Moreover, it can only be observed in discretized form: where the natural number N is the number of steps of discrete observation and φ j , j = 1, N, are defined in the next section.For more details on this random model, we refer to [1,4].This problem is well-known to be severely ill-posed and regularization methods for it are required.Strongly damped wave equation (SDWE) occurs in a wide range of applications such as modeling motion of viscoelastic materials [2,7,8].From both the theoretical and numerical points of view, the initial value problem has been extensively studied (see e.g., [3,5,9]).However, to the best of our knowledge, the final value problem (1.1)-(1.3)with Gaussian white noise data has not been studied.In [6], Lions and Lattes introduced Problem (1.1)-(1.3),but regularization methods haven't been mentioned.The major object of this paper is to propose a stable regularized solution for the problem (1.1)-(1.3)using the Fourier truncation method.

Preliminaries
Let us recall that the problem admits the eigenvalues λ j = j 2 , j = 1, ∞, and φ j (x) = 2 π sin( jx) are the corresponding eigenfunctions, which form an orthonormal basis of L 2 (D).Definition 2.1 Let H be a Hilbert space.The stochastic error is a Hilbert space process, i.e. a bounded linear operator ξ : H → L 2 (Ω, A, P), where (Ω, A, P) is the underlying probability space and L 2 (•) is the space of all square integrable measurable functions.
ITM Web of Conferences 20, 02003 (2018) https://doi.org/10.1051/itmconf/20182002003ICM 2018 For p > 0, we denote by H p (D) the closed subspace of L 2 (D), given by and equipped with the norm where •, • denotes the inner product in L 2 (D).Throughout this paper, denote by • the norm in L 2 (D).
Assume that g ∈ H γ (D) for γ > 0.Then, we have the following estimate Here, N depends on and satisfies that lim Proof.By the usual MISE decomposition which involves a variance term and a bias term, we get Since

Main results
Assume that Problem (1.1)-(1.3)has a unique true solution, we first find its Fourier series representation.From (1.1)-( 1.3), we can derive the following ODE with given data at t = T where u j (t) = π 0 u(x, t)φ j (x)dx, g j = π 0 g(x)φ j (x)dx.For fixed damping α > 2, the solution of the latter equation is given by Hence, the solution of problem (1.1)-(1.3)can be represented as For fixed t and β N( ) , let us define the operator Now, we sate and prove the following lemma, which is needed for our analysis.Lemma 3.1 For fixed t ∈ [0, T ], we have Moreover, α 2 e αtβ N( ) ϕ . (3.9) Proof.Using Parseval's equality, we obtain (3.10) From the equality A j (t) can be rewritten as ITM Web of Conferences 20, 02003 (2018) https://doi.org/10.1051/itmconf/20182002003ICM 2018 Applying the inequalities (a + b) 2 ≤ 2a 2 + 2b 2 and |e a − e b | ≤ max{e a , e b }|a − b|, we obtain For the second statement of Lemma 3.1, from (3.8), we have The proof of Lemma 3.1 is complete.
Define the truncation operator Next, we formulate a regularized problem for Problem (1.1)-(1.3)as follows where β N( ) plays the role of regularization parameter.Using the same techniques as the beginning of this section for the problem (1.1)-(1.3),we can easily show that Problem (3.12) has a unique solution We are now in position to state the main results of this paper.