Mathematical Problems in Engineering

Volume 2015, Article ID 796539, 8 pages

http://dx.doi.org/10.1155/2015/796539

## An Adaptive Observer-Based Algorithm for Solving Inverse Source Problem for the Wave Equation

Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science & Technology, P.O. Box 4700, Thuwal 23955-6900, Saudi Arabia

Received 12 February 2015; Revised 20 August 2015; Accepted 31 August 2015

Academic Editor: Herb Kunze

Copyright © 2015 Sharefa Asiri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Observers are well known in control theory. Originally designed to estimate the hidden states of dynamical systems given some measurements, the observers scope has been recently extended to the estimation of some unknowns, for systems governed by partial differential equations. In this paper, observers are used to solve inverse source problem for a one-dimensional wave equation. An adaptive observer is designed to estimate the state and source components for a fully discretized system. The effectiveness of the algorithm is emphasized in noise-free and noisy cases and an insight on the impact of measurements’ size and location is provided.

#### 1. Introduction

In this paper we are interested in an inverse source problem for the wave equation. This problem appears frequently in many fields, especially, in modern seismology [1]. One important application of this problem is to distinguish between different types of seismic events (e.g., earthquake, implosion, or explosion) [2]. It is also important in monitoring hydraulic fracturing by which fractures are created in rocks such that entrapped hydrocarbons can be released and extracted [3].

Inverse problems are usually solved using optimization techniques, where an appropriate cost function is minimized. However the ill-posedness of such problems generates instability. Regularization techniques are then used to restore the stability. Among the regularization techniques, Tikhonov regularization [4] is probably the most used one. For instance, it has been applied to the wave equation in [5, 6]. Other techniques have been also proposed to solve inverse problems; for example, in [7], a new minimization algorithm has been proposed to solve an inverse problem for the wave equation with unknown wave speed. Most of the proposed methods end up with an optimization step which generally turns to be computationally heavy, especially in the case of large number of unknowns, and may require an extensive storage.

The objective of this paper is to present an alternative algorithm, based on observers, to solve the inverse source problem for the wave equation. Observers are well known in control theory for state estimation in finite dimensional dynamical systems. Presenting the distinctive feature and main advantage of operating recursively on direct problems, observers are gaining more and more interest in a wide variety of problems, including partial differential equations (PDEs) systems. For instance, in [8] states and parameters are estimated using an observer depending on a discretized space for a mechanical system. In [9], the initial state of a distributed parameter system has been estimated using two observers: one for the forward time and the other for the backward time. A similar approach has been used in [10], using the forward-backward approach to solve inverse source problem for the wave equation. An adaptive observer was applied in [11] for parameter estimation and stabilization of one-dimensional wave equation where the boundary observation suffers from an unknown constant disturbance. A similar work was proposed in [12] with the state as unknown and the boundary observation suffers from an arbitrary long time delay.

Dealing with PDEs, either with observers or classical inverse problems methods, poses the challenge of approximating infinite dimensional systems. As regards observers, we can distinguish three approaches for studying such systems. The first approach considers the design of the observer in the continuous domain which requires mathematical analysis [13], and the application of the observer to real application will require some adaptation. The second approach consists in the semidiscretization of the equation in space. The result of this semidiscretization can be usually written in the standard state-space representation in the continuous domain (in time) which makes the extension of the known methods in control theory easier. The third approach is the full-discretization of the PDE in space and time. In this case we can write the system in a discrete state-space representation. We have chosen this latter approach since it is more suitable for real implementation. We show that it can give good results provided that some conditions, aimed at minimizing the effect of numerical errors resulting from discretization, are met.

Another challenge, related to solving inverse problems in general, arises when it comes to measurement constraints. Indeed, from a practical point of view, we usually do not have enough measurements to estimate all the unknowns. Dealing with this source of ill-posedness, means, in observers theory framework, satisfying the equivalent property of observability. Indeed, given the PDE system together with the measurements, we can test in a prior step whether the unknown variables can be estimated fully or partially, regardless of the kind of observer to be used. For instance, in [8, 9, 11, 12], the measurements were taken as the time derivative of the solution of the wave equation. This kind of measurements gives a typical observability condition which has a positive effect on the stabilization, but it is less readily available than field measurements. Hence, some authors sought to solve inverse problems for wave equation using observers based on partial filed measurements, that is, measurements taken from the solution of the wave equation, as in [14–16].

In this paper, we consider a fully discretized version of a one-dimensional wave equation and we propose a new algorithm for inverse source problem based on adaptive observer for the joint estimation of the states and the source term from partial measurements of the field. Adaptive observers are widely used in control theory for parameter estimation in adaptive control or fault estimation in fault detection and isolation [17, 18]. In Section 2, problem statement is detailed. Then, the observer design is presented in Section 3. Finally, numerical results are presented and discussed.

#### 2. Problem Statement

Consider the one-dimensional wave equation with Dirichlet boundary conditions defining in the domain :where is the space coordinate, is the time coordinate, and are the initial conditions in , is the source function which is assumed, for simplicity, to be independent of time, and is the velocity which is known. The notations and refer to the first and second derivatives of with respect to , respectively.

Our inverse problem falls in the estimation of the source in (1) using an adaptive observer with partial measurements of the field available. We first propose to rewrite (1) in a system of first order PDEs by introducing two auxiliary variables and and letwhere refers to transpose. Then (1) can be written as follows:where the operator is given by , , is the output, and is the observation operator such that , where is a restriction operator on the measured domain.

Discretizing system (3) using implicit Euler scheme in time and central finite difference discretization for the space gives the following discrete state-space representation:whereand is the identity matrix of dimension , where refers to the number of measurements, and is a term that includes the boundary conditions such thatThis system is linear multiple-input multiple-output discrete time invariant. If refers to the space grid size and refers to the number of measurements, then the state matrix is of dimension , the observer matrix is of dimension , and the input matrix is of dimension .

The numerical scheme (4) is consistent. In addition, it is stable if and only if (the CFL condition). Thus, if , scheme (4) converges as , to (3) and therefore to (1).

#### 3. Observer Design

We propose to use an adaptive observer for the joint estimation of the states and and the source . This observer has been proposed in [17] and it has been developed for joint estimation of the state and the parameters. However, we propose to generalize the idea behind this observer to estimate the input considering each spatial sample of the input as an independent parameter. The adaptive observer is given by the following system of equations:where is the observer gain matrix of dimension , and are the state and source estimates, respectively, is a matrix sequence obtained by linearly filtering , and is a scalar gain satisfying the following assumption as in [17].

*Assumption 1. *The scalar gain satisfies the following: (1);(2) for some constant , integer , and all .

*Remark 2. *In the proposed method, no particular form for the matrices , , and is required. However, all these matrices are assumed bounded. In our problem, wave equation with constant velocity, these matrices are actually constant and therefore always bounded.

*Remark 3. *From Remark 2, any consistent and stable numerical method can be used to discretize system (3) provided that it ends up with bounded matrices (, , and ).

Under Assumption 1, Algorithm (7) converges exponentially fast when tends to infinity in noise-free case, and the estimation errors remain bounded in the noisy case as long as the noises are bounded. Moreover, the estimation errors converge in the mean to zero if the noises have zero means; see Theorems and with their proofs in [17].

#### 4. Numerical Simulations

To test the performance of the observer, we generated a set of synthetic data using the following parameters: , , , and . Thus, and . The velocity is chosen to be , and the source is equal to . The matrix sequence and the scalar gain are chosen such that Assumption 1 is satisfied. The algorithm was implemented in Matlab and the tests were run for two main cases: noise-free and noisy datasets. In the noise-corrupted case, zero mean white Gaussian random noises were added to the states and to the measurements with standard deviations and , respectively. The gain matrix is selected to have fast and accurate convergence of the observer. We took advantage of the particular structure of to design the gain . Indeed the matrix is sparse, so we selected to be also a sparse matrix. The number of unknown entries is then reduced and we identified them such that the eigenvalues of are inside the unit circle. In general, standard pole placement can be used to select the gain matrix .

Figure 1 shows the error in the estimated state and Figure 2 presents the exact and the estimated source; both figures exhibit noise-free and noisy cases with respect to full and partial measurements. For the partial measurements, we supposed that the field is available on half of the space domain only. Tables 1 and 2 show the minimum square error (MSE) in the estimated source in noise-free and noisy cases, respectively. Both tables show the error in case of full measurements, partial measurements taken from the middle, and partial measurements taken from the end.