Quasi-reversibility Method Research Articles

PDE-based numerical method for a limited angle x-ray tomography

A new numerical method for x-ray tomography for a specific case of incomplete Radon data is proposed. Potential applications are in checking out bulky luggage in airports. This method is based on the analysis of the transport PDE governing the x-ray tomography rather than on the conventional integral formulation. The quasi-reversibility method is applied. Convergence analysis is performed using a new Carleman estimate. Numerical results are presented and compared with the inversion of the Radon transform using the well-known filtered back projection algorithm. In addition, it is shown how to use our method to study the inversion of the attenuated x-ray transform for the same case of incomplete data.

An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method

We propose in this paper a numerical method to solve a linear inverse source problem for general hyperbolic equations. This is the problem of reconstructing sources from the lateral Cauchy data of the wave field on the boundary of a domain. In order to achieve the goal, we derive an equation involving a Volterra integral, whose solution directly provides the desired solution of the inverse source problem. Due to the presence of such a Volterra integral, this equation is not in a standard form of partial differential equations. We employ the quasi-reversibility method to find its regularized solution. Using Carleman estimates, we show that the obtained regularized solution converges to the true solution with the Lipschitz-like convergence rate as the measurement noise tends to 0. This is one of the novelties of this paper since currently, convergence results for the quasi-reversibility method are only known for purely differential equations. Numerical tests demonstrate a good reconstruction accuracy.

A quasi-reversibility regularization method for a Cauchy problem of the modified Helmholtz-type equation

The Cauchy problem of the modified Helmholtz-type equation is severely ill-posed, i.e., the solution does not depend continuously on the given Cauchy data. Thus the regularization methods are required to recover the numerical stability. In this paper, we propose a quasi-reversibility regularization method to deal with this ill-posed problem. Convergence estimates are obtained under a-priori bound assumptions for the exact solution and the selection of regularization parameter. Some numerical results are given to show that this method is stable and feasible.

On an Inverse Source Problem for the Full Radiative Transfer Equation with Incomplete Data

A new numerical method to solve an inverse source problem for the radiative transfer equation involving the absorption and scattering terms, with incomplete data, is proposed. No restrictive assumption on those absorption and scattering coefficients is imposed. The original inverse source problem is reduced to boundary value problem for a system of coupled partial differential equations of the first order. The unknown source function is not a part of this system. Next, we write this system in the fully discrete form of finite differences. That discrete problem is solved via the quasi-reversibility method. We prove the existence and uniqueness of the regularized solution. Especially, we prove the convergence of regularized solutions to the exact one as the noise level in the data tends to zero via a new discrete Carleman estimate. Numerical simulations demonstrate good performance of this method even when the data are highly noisy.

A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media

A new numerical method to solve an inverse source problem for the Helmholtz equation in inhomogenous media is proposed. This method reduces the original inverse problem to a boundary value problem for a coupled system of elliptic PDEs, in which the unknown source function is not involved. The Dirichlet boundary condition is given on the entire boundary of the domain of interest and the Neumann boundary condition is given on a part of this boundary. To solve this problem, the quasi-reversibility method is applied. Uniqueness and existence of the minimizer are proven. A new Carleman estimate is established. Next, the convergence of those minimizers to the exact solution is proven using that Carleman estimate. Results of numerical tests are presented.

Analysis of a Quasi-Reversibility Method for a Terminal Value Quasi-Linear Parabolic Problem with Measurements

This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value parabolic problem with noisy data. Based on data measurements, we perturb the problem by the so-called filter regularized operator to design an approximate problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semilinear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there are likely myriad models derived from this simpler case, such as source localization problems for brain tumors and heat conduction problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the Faedo--Galerkin method to study the weak solvability of the approximate problem. Relying on the energy-like analysis, we provide detailed convergence rates in $L^2$-$H^1$ of the proposed method when the true solution is sufficiently smooth. Depending on the dimensions of the domain, we obtain an error estimate in $L^{r}$ for some $r>2$. Proof of the backward uniqueness for the quasi-linear system is also depicted in this work. To prove the regularity assumptions acceptable, several physical applications are discussed.

Об условиях квалифицированной сходимости разностных методов и метода квазиобращения для решения линейных некорректных задач Коши в гильбертовом пространстве

Об условиях квалифицированной сходимости разностных методов и метода квазиобращения для решения линейных некорректных задач Коши в гильбертовом пространстве

Determination of initial data for a reaction-diffusion system with variable coefficients

In this paper, we study a final value problem for a reaction-diffusion system with time and space dependent diffusion coefficients. In general, the inverse problem of identifying the initial data is not well-posed, and herein the Hadamard-instability occurs. Applying a new version of a modified quasi-reversibility method, we propose a stable approximate (regularized) problem. The existence, uniqueness and stability of the corresponding regularized problem are obtained. Furthermore, we also investigate the error estimate and show that the approximate solution converges to the exact solution in \begin{document}$L_2$\end{document} and \begin{document}$\stackrel{0}{H_1}$\end{document} norms. Our method can be applied to some concrete models that arise in biology, chemical engineering, etc.

On a backward problem for the Kirchhoff’s model of parabolic type

We study for the first time the backward problem for nonlocal nonlinear boundary value problem of Kirchhoff’s model of parabolic type. First, we show that the problem is severely ill-posed in the sense of Hadamard. We propose two methods: the Fourier truncation method for stabilizing the problem with homogeneous source and the quasi-reversibility method for regularizing the problem with nonlinear source. Under some priori assumptions on the exact solution, we establish some stability estimates in the H01 norm.

Regularization of the Solution of the Cauchy Problem: The Quasi-Reversibility Method

Some regularization algorithm is proposed related to the problem of continuation of the wave field from the planar boundary into the half-plane. We consider a hyperbolic equation whose main part coincideswith the wave operator, whereas the lowest term contains a coefficient depending on the two spatial variables. The regularization algorithm is based on the quasi-reversibility method proposed by Lattes and Lions. We consider the solution of an auxiliary regularizing equation with a small parameter; the existence, the uniqueness, and the stability of the solution in the Cauchy data are proved. The convergence is substantiated of this solution to the exact solution as the small parameter vanishes. A solution of an auxiliary problem is constructed with the Cauchy data having some error. It is proved that, for a suitable choice of a small parameter, the approximate solution converges to the exact solution.

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

In this paper, we find a regularized approximate solution for an inverse problem for Burgers’ equation. The solution of the inverse problem for 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.

Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction

The identification of the population density of a logistic equation backwards in time associated with nonlocal diffusion and nonlinear reaction, motivated by biology and ecology fields, is investigated. The diffusion depends on an integral average of the population density whilst the reaction term is a global or local Lipschitz function of the population density. After discussing the ill-posedness of the problem, we apply the quasi-reversibility method to construct stable approximation problems. It is shown that the regularized solutions stemming from such method not only depend continuously on the final data, but also strongly converge to the exact solution in L2-norm. New error estimates together with stability results are obtained. Furthermore, numerical examples are provided to illustrate the theoretical results.

On the simultaneous recovery of boundary heat transfer coefficient and initial heat status

Abstract Consider the heat conduction process with heat flux exchanges on the boundary of a 2-dimensional domain. The aim is to identify both the boundary heat exchange coefficient and the initial heat distribution simultaneously from the final measurement data of the heat field. We prove the uniqueness for this nonlinear inverse problem for the strictly positive exchange coefficient and initial heat distribution in terms of the maximal principle and the eigenfunction expansions. Then a regularizing scheme combining the data mollification and quasi-reversibility method along time direction together is established to recover two unknowns, with the choice strategies for the regularizing parameters and the error estimates on the regularizing solutions. The reconstruction implementations are carried out in terms of the potential representation of the heat field, with numerical examples showing the validity of the proposed scheme.

Quantification of the unique continuation property for the heat equation

In this paper we prove a logarithmic stability estimate in the whole domain for the solution to the heat equation with a source term and lateral Cauchy data. We also prove its optimality up to the exponent of the logarithm and show an application to the identification of the initial condition and to the convergence rate of the quasi-reversibility method.

Regularization of an inverse nonlinear parabolic problem with time-dependent coefficient and locally Lipschitz source term

We consider a backward problem of finding a function u satisfying a nonlinear parabolic equation in the form ut+a(t)Au(t)=f(t,u(t)) subject to the final condition u(T)=φ. Here A is a positive self-adjoint unbounded operator in a Hilbert space H and f satisfies a locally Lipschitz condition. This problem is ill-posed. Using quasi-reversibility method, we shall construct a regularized solution uε from the measured data aε and φε. We show that the regularized problems are well-posed and that their solutions converge to the exact solutions. Error estimates of logarithmic type are given and a simple numerical example is presented to illustrate the method as well as verify the error estimates given in the theoretical parts.

On an inverse problem for fractional evolution equation

In this paper, we investigate a backward problem for a fractional abstract evolution equation for which we wants to extract the initial distribution from the observation data provided along the final time $t = T.$ This problem is well-known to be ill-posed due to the rapid decay of the forward process. We consider a final value problem for fractional evolution process with respect to time. For this ill-posed problem, we construct two regularized solutions using quasi-reversibility method and quasi-boundary value method. The well-posedness of the regularized solutions as well as the convergence property is analyzed. The advantage of the proposed methods is that the regularized solution is given analytically and therefore is easy to be implemented. A numerical example is presented to show the validity of the proposed methods.

Quasi-Reversibility Method for an Ill-Posed Nonhomogeneous Parabolic Problem

ABSTRACTThe quasi-reversibility method is considered for the non-homogeneous backward Cauchy problem ut+Au = f(t), u(τ) = ϕ for 0≤t<τ, which is known to be an ill-posed problem. Here, A is a densely defined positive self-adjoint unbounded operator on a Hilbert space H with given data f∈L1([0,τ],H) and ϕ∈H. Error analysis is considered when the data ϕ, f are exact and also when they are noisy. The results obtained generalize and simplify many of the results available in the literature.

The “Exterior Approach" Applied to the Inverse Obstacle Problem for the Heat Equation

In this paper we consider the " exterior approach " to solve the inverse obstacle problem for the heat equation. This iterated approach is based on a quasi-reversibility method to compute the solution from the Cauchy data while a simple level set method is used to characterize the obstacle. We present several mixed formulations of quasi-reversibility that enable us to use some classical conforming finite elements. Among these, an iterated formulation that takes the noisy Cauchy data into account in a weak way is selected to serve in some numerical experiments and show the feasibility of our strategy of identification.

Stability estimates for Navier-Stokes equations and application to inverse problems

In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method.

Stable explicit stepwise marching scheme in ill-posed time-reversed viscous wave equations

The numerical computation of ill-posed, nonlinear, multidimensional initial value problems presents considerable difficulties. Conventional stepwise marching schemes for such problems, whether explicit or implicit, are necessarily unconditionally unstable and result in explosive noise amplification. Following previous work on backward parabolic equations, this paper develops and analyses a stabilized explicit marching scheme for ill-posed time-reversed viscous wave equations. The method uses easily synthesized linear smoothing operators at each time step to quench the instability. Smoothing operators based on positive real powers p of the negative Laplacian are helpful, and can be realized efficiently on rectangular domains using FFT algorithms. The stabilized explicit scheme is unconditionally stable, marching forward or backward in time, and can be applied to nonlinear viscous wave equations by simply lagging the nonlinearity at the previous time step. However, the smoothing operation at each step leads to a distortion away from the true solution. This is the stabilization penalty. It is shown that in many problems of interest, that distortion is often small enough to allow for useful results. In the canonical case of linear autonomous selfadjoint time-reversed viscous wave equations, with solutions satisfying prescribed bounds, it is proved that the stabilized explicit scheme leads to an error estimate differing from the best-possible estimate, only by the stabilization penalty. The procedure is a valuable complement to the well-known quasi-reversibility method. As illustrative examples, the paper uses fictitiously blurred pixel images, obtained using sharp images as initial values in linear or nonlinear well-posed, forward viscous wave equations. Such images are associated with highly irregular underlying data intensity surfaces that can severely challenge reconstruction procedures. Deblurring these images proceeds by applying the stabilized explicit scheme on the corresponding ill-posed, time-reversed equation. Instructive computational experiments demonstrate the capabilities of the method on 2D rectangular regions.