Noisy Cauchy Data Research Articles

Analysis and simulation of a variational stabilization for the Helmholtz equation with noisy Cauchy data

This article considers a Cauchy problem of Helmholtz equations whose solution is well known to be exponentially unstable with respect to the inputs. In the framework of variational quasi-reversibility method, a Fourier truncation is applied to appropriately perturb the underlying problem, which allows us to obtain a stable approximate solution. The corresponding approximate problem is of a hyperbolic equation, which is also a crucial aspect of this approach. Error estimates between the approximate and true solutions are derived with respect to the noise level. From this analysis, the Lipschitz stability with respect to the noise level follows. Some numerical examples are provided to see how our numerical algorithm works well.

A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain

The Cauchy problem of the Laplace equation is investigated for both exact and perturbed data on a doubly connected domain, i.e., the numerical reconstruction of the function value and the normal derivative value on a part of the boundary from the knowledge of exact or noisy Cauchy data on the remaining and accessible boundary, which is completely different from the Cauchy problem on a simply connected bounded region. We first establish the existence of a solution through the potential theory. By expressing the solution as a sum of single-layer potentials using boundary value condition, we get the integral equation systems about the density function on the boundary, and by applying local regularization scheme to the obtained integral equation systems, we get the regularization solution of the original problem. Some numerical results are presented to validate the applicability and effectiveness of the proposed method.

The Gradient Descent Method for the Convexification to Solve Boundary Value Problems of Quasi-Linear PDEs and a Coefficient Inverse Problem

We study the global convergence of the gradient descent method of the minimization of strictly convex functionals on an open and bounded set of a Hilbert space. Such results are unknown for this type of sets, unlike the case of the entire Hilbert space. Then, we use our result to establish a general framework to numerically solve boundary value problems for quasi-linear partial differential equations with noisy Cauchy data. The procedure involves the use of Carleman weight functions to convexify a cost functional arising from the given boundary value problem and thus to ensure the convergence of the gradient descent method above. We prove the global convergence of the method as the noise tends to 0. The convergence rate is Lipschitz. Next, we apply this method to solve a highly nonlinear and severely ill-posed coefficient inverse problem, which is the so-called back scattering inverse problem. This problem has many real-world applications. Numerical examples are presented.

BEM-Fading regularization algorithm for Cauchy problems in 2D anisotropic heat conduction

We investigate the numerical reconstruction of the missing thermal boundary data on a part of the boundary for the steady-state heat conduction equation in anisotropic solids from the knowledge of exact or noisy Cauchy data on the remaining and accessible boundary. This inverse boundary value problem is tackled by applying and adapting to the anisotropic case the algorithm based on the fading regularization method, originally proposed by Cimetiere, Delvare, and Pons (Comptes Rendus de l’Academie des Sciences - Serie IIb - Mecanique, 328 639–644 2000), and Cimetiere, Delvare, et al. (Inverse Probl., 17 553–570 2001) for the isotropic heat conduction equation. The numerical implementation is realised for 2D homogeneous solids by using the boundary element method, whilst the numerical solution is stabilized/regularized by stopping the iterative process based on an L-curve type criterion (Hansen 1998).

Determining two coefficients in diffuse optical tomography with incomplete and noisy Cauchy data

In this paper we investigate the non-linear and ill-posed inverse problem of simultaneously identifying the conductivity and the reaction in diffuse optical tomography with noisy measurement data available on an accessible part of the boundary. We propose an energy functional method and the total variational regularization combining with the quadratic stabilizing term to formulate the identification problem to a PDE constrained optimization problem. We show the stability of the proposed regularization method and the convergence of the finite element regularized solutions to the identification in the L s -norm for all s ∈ [0, ∞) and in the sense of the Bregman distance with respect to the total variation semi-norm. To illustrate the theoretical results, a numerical case study is presented which supports our analytical findings.

On the alternating method and boundary-domain integrals for elliptic Cauchy problems

A numerical implementation of the alternating iterative method is presented for the Cauchy problem of the conductivity equation in planar annular domains. The conductivity is space dependent as opposed to earlier works, where it is assumed constant. In the constant conductivity case, the mixed problems in the iterations are solved efficiently using boundary integrals. Following on to this, for a space dependent conductivity, it is outlined how to solve these mixed problems using a recent boundary-domain integral equations approach involving the parametrix. For a mixed problem the solution is written as a combination of boundary integrals and a domain integral, with densities to be determined. The densities needed over the boundary and domain are identified by matching against the given mixed boundary data and by requiring the governing equation to hold in the domain. An efficient Nyström scheme is applied for the discretisation. Numerical results are presented for several domains and conductivities, using exact as well as noisy Cauchy data, showing that a stable solution can be obtained with good accuracy and small computational cost.

Regularization of a Cauchy problem for the heat equation

In this paper, we study a Cauchy problem for the heat equation with linear source in the form ut(x,t)= uxx(x,t)+f(x,t), u(L,t)= φ(t), u(L,t)= Ψ (t), (x,t) ∈ (0,L) ×(0, 2π). This problem is ill-posed in the sense of Hadamard. To regularize the problem, the truncation method is proposed to solve the problem in the presence of noisy Cauchy data φε and Ψε satisfying ‖ φε - φ ‖+‖ Ψε - Ψ ‖ ≤ ε and that fε satisfying ‖ fε(x,. ) - f(x,.) ‖ ≤ ε . We give some error estimates between the regularized solution and the exact solution under some different a-priori conditions of exact solution.

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.

Numerical analysis of an energy-like minimization method to solve a parabolic Cauchy problem with noisy data

This paper is concerned with solving the Cauchy problem for the parabolic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to an adapted stopping criteria depending on noise rate for the minimization process. Numerical experiments are performed and confirm the theoretical convergence order and the good behavior of the minimization process.

Singular boundary method for inverse heat conduction problems in general anisotropic media

This study documents the first attempt to apply the singular boundary method, a recently developed meshless boundary collocation method, to the solution of inverse anisotropic heat conduction problems. The method cures the perplexing fictitious boundary issue associated with the method of fundamental solutions while inheriting the merits of the latter of being truly meshless, integration-free and easy to programme. Thanks to its boundary-only discretization and semi-analytical nature, the method can be viewed as an ideal candidate for the solution of inverse problems. Four benchmark examples indicate that the proposed method, in connection with proper regularization techniques, is accurate, computationally efficient and numerically stable for the solution of inverse problems with various levels of noisy Cauchy data.

Combined energy method and regularization to solve the Cauchy problem for the heat equation

This paper deals with an energy method coupled with total variation regularization and an adequate stopping criterion in order to solve a Cauchy problem for the heat equation when using noisy data. First, the Cauchy problem is written as a data completion one, then it is split into two well-posed thermal problems. Therefore, a pseudo-energy functional measuring the gap between solutions of these two problems is introduced and minimized. The problem is then converted into one of constrained optimization; the computation of the gradient of this functional is given for the full time–space discrete problems by means of the adjoint method. In order to deal with noisy data, two regularization techniques were used, the first one which fits into the optimization procedure is an adequate stopping criterion depending on the noise level and it avoids numerical instability. The second one is a total variation regularization method which can be carried out a priori and/or a posteriori of the optimization procedure. Numerical experiments are performed on the noisy Cauchy data and/or the identified data. Numerical experiments highlight the efficiency and weakness of the coupled methods.

Regularization of the noisy Cauchy problem solution approximated by an energy‐like method

SUMMARYThis paper focuses on the regularization of noisy Cauchy data and unknown boundary conditions approximated by the energy‐like minimization method. After providing considerations necessary for the numerical analysis leading to an adequate stopping criterion for the minimization process, we describe the denoising procedure proposed. Numerical experiments involving singular data are presented.Copyright © 2013 John Wiley & Sons, Ltd.

Kernel-based approximation for Cauchy problem of the time-fractional diffusion equation

Abstract We investigate in this paper a Cauchy problem for the time-fractional diffusion equation (TFDE). Based on the idea of kernel-based approximation, we construct an efficient numerical scheme for obtaining the solution of a Cauchy problem of TFDE. The use of M -Wright functions as the kernel functions for the approximation space allows us to express the solution in terms of M -Wright functions, whose numerical evaluation can be accurately achieved by applying the inverse Laplace transform technique. To handle the ill-posedness of the resultant coefficient matrix due to the noisy Cauchy data, we adapt the standard Tikhonov regularization technique with the L -curve method for obtaining the optimal regularization parameter to give a stable numerical reconstruction of the solution. Numerical results indicate the efficiency and effectiveness of the proposed scheme.

Numerical analysis of an energy-like minimization method to solve the Cauchy problem with noisy data

This paper is concerned with solving the Cauchy problem for an elliptic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to adapted stopping criteria for the minimization process depending on the noise rate. Numerical examples involving smooth and singular data are presented.

A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data

In this paper, we introduce a new version of the method of quasi-reversibility to solve the ill-posed Cauchy problems for the Laplace's equation in the presence of noisy data. It enables one to regularize the noisy Cauchy data and to select a relevant value of the regularization parameter in order to use the standard method of quasi-reversibility. Our method is based on duality in optimization and is inspired by the Morozov's discrepancy principle. Its efficiency is shown with the help of some numerical experiments in two dimensions.

Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions

In this paper, we consider the Cauchy problem of Laplace's equation in the neighborhood of a circle. The method of fundamental solutions (MFS) combined with the discrete Tikhonov regularization is applied to obtain a regularized solution from noisy Cauchy data. Under the suitable choices of a regularization parameter and an a priori assumption to the Cauchy data, we obtain a convergence result for the regularized solution. Numerical experiments are presented to show the effectiveness of the proposed method.

Generalized time fractional IHCP with Caputo fractional derivatives

The numerical solution of the generalized time fractional inverse heat conduction problem (GTFIHCP) on a finite slab is investigated in the presence of measured (noisy) data when the time fractional derivative is interpreted in the sense of Caputo. The GTFIHCP involves the simultaneous identification of the heat flux and temperature transient functions at one of the boundaries of the finite slab together with the initial condition of the original direct problem from noisy Cauchy data at a discrete set of points on the opposite (active) boundary. A finite difference space marching scheme with adaptive regularization, using trigonometric mollification techniques and generalized cross validation is introduced. Error estimates for the numerical solution of the mollified problem and numerical examples are provided.

Electrical impedance tomography and Mittag-Leffler's function

Given a two-dimensional bounded domain and a conductivity distribution, how to extract information about unknown inclusions in conductivity from finitely many noisy Cauchy data? (In practice, the data are given by electrical impedance tomography (EIT) measurements.) First, a direct approach is presented, based on an extraction formula of the set of all points in the domain that can be connected with infinity by a straight line without intersecting the closure of the inclusions. The formula uses an indicator function that depends on a large parameter τ and can be calculated from infinitely many non-noisy Cauchy data. Second, a modified indicator function is defined. It can be calculated from finitely many noisy Cauchy data. Its properties are studied; the results suggest how many data are needed and how to choose τ. Third, a regularized algorithm that is based on the theoretical study is given for extracting inclusions approximately. The choice of all parameters is described explicitly. Numerical examples using simulated continuum data show that a reliable estimate for the number and location of inclusions is achieved. However, the shape of the inclusions is not recovered.