Dinh Nho Hào

Stability and Regularization of a Discrete Approximation to the Cauchy Problem for Laplace's Equation

The standard five-point difference approximation to the Cauchy problem for Laplaces equation satisfies stability estimates---and hence turns out to be a well-posed problem---when a certain boundedness requirement is fulfilled. The estimates are of logarithmic convexity type. Herewith, a regularization method will be proposed and associated error bounds can be derived. Moreover, the error between the given (continuous) Cauchy problem and the difference approximation obtained via a suitable minimization problem can be estimated by a discretization and a regularization term.

On a sideways parabolic equation

The sideways parabolic equation in the quarter plane is considered. This is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but one side is inaccessible to measurements. This problem is well known to be severely ill-posed: a small perturbation in the data, g, may cause dramatically large errors in the solution. The results available in the literature are mainly devoted to the case of constant coefficients, where one can find an explicit representation for the solution of the problem. In this paper a stability estimate of the Holder type for the solution of this general problem is established, it is also shown how to apply the mollification method recently proposed by Dinh Nho Hao to solve the problem in a stable way.

A non-local boundary value problem method for the Cauchy problem for elliptic equations

Let H be a Hilbert space with norm || ||, A:D(A) ⊂ H → H a positive definite, self-adjoint operator with compact inverse on H, and T and given positive numbers. The ill-posed Cauchy problem for elliptic equations is regularized by the well-posed non-local boundary value problem with a ≥ 1 being given and α > 0 the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.

Regularization of a non-characteristic Cauchy problem for a parabolic equation

In this paper the non-characteristic Cauchy problem ut- alpha (x)uxx-b(x)ux-c(x)u=0, x in (0,l), t in R; u(0,t)= phi (t), t in R; ux(0,t)=0, t in R; is considered. The problem is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. In this paper the following mollification method is suggested for this problem: if the Cauchy data are given inexactly then we mollify them by elements of well-posedness classes of the problem, namely by elements of an appropriate co-regular multiresolution approximation {Vj}j in Z of L2(R) which is generated by the father wavelet of Meyer (1992). Within VJ the problem is well posed, and we can find a mollification parameter J depending on the noise level epsilon in the Cauchy data such that the error estimation between the exact solution and the mollified solution is of Holder type. The method can be numerically implemented using fundamental results by Beylkin, Coifman and Rokhlin (1991) on representing (pseudo)differential operators in wavelet bases. A stable marching difference scheme based on this method is suggested. Several numerical examples are given.

A noncharacteristic Cauchy problem for linear parabolic equations. II : A variational method

Many inverse heat conduction problems lead us to consider the following noncharacteristic Cauchy problem for parabolic equations of the form “surface temperature” “surface heat flux” where p is an elliptic operator, ϕ and g are given functions. This problem is well-known to be severely ill-posed, and up to now there have been many approaches for solving it in a stable way. However, most of them need a supplementary condition: either the initial condition, or a boundary condition, etcIn this paper a variational method for this problem is suggested. In contrast to the other works, in the paper the initial condition is not assumed to be known. A short discussion on using the gradient methods is also given.

Infrared Thermography for Buried Landmine Detection: Inverse Problem Setting

This paper deals with an inverse problem arising in infrared (IR) thermography for buried landmine detection. It is aimed at using a thermal model and measured IR images to detect the presence of buried objects and characterize them in terms of thermal and geometrical properties. The inverse problem is mathematically stated as an optimization one using the well-known least-square approach. The main difficulty in solving this problem comes from the fact that it is severely ill posed due to lack of information in measured data. A two-step algorithm is proposed for solving it. The performance of the algorithm is illustrated using some simulated and real experimental data. The sensitivity of the proposed algorithm to various factors is analyzed. A data processing chain including anomaly detection and characterization is also introduced and discussed.

Gradient methods for inverse heat conduction problems

A variational formulation for inverse heat conduction problems (IHCP) is studied. Various fast and efficient gradient methods based on this formulation are presented. Several numerical examples are discussed. The present paper consists in a more detailed study and improvement of the method previously developed by the authors in [22].

A boundary element method for a multi-dimensional inverse heat conduction problem

In this paper, we investigate a variational method for a multi-dimensional inverse heat conduction problem in Lipschitz domains. We regularize the problem by using the boundary element method coupled with the conjugate gradient method. We prove the convergence of this scheme with and without Tikhonov regularization. Numerical examples are given to show the efficiency of the scheme.

Convergence rates for total variation regularization of coefficient identification problems in elliptic equations I

We investigate the convergence rates for total variation regularization of the problem of identifying (i) the coefficient q in the Neumann problem for the elliptic equation , and (ii) the coefficient a in the Neumann problem for the elliptic equation , when u is imprecisely given by z? in . We regularize these problems by correspondingly minimizing the convex functionals and over the admissible sets, where U(q) (U(a)) is the solution of the first (second) Neumann boundary value problem; ? > 0 is the regularization parameter. Taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems, we obtain the convergence rates of them to a total variation-minimizing solution in the sense of the Bregman distance under relatively simple source conditions without the smallness requirement on the source functions.

Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations

We investigate the convergence rates for Tikhonov regularization of the problem of identifying (1) the coefficient q L fty(?) in the Dirichlet problem ?div(q?u) = f in ?, u = 0 on ??, and (2) the coefficient a L fty(?) in the Dirichlet problem ??u + au = f in ?, u = 0 on ??, when u is imprecisely given by z? H10(?), , We regularize these problems by correspondingly minimizing the strictly convex functionals and where U(q) (U(a)) is the solution of the first (second) Dirichlet problem, ? > 0 is the regularization parameter and q* (or a*) is an a priori estimate of q (or a). We prove that these functionals attain a unique global minimizer on the admissible sets. Further, we give very simple source conditions without the smallness requirement on the source functions which provide the convergence rate for the regularized solutions.