Landweber–Fridman algorithms for the Cauchy problem in steady-state anisotropic heat conduction

We investigate the numerical reconstruction of the missing thermal boundary conditions on an inaccessible part of the boundary in the case of steady-state heat conduction in anisotropic solids from the knowledge of over-prescribed noisy data on the remaining accessible boundary. This inverse problem is tackled by employing a variational formulation that transforms it into an equivalent control problem; four such approaches are discussed thoroughly. The numerical implementation is realised for the 2D case via the boundary element method for perturbed Cauchy data, whilst the numerical solution is stabilised/regularised by stopping the iterative procedure according to Morozov’s discrepancy principle (Morozov, VA. On the solution of functional equations by the method of regularization. Doklady Mathematics 1966; 7: 414–417).

An alternative algorithm for regularization of noisy volatility calibration in Finance

International audience This contribution is an extension of the work initiated in [1], presenting a strategy for the calibration of the local volatility. Due to Morozov's discrepancy principle [6], the Tikhonov regularization problem introduced in [7] is understood as an inequality-constrained minimization problem. An Uzawa procedure is proposed to replace this latter by a sequence of unconstrained problems dealt with in the modified Thikonov regularization procedure in [1]. Numerical tests confirm the consistency of the approach and the significant speed-up of the process of local volatility determination. Cette contribution dans ce papier est une extension des travaux initiés dans [1], qui pré-sente une stratégie pour l'estimation de la volatilité locale. En raison du principe de la différence de Morozov [6], le problème de la régularisation de Tikhonov introduite dans [7] est reformulé comme un problème de minimisation de l'inégalité des contraintes. Une procédure Uzawa est proposé de remplacer ce dernier par une séquence de problèmes non contraints traités dans la procédure de régularisation Thikonov modifié dans [1]. Des tests numériques confirment la cohérence de l'approche et l'importante accélérer le processus de détermination de la volatilité locale.

ON MOROZOV’S DISCREPANCY PRINCIPLE FOR NONLINEAR ILL-POSED EQUATIONS

Morozov’s discrepancy principle is one of the simplest and most widely used parameter choice strategies in the context of regularization of ill-posed operator equations. Although many authors have considered this principle under general source conditions for linear ill-posed problems, such study for nonlinear problems is restricted to only a few papers. The aim of this paper is to apply Morozov’s discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems under general source conditions.

Morozov’s Discrepancy Principle For The Tikhonov Regularization Of Exponentially Ill-Posed Problems

The problem of approximate solution of severely ill-posed problems given in the form of linear operator equations of the first kind with approximately known right-hand sides was considered. We have studied a strategy for solving this type of problems, which consists in combinating of Morozov’s discrepancy principle and a finite-dimensional version of the Tikhonov regularization. It is shown that this combination provides an optimal order of accuracy on source sets

A SIMPLE REGULARIZATION METHOD FOR SOLVING ACOUSTICAL INVERSE SCATTERING PROBLEMS

The problem of determining the shape of an object from far-field data is considered. We present a method, originally formulated in Ref. 1 and furtherly modified in Ref. 3, for the solution of this ill-posed nonlinear inverse problem whose main features are: • the method is exact, that is no low- or high-frequency approximation is considered; • it is not necessary to know the number of scatterers and whether or not the scatterers are penetrable by the waves; • if the medium is not penetrable, it is not necessary to know whether the obstacle is sound-hard or sound-soft; • in the case of an inhomogeneous scatterer, the method provides the shape of the inhomogeneity. The method is particularly simple since it requires only the solution of a linear Fredholm integral equation of the first kind whose integral kernel is the far-field pattern. The numerical instability due to ill-conditioning can be reduced by using regularization algorithms such as Tikhonov method where the regularization parameter is chosen by using Morozov's discrepancy principle generalized to the case where the noise affects the kernel of the integral operator.