We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 ≤ x < 1.The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge-Kutta method.We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

The Tikhonov regularization method for discrete ill-posed problems is considered. For the practical choice of the regularization parameter c, some authors use a plot of the norm of the regularized solution versus the norm of the residual vector for all a considered. This paper contains an analysis of the shape of this plot and gives a theoreticaljustification for choosing the regularization parameter so it is related to the "L-comer" ofthe plot considered in the logarithmic scale. Moreover, a new criterion for choosing c is introduced (independent of the shape of the plot) which gives a new interpretation of the "comer criterion" mentioned above. The existence of"L-comer" is discussed.

A problem of reconstruction of the radiation field in a domain Ω ⊂ R 3 from experimental data given on a part of boundary is considered. For the model problem described by a Cauchy problem for the Helmholtz equation, an approximate method based on regularization in the frequency space is analyzed. Convergence and stability are proved under a suitable choice of regularization parameter. Numerical implementation of the method is discussed.

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We consider a Cauchy problem for the heat equation in the quarter plane, where data are given at x = 1 and a solution is sought in the interval 0 < x < 1. This sideways heat equation is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. Meyer wavelets have the property that their Fourier transform has compact support. Therefore, by expanding the data and the solution in a basis of Meyer wavelets, high-frequency components can be filtered away. We show that using a wavelet-Galerkin approach, we restore continuous dependence on the data, and we give a recipe for choosing the coarse level resolution in the wavelet representation, depending on the noise level of the data. Furthermore, we solve the sideways problem numerically in the coarse level representation, as an ordinary differential equation in the space variable, where the time derivative is replaced by its wavelet representation. Numerical examples are given. † In the literature this is sometimes referred to as the inverse heat conduction problem (IHCP). Since there are several inverse problems for parabolic equations, we prefer the term 'sideways heat equation' (cf the 'backwards heat equation', which is a classical inverse problem).‡ Since we are only given g m , satisfying (1.3), it seems inappropriate to require that the given data be matched more accurately than is warranted by the precision of the data.

The paper is concerned with the problem of reconstruction of acoustic or electromagnetic field from inexact data given on an open part of the boundary of a given domain. A regularization concept is presented for the moment problem that is equivalent to a Cauchy problem for the Helmholtz equation. A method of regularization by projection with application of the Meyer wavelet subspaces is introduced and analyzed. The derived formula, describing the projection level in terms of the error bound of the inexact Cauchy data, allows us to prove the convergence and stability of the method.

- We consider an inverse heat conduction problem, the Sideways Heat Equation. This is a Cauchy problem for the heat equation in a quarter-plane, with data given along the line x = 1, where the solution is sought for 0 ≤ x < 1. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We discuss the stability and convergence properties of the wavelet- Galerkin method for solving the sideways heat equation. The wavelets are of Meyer type that have compact support in frequency space. Previous stability results for this method were suboptimal. We show that with additional assumptions concerning the smoothness of the solution, and concerning the definition of the wavelets, we can obtain almost optimal error estimates.