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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