Two numerical methods for solving a backward heat conduction problem

Abstract We introduce a central difference method and a quasi-reversibility method for solving a backward heat conduction problem (BHCP) numerically. For these two numerical methods, we give the stability analysis. Meanwhile, we investigate the roles of regularization parameters in these two methods. Numerical results show that our algorithm is effective

A source identification problem in a bi-parabolic equation: convergence rates and some optimal results

This paper is concerned with identification of a spatial source function from final time observation in a bi-parabolic equation, where the full source function is assumed to be a product of time dependent and a space dependent function. Due to the ill-posedness of the problem, recently some authors have employed different regularization method and analysed the convergence rates. But, to the best of our knowledge, the quasi-reversibility method is not explored yet, and thus we study that in this paper. As an important implication, the H{\"o}lder rates for the apriori and aposteriori error estimates obtained in this paper improve upon the rates obtained in earlier works. Also, in some cases we show that the rates obtained are of optimal order. Further, this work seems to be the first one that has broaden the applicability of the problem by allowing the time dependent component of the source function to change sign. To the best of our knowledge, the earlier known work assumed the fixed sign of the time dependent component by assuming some bounded below condition.Comment: Comments are welcome. Typos and some mistakes with sign in the PDE are rectified. Section 4 and 5 are majorly revise

Partial Volume Reduction by Interpolation with Reverse Diffusion

Many medical images suffer from the partial volume effect where a boundary between two structures of interest falls in the midst of a voxel giving a signal value that is a mixture of the two. We propose a method to restore the ideal boundary by splitting a voxel into subvoxels and reapportioning the signal into the subvoxels. Each voxel is divided by nearest neighbor interpolation. The gray level of each subvoxel is considered as “material” able to move between subvoxels but not between voxels. A partial differential equation is written to allow the material to flow towards the highest gradient direction, creating a “reverse” diffusion process. Flow is subject to constraints that tend to create step edges. Material is conserved in the process thereby conserving signal. The method proceeds until the flow decreases to a low value. To test the method, synthetic images were downsampled to simulate the partial volume artifact and restored. Corrected images were remarkably closer both visually and quantitatively to the original images than those obtained from common interpolation methods: on simulated data standard deviation of the errors were 3.8%, 6.6%, and 7.1% of the dynamic range for the proposed method, bicubic, and bilinear interpolation, respectively. The method was relatively insensitive to noise. On gray level, scanned text, MRI physical phantom, and brain images, restored images processed with the new method were visually much closer to high-resolution counterparts than those obtained with common interpolation methods

A comparison of regularizations for an ill-posed problem

Abstract. We consider numerical methods for a “quasi-boundary value ” regularization of the backward parabolic problem given by ut + Au =0, 0<t<T u(T)=f, where A is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value u(T) by adding αu(0), where α is a small parameter. We show how this leads very naturally to a reformulation of the problem as a second-kind Fredholm integral equation, which can be very easily approximated using methods previously developed by Ames and Epperson. Error estimates and examples are provided. We also compare the regularization used here with that from Ames and Epperson. 1