An Iterative Compensative Method For Solving A Backward Heat Conduction Problem

The inverse problems of mathematical physics experience a rapid growth in the development of modern mathematics. In this paper, we consider a clas-sical inverse problem arising in the research of heat conduction equation, i.e., the Backward Heat Conduction Problem (BHCP). To be specific, we mainly focus on the problem of the following form, where A is a pseudo-differential operator.During recent years, the mathematicians have proposed many available numerical methods to solve the problem above, e.g., the Conjugate Gradi-ent Method[24], the Fourier Regularization Method[14], the Entropy-based Regularization Method[15], the Variational Method[6], the Group Preserving Method[30,31], the Enclosure Method and the Iterative Boundary Element Method[12]. Here we introduce a new kind of iterative regularization method, which has its origin in the Quasi-reversibility method by Lattes and Lions and the (weakly) convergent algorithm by Gajewski and Zacharias. We shall improve the error estimate of our numerical solutions by some compensative implementations, i.e., consider the following iterations and when n is chosen large enough, the iterated solutions would approximate the exact solution in (1.1). With no doubt, we demand the well-posedness and convergence for every iteration.Since there is no explicit description and proof of the optimal logarithmic error estimate of BHCP in the extant references, we shall introduce the concept of the Variable Hilbert Scales, and in this context, give a brief discussion of the worst case error according to the a priori information in terms of||·||p which is the so-called ’stronger’norm.For simplicity, we would confine our discussion to the one-dimensional case of our iterative scheme (1.4) and (1.5). We shall employ the Fourier transform to give the theoretic error analysis, from which it is easily to verify the optimal convergent rates in the theoretical background. In other words, we could achieve the logarithmic stability according to the a priori information in terms of||·||p, i.e., We would also carry out the numerical experiments to show the validity and effectiveness of our iteration strategy. Literally, the numerical results in our paper exhibit a sharp approximation to the exact solution of the BHCP.