A Class Of New-type Regularization Methods For Several Inverse Source Problems From Multispectral Bioluminescence Tomography

Over the past twenty years, inverse problems with applications have become one of the fastest developing branch in the field of applied and computational mathematics. It is of both theoretical and practical importance to investigate theories and numerical methods for inverse problems, which frequently occur in the areas of biomedical imaging, molecu-lar image processing, non-invasive detection and so on. Especially in molecular imaging, the newly emerging BLT (Bioluminescence Tomography) problem [1-10] and multispec-tral BLT problem [11-17] can be formulated as a type of ill-posed inverse source problems [18,19,11,20-22,13,23,24], which have been attacked by a series of numerical methods [19-23,25]. In this thesis, we intend to propose a new method for reconstructing the source function in BLT. The convergence property and error estimate are discussed when the admis-sible set of source functions is non-discrete or discretized into a bounded set of the piecewise constant function space. If the admissible set of the source is discretized into piecewise linear function space, we show the convergence property of the corresponding method and derive an improved error estimate. For the multispectral BLT problem, a novel approach is pre-sented for reconstructing the source functions within all spectral bands in different regions of the experimental organism. We prove the convergence property and error estimate when the admissible set of the source functions is non-discrete, or discretized into a bounded set of piecewise constant function space or piecewise linear function space. Moreover, a series of numerical experiments are performed to illustrate the computational performance of our methods.First of all, a new method is proposed for reconstructing the source function in BLT. In contrast with several existing methods, the main difference of our method is that the func-tional used for measuring the data fitting is a certain quantity defined over the whole domain under consideration, so that the finite element approximation is quite convenient to imple-ment in practical applications. The main idea behind our method is to reconstruct the source for the ill-posed inverse source problem by solving a certain minimization problem through Tikhonov regularization method. The unique existence of the solution for the minimization problem is ensured by strict convexity of the objective functional. The convergence property and error estimate of the finite element solution for the minimization problem are also dis-cussed. It is shown that the error of the numerical solution is bounded by O(h). Moreover, we discretize the admissible set of the source function into a bounded set of the piecewise constant function space. The convergence property and error estimate of the solution for the discrete minimization problem are derived by careful and subtle finite element analysis, the error of the numerical solution being O(h+(h+H1/2)EII(pε)1/2. The computational efficiency of the new method is illustrated by several numerical tests.Secondly, as a complement and development of the last method, we discretize the ad-missible set of the source by a set of piecewise linear functions. We show the convergence property and error estimate in this case, and the error of the finite element solution is bounded by O(h+(h+h1/2H1/2+H)EH(pε)1/2). Also, several numerical examples are included to show the numerical performance of the method.Finally, a novel approach is presented for reconstructing the source functions of mul-tispectral BLT problem within all spectral bands in different regions of the experimental organism. The reconstruction of the sources in different spectral bands and regions is de-scribed by a series of ill-posed inverse source problems which are transformed into a certain minimization problem through Tikhonov regularization method. The unique existence of the solution for the minimization problems is ensured by strict convexity of the objective functional. We develop the convergence property and error estimate for the numerical so-lution of the corresponding minimization problem. It is shown that the error is bounded by O(h). Then, we discretize the admissible set of the sources into a bounded set of piece-wise constant function space or piecewise linear function space. The convergence prop-erty and error estimate of the solution for the discrete minimization problems are obtained, and the errors for the two cases are bounded by O(h1/2+H1/2‖pεM—∏1HpεM‖Q1/2) and O(h+(h+h1/2H1/2+H)‖pεM—∏2IIpεM‖Q1/2), respectively. A series of numerical results are given to show the computational performance of the method.