The Inverse Blackbody Radiation Problem

The inverse blackbody radiation problem is a kind of significant problems in mathematical physics. The inverse problem of blackbody radiation is to inverse the internal temperature distribution based on the given energy spectrum data. Its biggest character is ill-posedness. Specifically, the main operator of blackbody radiation equations is a compact operator, and the inverse operator is an unbounded operator. We let the unbounded operator effect on the measured energy spectrum, and then the measurement error will be magnified indefinitely, leading to inversion result distorted.This paper mainly studies the numerical solution of the inverse problem of the blackbody radiation.From the mathematical expressions, we find that the inverse blackbody radiation problem is a kind of Fredholm integral equations defined in infinite interval. Due to its ill-posedness, regularization procedure must be used. This thesis consists of two parts. In the first part, we carry on the theory research about the first kind of Fredholm integral equations in infinite intervals and give the qualitative results of inverse problem. In the second part, we construct a simple and very effective method for numerical inversion. As follows:In Chapter Two, we directly discuss the general inverse problem of Fredhom equations in infinite interval. Because of its ill-posedness, we need to adopt the regularization method. Here we introduced the Tikhonov regularization method to conduct the research, which is the best method to solve ill-posedness. The basic idea of the method is to instruct a cluster of posed problems to approach to the original problem. Then we hope to seek the solution of posed problem as the approximate solution. The flattened functional is constructed with the standardized Sobolev space norm. And we apply the posterior Morozov discrepancy principle to determine the regularization parameter. Finally we will prove the existence of the solution and then discuss how to solve the problem using the variational method.In Chapter Three, on the basis of the theoretical analysis to the front part, we construct a numerical method to calculate the inverse blackbody radiation problem. The numerical calculation of the inverse blackbody radiation problem requires to discrete the integral term. Previous researchers adopt numerical integral formula with many nodes to discrete the integral, just in order to make the integral calculation accurate. But much more discrete nodes result in larger dimension of the coefficient matrix, larger condition number, exacerbating morbidity and more serious ill-posedness. Even worse, the approximate solution may not be close to the true solution. These methods bring so many calculations and are not convenient for computer implementation. Through going deep into analysis, we finally decide to discrete integral term with Gauss type quadrature formulas which have high accuracy. Due to the integral interval is infinite, so we choose the Gauss-Laguerre quadrature formula to discrete the integral term. After discretization, we get linear system which is not posed problem. Therefore, the regularized equation is regularized to be a linear system. Furthermore, we adapt L-curve to select the regularization parameter. Then we can solve the model directly. After this, we simulate the approximate solution by numerical experiments. Numerical simulation results show that our algorithm has higher precision, stronger operability and is quite effective and simple.