| In this thesis, we shall propose some numerical methods for solving two important classes of application problems, namely the inverse heat source problems and the backward stochastic differential equations.;The inverse heat source problems are to recover the source terms in a convection-diffusion-reaction system. These inverse problems have wide applications in many areas, such as engineering, chemistry, biology, pollutant tracking, and so on. We shall first investigate the simultaneous reconstruction of the location and strength of a static singular source. An adjoint probabilistic algorithm is proposed, which turns the inverse heat source problem into an inverse Volterra integral problem. The identifiability of the location and strength of a singular source is also discussed, and numerical results are presented to show the robustness and effectiveness of the method. Then we extend the adjoint probabilistic method to reconstruct the source trace and release history of a singular moving point source. Numerical examples show that the adjoint probabilistic method is more efficient and less expensive than most existing efficient numerical methods.;The second part of the thesis is devoted to numerical solutions of some nonlinear backward stochastic differential equations (BSDEs). BSDEs are widely used in various fields like stochastic control, biology, chemistry reaction, especially mathematical finance. Our numerical methods are based on a new framework about the transposition solution to BSDEs. The proof of the well-posedness of the transposition solution does not involve Martingale representation, neither does our error analysis for the numerical schemes proposed in this thesis. For general BSDEs, we first propose a simple backward scheme, which is proved to have an accuracy of half order. However, in the real application of the scheme, it is only possible to choose a finite subset of basis functions, which will generate truncation error. The truncation error accumulates backward in time, leading to the increment of the numerical error up to a half order. To overcome this drawback, we propose a new numerical scheme without Picard iterations and prove that the truncation error is bounded independent of time partitions. Afterwards, we propose some higher order schemes for Markovian BSDEs with rigorous error analysis. Finally, numerical simulations are presented to demonstrate that the proposed methods are accurate, stable and less expensive than most existing ones. |
