| Summary: | 博士 === 國立中興大學 === 應用數學系所 === 104 === In this dissertation, we study the tailored finite point method (TFPM) and polynomial chaos expansion (PCE) scheme for solving partial differential equations (PDEs). These PDEs are related to fluid dynamics, imaging processing and finance problems.
In the first part, we concern on quasilinear time-dependent Burgers'' equations with small coefficients of viscosity. The selected basis functions for the TFPM method automatically fit the properties of the local solution in time and space simultaneously. We apply the Hopf-Cole transformation to derive the first TFPM-I scheme. For the second scheme, we approximate the solution by using local exact solutions and consider iterated processes to attain numerical solutions to the original form of the Burgers'' equation. The TFPM-II is particularly suitable for a solution with steep gradients or discontinuities. More importantly, the TFPM obtained numerical solutions with reasonable accuracy even on relatively coarse meshes for Burgers'' equations.
In the second part, we employ the application of the TFPM in an anisotropic convection-diffusion (ACD) filter for image denoising. A quadtree structure is implemented in order to allow multi-level storage during the denoising and compression process. The ACD filter exhibits the potential to get a more accurate approximated solution to the PDEs.
In the third part, we regard the TFPM for Black-Scholes equations, European option pricing. We compare the performance of our algorithm with other popular numerical schemes. The numerical experiments using the TFPM is more efficient and accurate compared to other well-known methods.
In the last part, we present the polynomial chaos expansion (PCE) for stochastic PDEs. We provide a review of the theory of generalized polynomial chaos expansion (gPCE) and arbitrary polynomial chaos expansion (aPCE) including the case analysis of test problems. We demonstrate the accuracy of the gPCE and aPCE for the Black-Scholes model with the log-normal random volatilities. Furthermore, we employ the aPCE scheme for arbitrary distributions of uncertainty volatilities with short term price data. This is the forefront of adopting the polynomial chaos expansion in the randomness of volatilities in financial mathematics.
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