Application And Algorithm Of SDE

In this thesis, we study some applications and algorithms of stochastic differentialequations (SDEs). There are two parts: the first one is about the application of SDEon the elastic behavior of the heterogeneous, the second is concerning with a algo-rithm for flow in randomly heterogeneous porous media, the adaptive Stroud StochasticCollocation Method.In the first part, two different results concerning the elastic behavior of the hetero-geneous wormlike chain (WLC) [D. Bensimon et al., Europhys. Lett., 42 (1998), pp.97] and rod like chain (RLC) [P. Nelson, Phys. Rev. Lett., 80 (1998), pp. 5810] are com-pared in this paper. We argue that the RLC is a more suitable model for double-stranded(ds-) DNA. As the hetero-RLC is the basic model for studying sequence-dependent ds-DNA, a rigorous path integral analysis for the effective bending persistence length isperformed in the weak disorder limit. The novelty of the paper is to analyze a pathintegral on the Lie group SO(3) with random forcing, which supplies a rigorous basisfor the analysis of RLC type models.In the second part, we apply the Stochastic Collocation Method (SCM) to steadyrandom porous media ?ow. At first, the Karhunen-Loe`ve expansion is taken to the logtransformed hydraulic conductivity field of ?ow, which introduces a stochastic PDEonly depends on finite Gaussian random variables. Based upon an error analysis forStroud cubature in SCM and observing the rapid decay of the coefficients for ran-dom variables in the Karhunen-Loe`ve expansion, we propose to subdivide the frontdimensions in the integration space for random variables to increase the accuracy. Wecall this approach adaptive Stroud SCM. One-dimensional and two-dimensional steady-state ?ows in saturated random porous media are simulated with the new method, andcomparisons are made with other stochastic methods: the Monte Carlo method, the ten-sor product SCM, and the quasi-Monte Carlo SCM. The result shows that the adaptiveStroud SCM is more efficient, and the statistical moments of the hydraulic head can bemore accurately estimated.