Orthogonal expansion of ground motion and PDEM-based seismic response analysis of nonlinear structures

This paper introduces an orthogonal expansion method for general stochastic processes. In the method, a normalized orthogonal function of time variable t is first introduced to carry out the decomposition of a stochastic process and then a correlated matrix decomposition technique, which transforms a correlated random vector into a vector of standard uncorrelated random variables, is used to complete a double orthogonal decomposition of the stochastic processes. Considering the relationship between the Hartley transform and Fourier transform of a real-valued function, it is suggested that the first orthogonal expansion in the above process is carried out using the Hartley basis function instead of the trigonometric basis function in practical applications. The seismic ground motion is investigated using the above method. In order to capture the main probabilistic characteristics of the seismic ground motion, it is proposed to directly carry out the orthogonal expansion of the seismic displacements. The case study shows that the proposed method is feasible to represent the seismic ground motion with only a few random variables. In the second part of the paper, the probability density evolution method （PDEM） is employed to study the stochastic response of nonlinear structures subjected to earthquake excitations. In the PDEM, a completely uncoupled one-dimensional partial differential equation, the generalized density evolution equation, plays a central role in governing the stochastic seismic responses of the nonlinear structure. The solution to this equation will yield the instantaneous probability density function of the responses. Computational algorithms to solve the probability density evolution equation are described. An example, which deals with a nonlinear frame structure subjected to stochastic ground motions, is illustrated to validate the above approach.

NECESSARY AND SUFFICIENT CONDITIONS FOR EXPANSIONS OF GABOR TYPE

In this paper, we give necessary and sufficient conditions for two families of Gabor functions of a certain type to yield a reproducing identity on L^2（R^n）. As applications, we characterize when such families yield orthonormal or bi-orthogonal expansions. We also obtain a generalization of the Balian-Low theorem for general reprodueing identities （not necessary coming from a frame）.

Orthogonal polynomial expansions for the valuation ofoptions under the stochastic volatility models with stochastic correlation

This work provides a new method for pricing options under the generalized stochastic volatility models with Jacobi stochastic correlation.Our method is based on the observation that the generalized models belong to the class of polynomial diffusions and therefore the option prices can be efficiently computed via orthogonal polynomial expansions.We take the Heston and Schöbel-Zhu models with stochastic correlation as two specific examples and are able to derive the analytical formulas for the option prices.We also illustrate the accuracy of the proposed method through a number of numerical experiments.

CONVERGENCE AND SUPERCONVERGENCE ANALYSIS OF LAGRANGE RECTANGULAR ELEMENTS WITH ANY ORDER ON ARBITRARY RECTANGULAR MESHES

This paper is to study the convergence and superconvergence of rectangular finite elements under anisotropic meshes. By using of the orthogonal expansion method, an anisotropic Lagrange interpolation is presented. The family of Lagrange rectangular elements with all the possible shape function spaces are considered, which cover the Intermediate families, Tensor-product families and Serendipity families. It is shown that the anisotropic interpolation error estimates hold for any order Sobolev norm. We extend the convergence and superconvergence result of rectangular finite elements to arbitrary rectangular meshes in a unified way.