Dimension reduction techniques for multi-dimensional numerical integrations based on Fourier-cosine series expansion

Abstract

A wide range of practical problems involve computing multi-dimensional integrations. However, in most cases, it is hard to find analytical solutions to these multi-dimensional integrations. Their numerical solutions always suffer from the `curse of dimension', which means the computational complexity grows exponentially with respect to the dimension.

There is an existing approach that approximates multivariate functions by a tensor of truncated multi-dimensional Fourier series coefficients, and uses the Stochastic Gradient Descent method to solve the lower-rank CPD model, which is used to reduce the computational complexity of the coefficient tensor. In contrast to this work, this thesis project extends its application to solve multi-dimensional integrations, utilizing the Fourier-cosine series expansion to represent the integrand. This project also replaces the SGD method with the Conjugate Gradient method, which improves the function matching accuracy significantly and also has great integration accuracy. The computational cost is reduced regardingly as well.

This thesis also tests an expectation operator related to the COS method, which can be used to compute the expectation of functions of several random variables with much less computational complexity. This method filters out insignificant Fourier-cosine basis functions of the marginal distribution functions, and uses the selected `principal' basis functions to compute Fourier-cosine coefficients of the joint density function by the high-dimensional COS method. The test results show that more than half of the Fourier-cosine series terms can be dropped per dimension while the expectation accuracy is kept, and the correlation does not influence the expectation accuracy significantly for target functions of normally distributed random variables.