# Hermite polynomial expansions : theory and computation

- Responsibility
- Laura A. Lyman.
- Publication
- [Stanford, California] : [Stanford University], 2023.
- Copyright notice
- ©2023
- Physical description
- 1 online resource.

## Digital content

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## Description

### Creators/Contributors

- Author/Creator
- Lyman, Laura Abigail, author.
- Contributor
- Iaccarino, Gianluca, degree supervisor. Thesis advisor
- Gerritsen, Margot, degree committee member. Thesis advisor
- Owen, Art, degree committee member. Thesis advisor
- Stanford University. School of Engineering.
- Stanford University. Institute for Computational and Mathematical Engineering.

### Contents/Summary

- Summary
- In uncertainty quantification, polynomial chaos is a non-sampling technique that can be used to approximate the solution of a partial differential equation (PDE) with random variable parameters. The stochastic PDE solution is expressed as a countably infinite polynomial expansion that is truncated via an M bound for the sake of approximation. The approach is then to derive a resulting system of deterministic (often coupled) PDEs, called an M system, with Galerkin projection and solve the M system for the expansion coefficients with standard numeric techniques. Some challenges with traditional computational methods applied in this context are as follows: (1) the solution to a polynomial chaos M system cannot easily reuse an already existing computer solution to an M_0 system for some M_0 < M, and (2) there is no flexibility around choosing which variables in an M system are more advantageous to estimate accurately. This latter point is especially relevant when, rather than obtaining the PDE solution itself, the objective is to approximate some function of the solution that weights the coefficient variables with relative levels of importance. In the first portion of this thesis, we present a promising iterative algorithm (bluff-and-fix) to address challenges (1) and (2). For the former, we find that numerical estimates of the accuracy and efficiency demonstrate that bluff-and-fix can be more effective than using conventional methods to solve a series of M systems directly. The work then showcases how bluff-and-fix successfully addresses challenge (2) by allowing for choice in which variables are better approximated, in particular when estimating statistical properties such as the mean and variance of an M system solution. In the presence of multiple uncertainties, Galerkin projection introduces products (Nth order moments) of the basis polynomials. Prior to this dissertation, when uncertainties were represented by correlated random variables, there was no closed-form expression for these products for any N > 2 even when the uncertainties had a joint Gaussian distribution. Consequently, the products were typically computed via sampling methods, which can (a) become computationally expensive to implement, and (b) introduce errors if the sample count is insufficient. In the second portion of this thesis, we offer a new expression by introducing multiset notation for the polynomial indexing that allows for the simple and efficient evaluation of the double products (N = 2). We then present a number of theoretical results for correlated multivariate Hermite polynomials, including a formula for the Nth order product in terms of the double product computations. These contributions will allow polynomial chaos methods to be more readily applied to solving stochastic partial differential equations with dependent Gaussian parameters.

### Bibliographic information

- Publication date
- 2023
- Copyright date
- 2023
- Note
- Submitted to the Institute for Computational and Mathematical Engineering.
- Note
- Thesis Ph.D. Stanford University 2023.