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The interplay between the algebraic and analytic properties of a matrix and the geometric properties of its pseudospectrum is investigated. It is shown that one can characterize Hermitian matrices, positive semi-definite matrices, orthogonal projections, unitary matrices, etc. in terms of the pseudospectrum. Also, characterizations are given to maps on… (More)

The reduced basis methodology is an efficient approach to solve parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space and an online step utilizes this approximation, the reduced basis, to solve a smaller reduced problem, which provides an accurate estimate… (More)

- Howard C. Elman, Virginia Forstall
- SIAM J. Scientific Computing
- 2015

The reduced basis methodology is an efficient approach to solve parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space, and an online step utilizes this approximation, the reduced basis, to solve a smaller reduced problem, which provides an accurate estimate… (More)

- Virginia Forstall, Aaron Herman, Chi-Kwong Li, Nung-Sing Sze, Vincent Yannello
- 2010

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gersh-gorin region, the union of Cassini ovals, and the Ostrowski's set. Characterization is obtained for maps Φ on n × n matrices satisfying S(Φ(A)Φ(B)) = S(AB) for all matrices A and B.

The goal of this project is to explore computational methods designed to solve the steady-state diffusion equation with a random coefficient. Although such equations can be solved using Monte-Carlo methods, the lengthy computation time can be constraining. Using a Karhunen-Lò eve expansion allows the random coefficient to be approximated with a finite sum… (More)

- Kevin Carlberg, Virginia Forstall, Ray S. Tuminaro
- SIAM J. Matrix Analysis Applications
- 2016

Goal: to efficiently solve a steady state diffusion equation with a random coefficient. Monte-Carlo methods are time intensive. Using principal components analysis (also known as the Karhunen-Loéve expansion) allows the random coefficient to be approximated with a finite sum of random variables. This expansion combined with a stochastic finite element… (More)

The goal of this project is to efficiently solve a steady-state diffusion equation with a random diffusion coefficient. Although, such equations can be solved using Monte-Carlo methods, the lengthy computation time can be constraining. Using a Karhunen-Loéve expansion allows the random coefficient to be approximated with a finite sum of random variables.… (More)

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