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- 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)

Dedicated to Professor Hans Schneider. Abstract 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.… (More)

This work presents a new Krylov-subspace-recycling method for efficiently solving sequences of linear systems of equations characterized by a non-invariant symmetric-positive-definite matrix. As opposed to typical truncation strategies used in recycling such as deflation, we propose a truncation method inspired by goal-oriented proper orthogonal… (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)

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin 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 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)

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

The goal of this project is to efficiently solve a 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-Loéve expansion allows the random coefficient to be approximated with a finite sum of random variables. This… (More)

Title of dissertation: ITERATIVE SOLUTION METHODS FOR REDUCED-ORDER MODELS OF PARAMETERIZED PARTIAL DIFFERENTIAL EQUATIONS Virginia Forstall, Doctor of Philosophy, 2015 Dissertation directed by: Professor Howard Elman Department of Computer Science This dissertation considers efficient computational algorithms for solving parameterized discrete partial… (More)

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-Loève expansion allows the random coefficient to be approximated with a finite sum of… (More)

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