Reduced order medeling of some nonlinear stochastic partial differential equations
John Burkardt 1, Max D. Gunzburger 1, Clayton Webster 21 School of Computational Sciences, Florida State University, Tallahassee, FL, 32306-4120, USA
2 School of Computational Sciences and Department of Mathematics, Florida State University, Tallahassee, FL, 32306-4120, USA
Received by the editors February 26, 2006 and, in revised form, April 25, 2006
Determining accurate statistical information about outputs from ensembles of realizations is not generally possible whenever the input-output map involves the (computational) solution of systems of nonlinear partial differential equations (PDEs). This is due to the high cost of effecting each realization. Recently, in applications such as control and optimization that also require multiple solutions of PDEs, there has been much interest in reduced-order models (ROMs) that greatly reduce the cost of determining approximate solutions. We explore the use of ROMs for determining outputs that depend on solutions of stochastic PDEs. One is then able to cheaply determine much larger ensembles, but this increase in sample size is countered by the lower fidelity of the ROM used to approximate the state. In the contexts of proper orthogonal decomposition-based ROMs, we explore these counteracting effects on the accuracy of statistical information about outputs determined from ensembles of solutions.
AMS subject classifications: 35R35, 49J40, 60G40
Key words: reduced order modeling; stochastic differential equations; Brownian motion; Monte Carlo methods; finite element methods
Email: firstname.lastname@example.org (J. Burkardt), email@example.com (Max D. Gunzburger), firstname.lastname@example.org (C. Webster)