Volume 4, Issue 3-4
Reduced Order Medeling of Some Nonlinear Stochastic Partial Differential Equations.

J. Burkardt, Max D. Gunzburger & C. Webster

DOI:

Int. J. Numer. Anal. Mod., 4 (2007), pp. 368-391

Published online: 2007-04

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  • Abstract

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.

  • Keywords

reduced order modeling stochastic differential equations Brownian motion Monte Carlo methods finite element methods

  • AMS Subject Headings

35R35 49J40 60G40

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-4-368, author = {J. Burkardt, Max D. Gunzburger and C. Webster}, title = {Reduced Order Medeling of Some Nonlinear Stochastic Partial Differential Equations.}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2007}, volume = {4}, number = {3-4}, pages = {368--391}, abstract = {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. }, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/867.html} }
TY - JOUR T1 - Reduced Order Medeling of Some Nonlinear Stochastic Partial Differential Equations. AU - J. Burkardt, Max D. Gunzburger & C. Webster JO - International Journal of Numerical Analysis and Modeling VL - 3-4 SP - 368 EP - 391 PY - 2007 DA - 2007/04 SN - 4 DO - http://dor.org/ UR - https://global-sci.org/intro/article_detail/ijnam/867.html KW - reduced order modeling KW - stochastic differential equations KW - Brownian motion KW - Monte Carlo methods KW - finite element methods AB - 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.
J. Burkardt, Max D. Gunzburger & C. Webster. (1970). Reduced Order Medeling of Some Nonlinear Stochastic Partial Differential Equations.. International Journal of Numerical Analysis and Modeling. 4 (3-4). 368-391. doi:
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