Hessian‐based model reduction: large‐scale inversion and prediction. (18th January 2012)

• Record Type: Journal Article
• Authors: Lieberman, C.; Fidkowski, K.; Willcox, K.; van, B.
• Is Part Of: International journal for numerical methods in fluids. Volume 71:Number 2(2013:Jan.)
• Page Start: 135
• Page End: 150
• Publication Date: 2012-01-18
• Subjects: Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.3650
• Languages: English
• ISSNs: 0271-2091

Abstract

<abstract abstract-type="main" id="fld3650-abs-0001"> <title>SUMMARY</title> <p id="fld3650-para-0001">Hessian‐based model reduction was previously proposed as an approach in deriving reduced models for the solution of large‐scale linear inverse problems by targeting accuracy in observation outputs. A control‐theoretic view of Hessian‐based model reduction that hinges on the equality between the Hessian and the transient observability gramian of the underlying linear system is presented. The model reduction strategy is applied to a large‐scale (<inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1n1k0dz7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:02712091:media:fld3650:fld3650-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">O</mml:mi><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow></mml:math> degrees of freedom) three‐dimensional contaminant transport problem in an urban environment, an application that requires real‐time computation. In addition to the inversion accuracy, the ability of reduced models of varying dimension to make predictions of the contaminant evolution beyond the time horizon of observations is studied. Results indicate that the reduced models have a factor <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1n1k0f09" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:02712091:media:fld3650:fld3650-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">O</mml:mi><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mn>0</mml:mn><mml:mn>0</mml:mn><mml:mn>0</mml:mn></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow></mml:math> speedup in computing time for the same level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.</p> </abstract>