# Non-Intrusive Solution of Stochastic and Parametric Equations

Handle URI:
http://hdl.handle.net/10754/624116
Title:
Non-Intrusive Solution of Stochastic and Parametric Equations
Authors:
Matthies, Hermann
Abstract:
Many problems depend on parameters, which may be a finite set of numerical values, or mathematically more complicated objects like for example processes or fields. We address the situation where we have an equation which depends on parameters; stochastic equations are a special case of such parametric problems where the parameters are elements from a probability space. One common way to represent this dependability on parameters is by evaluating the state (or solution) of the system under investigation for different values of the parameters. But often one wants to evaluate the solution quickly for a new set of parameters where it has not been sampled. In this situation it may be advantageous to express the parameter dependent solution with an approximation which allows for rapid evaluation of the solution. Such approximations are also called proxy or surrogate models, response functions, or emulators. All these methods may be seen as functional approximations—representations of the solution by an “easily computable” function of the parameters, as opposed to pure samples. The most obvious methods of approximation used are based on interpolation, in this context often labelled as collocation. In the frequent situation where one has a “solver” for the equation for a given parameter value, i.e. a software component or a program, it is evident that this can be used to independently—if desired in parallel—solve for all the parameter values which subsequently may be used either for the interpolation or in the quadrature for the projection. Such methods are therefore uncoupled for each parameter value, and they additionally often carry the label “non-intrusive”. Without much argument all other methods— which produce a coupled system of equations–are almost always labelled as “intrusive”, meaning that one cannot use the original solver. We want to show here that this not necessarily the case. Another approach is to choose some other projection onto the subspace spanned by the approximating functions. Usually this will involve minimising some norm of the difference between the true parametric solution and the approximation. Such methods are sometimes called pseudo-spectral projections, or regression solutions. On the other hand, methods which try to ensure that the approximation satisfies the parametric equation as well as possible are often based on a Rayleigh-Ritz or Galerkin type of “ansatz”, which leads to a coupled system for the unknown coefficients. This is often taken as an indication that the original solver can not be used, i.e. that these methods are “intrusive”. But in many circumstances these methods may as well be used in a non-intrusive fashion. Some very effective new methods based on low-rank approximations fall in the class of “not obviously non-intrusive” methods; hence it is important to show here how this may be computed non-intrusively.
Conference/Event name:
Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2015)
Issue Date:
7-Jan-2015
Type:
Presentation
http://mediasite.kaust.edu.sa/Mediasite/Play/28c95e1b13874b41ac915c04bc8d64e21d?catalog=ca65101c-a4eb-4057-9444-45f799bd9c52
Appears in Collections:
Conference on Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2015)

DC FieldValue Language
dc.contributor.authorMatthies, Hermannen
dc.date.accessioned2017-06-05T08:35:49Z-
dc.date.available2017-06-05T08:35:49Z-
dc.date.issued2015-01-07-
dc.identifier.urihttp://hdl.handle.net/10754/624116-
dc.description.abstractMany problems depend on parameters, which may be a finite set of numerical values, or mathematically more complicated objects like for example processes or fields. We address the situation where we have an equation which depends on parameters; stochastic equations are a special case of such parametric problems where the parameters are elements from a probability space. One common way to represent this dependability on parameters is by evaluating the state (or solution) of the system under investigation for different values of the parameters. But often one wants to evaluate the solution quickly for a new set of parameters where it has not been sampled. In this situation it may be advantageous to express the parameter dependent solution with an approximation which allows for rapid evaluation of the solution. Such approximations are also called proxy or surrogate models, response functions, or emulators. All these methods may be seen as functional approximations—representations of the solution by an “easily computable” function of the parameters, as opposed to pure samples. The most obvious methods of approximation used are based on interpolation, in this context often labelled as collocation. In the frequent situation where one has a “solver” for the equation for a given parameter value, i.e. a software component or a program, it is evident that this can be used to independently—if desired in parallel—solve for all the parameter values which subsequently may be used either for the interpolation or in the quadrature for the projection. Such methods are therefore uncoupled for each parameter value, and they additionally often carry the label “non-intrusive”. Without much argument all other methods— which produce a coupled system of equations–are almost always labelled as “intrusive”, meaning that one cannot use the original solver. We want to show here that this not necessarily the case. Another approach is to choose some other projection onto the subspace spanned by the approximating functions. Usually this will involve minimising some norm of the difference between the true parametric solution and the approximation. Such methods are sometimes called pseudo-spectral projections, or regression solutions. On the other hand, methods which try to ensure that the approximation satisfies the parametric equation as well as possible are often based on a Rayleigh-Ritz or Galerkin type of “ansatz”, which leads to a coupled system for the unknown coefficients. This is often taken as an indication that the original solver can not be used, i.e. that these methods are “intrusive”. But in many circumstances these methods may as well be used in a non-intrusive fashion. Some very effective new methods based on low-rank approximations fall in the class of “not obviously non-intrusive” methods; hence it is important to show here how this may be computed non-intrusively.en
dc.relation.urlhttp://mediasite.kaust.edu.sa/Mediasite/Play/28c95e1b13874b41ac915c04bc8d64e21d?catalog=ca65101c-a4eb-4057-9444-45f799bd9c52en
dc.titleNon-Intrusive Solution of Stochastic and Parametric Equationsen
dc.typePresentationen
dc.conference.dateJanuary 6-9, 2015en
dc.conference.nameAdvances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2015)en
dc.conference.locationKAUSTen
dc.contributor.institutionTU Braunschweigen