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14
Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
, 2014
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Model Order Reduction by ParameterVarying Oblique Projection
"... Abstract — A method to reduce the dynamic order of linear parametervarying (LPV) systems in grid representation is developed in this paper. It approximates balancing and truncation by an oblique projection onto a dominant subspace. The approach is novel in its use of a parametervarying kernel to ..."
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Abstract — A method to reduce the dynamic order of linear parametervarying (LPV) systems in grid representation is developed in this paper. It approximates balancing and truncation by an oblique projection onto a dominant subspace. The approach is novel in its use of a parametervarying kernel to define the direction of this projection. Parametervarying state transformations in general lead to parameter rate dependence in the model. The proposed projection avoids this dependence and maintains a consistent state space basis for the reducedorder system. The method is compared with LPV balancing and truncation for a nonlinear massspringdamper system. It is shown to yield similar accuracy while the required computation time is reduced by a factor of almost 100,000. I.
Model reduction with mapreduceenabled tall and skinny singular value decomposition
 SIAM Journal on Scientific Computing
"... Abstract. We present a method for computing reducedorder models of parameterized partial differential equation solutions. The key analytical tool is the singular value expansion of the parameterized solution, which we approximate with a singular value decomposition of a parameter snapshot matrix. ..."
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Abstract. We present a method for computing reducedorder models of parameterized partial differential equation solutions. The key analytical tool is the singular value expansion of the parameterized solution, which we approximate with a singular value decomposition of a parameter snapshot matrix. To evaluate the reducedorder model at a new parameter, we interpolate a subset of the right singular vectors to generate the reducedorder model’s coefficients. We employ a novel method to select this subset that uses the parameter gradient of the right singular vectors to split the terms in the expansion yielding a mean prediction and a prediction covariance—similar to a Gaussian process approximation. The covariance serves as a confidence measure for the reduce order model. We demonstrate the efficacy of the reducedorder model using a parameter study of heat transfer in random media. The highfidelity simulations produce more than 4TB of data; we compute the singular value decomposition and evaluate the reducedorder model using scalable MapReduce/Hadoop implementations. We compare the accuracy of our method with a scalar response surface on a set of temperature profile measurements and find that our model better captures sharp, local features in the parameter space.
Model Reduction by Rational Interpolation
"... Abstract. The last two decades have seen major progress in interpolatory methods for model reduction of largescale dynamical systems have. The ability to produce optimal (at least locally) interpolatory reduced models at a modest cost for linear and bilinear systems, extensions to reducing paramet ..."
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Abstract. The last two decades have seen major progress in interpolatory methods for model reduction of largescale dynamical systems have. The ability to produce optimal (at least locally) interpolatory reduced models at a modest cost for linear and bilinear systems, extensions to reducing parametric systems, and the ability to produce reducedmodels directly from input/output measurements are some examples of these new developments. This chapter will give a survey of interpolatory model reduction methods including a detailed analysis of basic principles together with a presentation of the more recent developments. Discussion will be supported by numerical examples.
Adaptive hrefinement for reducedorder models
 Computing Research Repository, abs/1404.0442
"... This work presents a method to adaptively refine reducedorder models a posteriori without requiring additional fullordermodel solves. The technique is analogous to meshadaptive hrefinement: it enriches the reducedbasis space by ‘splitting ’ selected basis vectors into several vectors with dis ..."
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This work presents a method to adaptively refine reducedorder models a posteriori without requiring additional fullordermodel solves. The technique is analogous to meshadaptive hrefinement: it enriches the reducedbasis space by ‘splitting ’ selected basis vectors into several vectors with disjoint support. The splitting scheme is defined by a tree structure constructed via recursive kmeans clustering of the state variables using snapshot data. The method identifies the vectors to split using a dualweighted residual approach that seeks to reduce error in an output quantity of interest. The resulting method generates a hierarchy of subspaces online without requiring largescale operations or highfidelity solves. Further, it enables the reducedorder model to satisfy any prescribed error tolerance online regardless of its original fidelity, as a completely refined reducedorder model is equivalent to the original fullorder model. Experiments on a parameterized inviscid Burgers equation highlight the ability of the method to capture phenomena (e.g., moving shocks) not contained in the span of the original reduced basis. Keywords: adaptive refinement, hrefinement, model reduction, dualweighted residual, adjoint error estimation, clustering 1.
MAX−PLANCK−INSTITUT FÜR DYNAMIK KOMPLEXER TECHNISCHER SYSTEME MAGDEBURG
, 2012
"... A fast solver for an H1 regularized PDEconstrained problem ..."
unknown title
"... Abstract. We present a method for computing reducedorder models of parameterized partial differential equation solutions. The key analytical tool is the singular value expansion of the parameterized solution, which we approximate with a singular value decomposition of a parameter snapshot matrix. ..."
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Abstract. We present a method for computing reducedorder models of parameterized partial differential equation solutions. The key analytical tool is the singular value expansion of the parameterized solution, which we approximate with a singular value decomposition of a parameter snapshot matrix. To evaluate the reducedorder model at a new parameter, we interpolate a subset of the right singular vectors to generate the reducedorder model’s coefficients. We employ a novel method to select this subset that uses the parameter gradient of the right singular vectors to split the terms in the expansion, yielding a mean prediction and a prediction covariance—similar to a Gaussian process approximation. The covariance serves as a confidence measure for the reducedorder model. We demonstrate the efficacy of the reducedorder model using a parameter study of heat transfer in random media. The highfidelity simulations produce more than 4TB of data; we compute the singular value decomposition and evaluate the reducedorder model using scalable MapReduce/Hadoop implementations. We compare the accuracy of our method with a scalar response surface on a set of temperature profile measurements and find that our model better captures sharp, local features in the parameter space.
unknown title
, 2015
"... Some a posteriori error bounds for reduced order modelling of (non)parametrized linear systems ..."
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Some a posteriori error bounds for reduced order modelling of (non)parametrized linear systems
FÜR DYNAMIK KOMPLEXER
, 2015
"... Lowrank solutions to an optimization problem constrained by the NavierStokes equations ..."
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Lowrank solutions to an optimization problem constrained by the NavierStokes equations
FAST ALGORITHMS FOR HYPERSPECTRAL DIFFUSE OPTICAL TOMOGRAPHY
"... The image reconstruction of chromophore concentrations using Diffuse Optical Tomography (DOT) data can be described mathematically as an illposed inverse problem. Recent work has shown that the use of hyperspectral DOT data, as opposed to data sets comprising of a single or, at most, a dozen wavele ..."
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The image reconstruction of chromophore concentrations using Diffuse Optical Tomography (DOT) data can be described mathematically as an illposed inverse problem. Recent work has shown that the use of hyperspectral DOT data, as opposed to data sets comprising of a single or, at most, a dozen wavelengths, has the potential for improving the quality of the reconstructions. The use of hyperspectral diffuse optical data in the formulation and solution of the inverse problem poses a significant computational burden. The forward operator is, in actuality, nonlinear. However, under certain assumptions, a linear approximation, called the Born approximation, provides a suitable surrogate for the forward operator, and we assume this to be true in the present work. Computation of the Born matrix requires the solution of thousands of large scale discrete PDEs and the reconstruction problem, requires matrixvector products with the (dense) Born matrix. In this paper, we address both of these difficulties, thus making the Born approach a computational viable approach for hyperspectral DOT (hyDOT) reconstruction. In this paper, we assume that the images we wish to reconstruct are anomalies of unknown shape and constant value, described using a parametric level set approach, (PaLS) [1] on a constant background. Specifically, to address the issue of the PDE solves, we develop a novel recyclingbased Krylov subspace approach that leverages certain system similarities across wavelengths. To address expense of using the Born operator in the inversion, we present a fast algorithm for compressing the Born operator that locally compresses across wavelengths for a given sourcedetector set and then recursively combines the lowrank factors to provide a global lowrank approximation. This lowrank approximation can be used implicitly to speed up the recovery of the shape parameters and the chromophore concentrations. We provide a detailed analysis of the accuracy and computational costs of the resulting algorithms and demonstrate the validity of our approach by detailed numerical experiments on a realistic geometry.