Computation of the singular value expansion

Hansen, R.C.

doi:10.1007/bf02251248

Cited by 47 publications

(44 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For such problems, the close relationship between the ill-posedness of the integral equation and the ill-conditioning of the matrix A are well understood [1,41,88]. In particular, it can be shown that the singular values of A decay in such a way that both criteria 1 and 2 above are satisfied.…”

Section: Discrete Ill-posed Problemsmentioning

confidence: 99%

“…The discrete Picard condition is not as "artificial" as it first may seem: it can be shown that if the underlying integral equation (2.1) satisfies the Picard condition, then the discrete ill-posed problem obtained by discretization of the integral equation satisfies the discrete Picard condition [41]. See also [83,84].…”

Section: The Discrete Picard Condition and Filter Factorsmentioning

confidence: 99%

See 1 more Smart Citation

REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems

1994

View full text Add to dashboard Cite

Section: Discrete Ill-posed Problemsmentioning

confidence: 99%

Section: The Discrete Picard Condition and Filter Factorsmentioning

confidence: 99%

REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems

1994

View full text Add to dashboard Cite

“…When T | D is not injective, computation of Λ requires computation of the complete singular value expansion of the kernel of the operator T . In theory, because T is known and is not estimated, a researcher can compute the SVD of T , calculate the elements {φ j } j∈J 0 by a simple procedure of basis completion, like the Gram-Schmidt orthonormalization, and then characterize the null space of the operator, see Hansen (1988). In practice, a researcher must truncate the expansion at some point and impose that all singular values not computed equal zero.…”

Section: Identification Of the Distribution Of Parametersmentioning

confidence: 99%

“…In practice, a researcher must truncate the expansion at some point and impose that all singular values not computed equal zero. The error of this approximation can be bounded using methods in Hansen (1988).…”

Section: Identification Of the Distribution Of Parametersmentioning

confidence: 99%

Semiparametric estimation of random coefficients in structural economic models

Hoderlein¹,

Nesheim²,

Simoni³

2012

View full text Add to dashboard Cite

This paper discusses nonparametric estimation of the distribution of random coefficients in a structural model that is nonlinear in the random coefficients. We establish that the problem of recovering the probability density function (pdf ) of random parameters falls into the class of convexly-constrained inverse problems. The framework offers an estimation method that separates computational solution of the structural model from estimation. We first discuss nonparametric identification. Then, we propose two alternative estimation procedures to estimate the density and derive their asymptotic properties. Our general framework allows us to deal with unobservable nuisance variables, e.g., measurement error, but also covers the case when there are no such nuisance variables. Finally, Monte Carlo experiments for several structural models are provided which illustrate the performance of our estimation procedure.

show abstract

“…We will employ his construction [18,Section 5], which ultimately uses point evaluations of the function f to construct a matrix suited for the SVD. Let x 1 , .…”

mentioning

confidence: 99%

Model Reduction With MapReduce-enabled Tall and Skinny Singular Value Decomposition

Constantine¹,

Gleich²,

Hou³

et al. 2014

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. We present a method for computing reduced-order 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 reduced-order model at a new parameter, we interpolate a subset of the right singular vectors to generate the reduced-order 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 reduced-order model using a parameter study of heat transfer in random media. The high-fidelity simulations produce more than 4TB of data; we compute the singular value decomposition and evaluate the reduced-order 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. Key words. model reduction, simulation informatics, MapReduce, Hadoop, tall and skinny SVD 1. Introduction & motivation. High-fidelity simulations of partial differential equations are typically too expensive for design optimization and uncertainty quantification, where many independent runs are necessary. Cheaper reduced-order models (ROMs) that approximate the map from simulation inputs to quantities of interest may replace expensive simulations to enable such parameter studies. These ROMs are constructed with a relatively small set of high-fidelity runs chosen to cover a range of input parameter values. Each evaluation of the ROM is a linear combination of basis functions derived from the outputs of the high-fidelity runs; ROM constructions differ in their choice of basis functions and method for computing the coefficients of the linear combination. Projection-based methods project the residual of the governing equations (i.e., a Galerkin projection) to create a relatively small system of equations for the coefficients; see the recent preprint [4] for a survey of projection-based techniques. Alternatively, one may derive a closely related optimization problem to compute the coefficients [9,10,14]. These two formulations can provide a measure of confidence along with the ROM. However, they are often difficult to implement in existing solvers since they need access to the equation's operators or residual.To bypass the implementation difficulties, one may use response surfaces-e.g., collocation [3,32] or Gaussian process regression [29]-which are essentially interpolation methods applied to the high-fidelity outputs. They do not need access to the differential operators or residuals and are therefore relatively easy to implement. However, measures of co...

show abstract

Computation of the singular value expansion

Cited by 47 publications

References 16 publications

REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems

REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems

Semiparametric estimation of random coefficients in structural economic models

Model Reduction With MapReduce-enabled Tall and Skinny Singular Value Decomposition

Contact Info

Product

Resources

About