Approximate singular value expansions and solutions of ill-posed problems Approximate singular value expansions and solutions of ill-posed problems With the advent of large (huge) scale linear and non-linear problems, it is crucial to understand the spectral properties of surrogate models that are used for dimension reduction. Without a clear theoretical basis for implementing a surrogate model, critical information may not be incorporated within the obtained solution. The focus here is on surrogate models that are obtained via approximation of an underlying system matrix by the dominant terms of the singular value decomposition (SVD). For generating the surrogate we analyze modifications (i) of the LSQR algorithm in which by extending the underlying Krylov subspace and then truncating the Ritz pairs for the truncated problem yield more accurate SVD approximations, (ii) of randomized algorithms for under determined problems, (iii) using the connections of the SVD and the continuous singular value expansion for dimension reduction while maintaining high resolution solutions for least squares and total least squares problems, and (iv) combinations of these techniques for effective algorithm design. Framing these algorithms within the common format of approximate singular value decompositions provides a method for analysis, studies of noise transfer to the underlying basis for the solution and for contrasting techniques. This yields enhancements to Krylov methods that have not previously been recognized, as well as insights into the performance of randomized algorithms for dimension reduction. AGEP Supplement Request: Approximate Singular Value Expansions and Solutions of Ill-Posed Problems
|Effective start/end date||7/1/19 → 6/30/23|
- National Science Foundation (NSF): $206,998.00
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