Efficient Algorithms for Solution of Regularized Total Least Squares
Error-contaminated systems $Ax \approx b$, for which A is ill-conditioned, are considered. Such systems may be solved using Tikhonov-like regularized total least squares (RTLS) methods. Golub, Hansen, and O'Leary [SIAM J. Matrix Anal. Appl., 21 (1999), pp. 185--194] presented a parameter-depend...
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| Published in: | SIAM journal on matrix analysis and applications Vol. 26; no. 2; pp. 457 - 476 |
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| Language: | English |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2004
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| ISSN: | 0895-4798, 1095-7162 |
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| Abstract | Error-contaminated systems $Ax \approx b$, for which A is ill-conditioned, are considered. Such systems may be solved using Tikhonov-like regularized total least squares (RTLS) methods. Golub, Hansen, and O'Leary [SIAM J. Matrix Anal. Appl., 21 (1999), pp. 185--194] presented a parameter-dependent direct algorithm for the solution of the augmented Lagrange formulation for the RTLS problem, and Sima, Van Huffel, and Golub [Regularized Total Least Squares Based on Quadratic Eigenvalue Problem Solvers, Tech. Report SCCM-03-03, SCCM, Stanford University, Stanford, CA, 2003] have introduced a technique for solution based on a quadratic eigenvalue problem, RTLSQEP. Guo and Renaut [A regularized total least squares algorithm, in Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications, S. Van Huffel and P. Lemmerling, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002, pp. 57--66] derived an eigenproblem for the RTLS which can be solved using the iterative inverse power method. Here we present an alternative derivation of the eigenproblem for constrained TLS through the augmented Lagrangian for the constrained normalized residual. This extends the analysis of the eigenproblem and leads to derivation of more efficient algorithms compared to the original formulation. Additional algorithms based on bisection search and a standard L-curve approach are presented. These algorithms vary with respect to the parameters that need to be prescribed. Numerical and convergence results supporting the different versions and contrasting with RTLSQEP are presented. |
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| AbstractList | Error-contaminated systems $Ax \approx b$, for which A is ill-conditioned, are considered. Such systems may be solved using Tikhonov-like regularized total least squares (RTLS) methods. Golub, Hansen, and O'Leary [SIAM J. Matrix Anal. Appl., 21 (1999), pp. 185--194] presented a parameter-dependent direct algorithm for the solution of the augmented Lagrange formulation for the RTLS problem, and Sima, Van Huffel, and Golub [Regularized Total Least Squares Based on Quadratic Eigenvalue Problem Solvers, Tech. Report SCCM-03-03, SCCM, Stanford University, Stanford, CA, 2003] have introduced a technique for solution based on a quadratic eigenvalue problem, RTLSQEP. Guo and Renaut [A regularized total least squares algorithm, in Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications, S. Van Huffel and P. Lemmerling, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002, pp. 57--66] derived an eigenproblem for the RTLS which can be solved using the iterative inverse power method. Here we present an alternative derivation of the eigenproblem for constrained TLS through the augmented Lagrangian for the constrained normalized residual. This extends the analysis of the eigenproblem and leads to derivation of more efficient algorithms compared to the original formulation. Additional algorithms based on bisection search and a standard L-curve approach are presented. These algorithms vary with respect to the parameters that need to be prescribed. Numerical and convergence results supporting the different versions and contrasting with RTLSQEP are presented. |
| Author | Guo, Hongbin Renaut, Rosemary A. |
| Author_xml | – sequence: 1 givenname: Rosemary A. surname: Renaut fullname: Renaut, Rosemary A. – sequence: 2 givenname: Hongbin surname: Guo fullname: Guo, Hongbin |
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| Keywords | Rayleigh quotient Lagrangian Error estimation total least squares Error in variable ill-posedness Iterative method Iteration Modeling Search algorithm Numerical analysis Least squares method 65F30 Linear algebra Rayleigh quotient iteration 65F22 Ill posed problem Eigenvalue problem Quotient Regularization Algorithm analysis Numerical convergence |
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| References | R2 R3 R4 R7 R8 R9 Tihonov A. (R15) 1963; 151 Tikhonov Andrey (R16) 1977 Wilkinson J. (R17) 1965 R10 R12 R11 Golub Gene (R5) 1996 Parlett Beresford (R13) 1980 R1 |
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| Snippet | Error-contaminated systems $Ax \approx b$, for which A is ill-conditioned, are considered. Such systems may be solved using Tikhonov-like regularized total... |
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| SubjectTerms | Algebra Algorithms Alzheimer's disease Eigenvalues Exact sciences and technology Linear and multilinear algebra, matrix theory Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Sciences and techniques of general use |
| Title | Efficient Algorithms for Solution of Regularized Total Least Squares |
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