Numerical methods for ill-posed, linear problems
<p>A means of assessing the effectiveness of methods used in the numerical solution of various linear ill-posed problems is outlined. Two methods: Tikhonov' s method of regularization and the quasireversibility method of Lattès and Lions are appraised from this point of view.</p>...
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ndltd-CALTECH-oai-thesis.library.caltech.edu-75672019-12-22T03:09:37Z Numerical methods for ill-posed, linear problems Stevens, Thomas <p>A means of assessing the effectiveness of methods used in the numerical solution of various linear ill-posed problems is outlined. Two methods: Tikhonov' s method of regularization and the quasireversibility method of Lattès and Lions are appraised from this point of view.</p> <p>In the former method, Tikhonov provides a useful means for incorporating a constraint into numerical algorithms. The analysis suggests that the approach can be generalized to embody constraints other than those employed by Tikhonov. This is effected and the general "T-method" is the result.</p> <p>A T-method is used on an extended version of the backwards heat equation with spatially variable coefficients. Numerical computations based upon it are performed.</p> <p>The statistical method developed by Franklin is shown to have an interpretation as a T-method. This interpretation, although somewhat loose, does explain some empirical convergence properties which are difficult to pin down via a purely statistical argument.</p> 1975 Thesis NonPeerReviewed application/pdf https://thesis.library.caltech.edu/7567/1/Stevens_t_1975.pdf https://resolver.caltech.edu/CaltechTHESIS:03292013-151040800 Stevens, Thomas (1975) Numerical methods for ill-posed, linear problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ZB0R-1F34. https://resolver.caltech.edu/CaltechTHESIS:03292013-151040800 <https://resolver.caltech.edu/CaltechTHESIS:03292013-151040800> https://thesis.library.caltech.edu/7567/ |
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<p>A means of assessing the effectiveness of methods used in the numerical solution of various linear ill-posed problems is outlined. Two methods: Tikhonov' s method of regularization and the quasireversibility method of Lattès and Lions are appraised from this point of view.</p>
<p>In the former method, Tikhonov provides a useful means for incorporating a constraint into numerical algorithms. The analysis suggests that the approach can be generalized to embody constraints other than those employed by Tikhonov. This is effected and the general "T-method" is the result.</p>
<p>A T-method is used on an extended version of the backwards heat equation with spatially variable coefficients. Numerical computations based upon it are performed.</p>
<p>The statistical method developed by Franklin is shown to have an interpretation as a T-method. This interpretation, although somewhat loose, does explain some empirical convergence properties which are difficult to pin down via a purely statistical argument.</p> |
author |
Stevens, Thomas |
spellingShingle |
Stevens, Thomas Numerical methods for ill-posed, linear problems |
author_facet |
Stevens, Thomas |
author_sort |
Stevens, Thomas |
title |
Numerical methods for ill-posed, linear problems |
title_short |
Numerical methods for ill-posed, linear problems |
title_full |
Numerical methods for ill-posed, linear problems |
title_fullStr |
Numerical methods for ill-posed, linear problems |
title_full_unstemmed |
Numerical methods for ill-posed, linear problems |
title_sort |
numerical methods for ill-posed, linear problems |
publishDate |
1975 |
url |
https://thesis.library.caltech.edu/7567/1/Stevens_t_1975.pdf Stevens, Thomas (1975) Numerical methods for ill-posed, linear problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ZB0R-1F34. https://resolver.caltech.edu/CaltechTHESIS:03292013-151040800 <https://resolver.caltech.edu/CaltechTHESIS:03292013-151040800> |
work_keys_str_mv |
AT stevensthomas numericalmethodsforillposedlinearproblems |
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1719305334663675904 |