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INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law o nature, a regularization of the universe. Carl Saga

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Page 1: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

INVERSE PROBLEMS and REGULARIZATION THEORY – Part II

AIP 2011Texas A&M University

MAY 22, 2011

CHUCK GROETSCH

Every restriction corresponds to a law of nature,a regularization of the universe.

Carl Sagan

Page 2: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

OUTLINE

What is Regularization?

Tikhonov Regularization

Iterative Regularization

Convergence Theory

Role of the SVD

Discretizations

Iterated T-regularization

Landweber Iteration

Page 3: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

REGULARIZATION

Page 4: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

Tikhonov Regularization

Page 5: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

Compact Operators K

Page 6: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of
Page 7: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

Discrepancy Principle

Page 8: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

Variational Characterization

Page 9: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

The Regularization Dance

Approximation Error:

Discretization Error:

Stability Error:

Page 10: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

RTSVD

Page 11: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

RTSVD

Page 12: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

RTSVD

Page 13: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

RTSVD

Page 14: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

Finite Elements

Page 15: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

Finite Rank - Quadrature

Page 16: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

Finite Rank - Quadrature

Page 17: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

Iterated Tikhonov Regularization

Page 18: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

Iterative Regularization

Page 19: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of
Page 20: INVERSE PROBLEMS and REGULARIZATION THEORY – Part II AIP 2011 Texas A&M University MAY 22, 2011 CHUCK GROETSCH Every restriction corresponds to a law of

INVERSE PROBLEMS and REGULARIZATION THEORY – Part I AIP 2011 Texas A&M University MAY 21, 2011 CHUCK GROETSCH

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