Computational Methods for Inverse Problems反问题的计算方法（影印版）

图书标准信息

• 作者 Curtis R.Vogel 著
• 出版社 清华大学出版社
• 出版时间 2011-03
• 版次 1
• ISBN 9787302245025
• 定价 28.00元
• 装帧 平装
• 开本 16开
• 纸张 胶版纸
• 页数 183页

【内容简介】

inverseproblemsariseinanumberofimportantpracticalapplications,rangingfrombiomedicalimagingtoseismicprospecting.thisbookprovidesthereaderwithabasicunderstandingofboththeunderlyingmathematicsandthecomputationalmethodsusedtosolveinverseproblems.italsoaddressesspecializedtopicslikeimagereconstruction,parameteridentification,totalvariationmethods,nonnegativityconstraints,andregularizationparameterselectionmethods.
becauseinverseproblemstypicallyinvolvetheestimationofcertainquantitiesbasedonindirectmeasurements,theestimationprocessisoftenill-posed.regularizationmethods,whichhavebeendevelopedtodealwiththisill-posedness,arecarefullyexplainedintheearlychaptersofcomputationalmethodsforinverseproblems.thebookalsointegratesmathematicalandstatisticaltheorywithapplicationsandpracticalcomputationalmethods,includingtopicslikemaximumlikelihoodestimationandBayesianestimation.

【作者简介】

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【目录】

foreword

preface

1introduction

1．1anillustrativeexample

1．2regularizationbyfiltering

1．2．1adeterministicerroranalysis

1．2．2ratesofconvergence

1．2．3aposterioriregularizationparameterselection

1．3variationalregularizationmethods

1．4iterativeregularizationmethods

exercises

2analyticaltools

2．1ill-posednessandregularization

2．1．1compactoperators，singularsystems，andthesvd

2．1．2leastsquaressolutionsandthepseudo-inverse

2．2regularizationtheory

2．3optimizationtheory

2．4generalizedtikhonovregularization

2．4．1penaltyfunctionals

2．4．2datadiscrepancyfunctionals

2．4．3someanalysis

exercises

3numericaloptimizationtools

3．1thesteepestdescentmethod

3．2theconjugategradientmethod

3．2．1preconditioning

3．2．2nonlinearcgmethod

3．3newton'smethod

3．3．1trustregionglobalizationofnewton'smethod

3．3．2thebfgsmethod

3．4inexactlinesearch

exercises

4statisticalestimationtheory

4．1preliminarydefinitionsandnotation

4．2maximumlikelihood'estimation

4．3bayesianestimation

4．4linearleastsquaresestimation

4．4．1bestlinearunbiasedestimation

4．4．2minimumvariancelinearestimation

4．5theemalgorithm

4．5．1anillustrativeexample

exercises

5imagedeblurring

5．1amathematicalmodelforimageblurring

5．1．1atwo-dimensionaltestproblem

5．2computationalmethodsfortoeplitzsystems

5．2．1discretefouriertransformandconvolution

5．2．2theffta，lgorithm

5．2．3toeplitzandcirculantmatrices

5．2．4bestcirculantapproximation

5．2．5blocktoeplitzandblockcirculantmatrices

5．3fourier-baseddeblurringmethods

5．3．1directfourierinversion

5．3．2cgforblocktoeplitzsystems

5．3．3blockcirculantpreconditioners

5．3．4acomparisonofblockcirculantpreconditioners

5．4multileveltechniques

exercises

6parameteridentification

6．1anabstractframework

6．1．1gradientcomputations

6．1．2adjoint，orcostate，methods

6．1．3hessiancomputations

6．1．4gauss-newtonhessianapproximation

6．2aone-dimensionalexample

6．3aconvergenceresult

exercises

7regularizationparameterselectionmethods

7．1theunbiasedpredictiveriskestimatormethod

7．1．1implementationoftheupremethod

7．1．2randomizedtraceestimation

7．1．3anumericalillustrationoftraceestimation

7．1．4nonlinearvariantsofupre

7．2generalizedcrossvalidation

7．2．1anumericalcomparisonofupreandgcv

7．3thediscrepancyprinciple

7．3．iimplementationofthediscrepancyprinciple

7．4thel-curvemethod

7．4．1anumericalillustrationofthel-curvemethod

7．5otherregularizationparameterselectionmethods

7．6analysisofregularizationparameterselectionmethods

7．6．1modelassumptionsandpreliminaryresults

7．6．2estimationandpredictiveerrorsfortsvd

7．6．3estimationandpredictiveerrorsfortikhonovregularization

7．6．4analysisofthediscrepancyprinciple

7．6．5analysisofgcv

7．6．6analysisofthel-curvemethod

7．7acomparisonofmethods

exercises

8totalvariationregularization

8．1motivation

8．2numericalmethodsfortotalvariation

8．2．1aone-dimensionaldiscretization

8．2．2atwo-dimensionaldiscretization

8．2．3steepestdescentandnewton'smethodfortotalvariation

8．2．4laggeddiffusivityfixedpointiteration

8．2．5aprimal-dualnewtonmethod

8．2．6othermethods

8．3numericalcomparisons

8．3．1resultsforaone-dimensionaltestproblem

8．3．2two-dimensionaltestresults

8．4mathematicalanalysisoftotalvariation

8．4．1approximationstothetvfunctional

exercises

9nonnegativityconstraints

9．1anillustrativeexample

9．2theoryofconstrainedoptimization

9．2．1nonnegativityconstraints

9．3numericalmethodsfornonnegativelyconstrainedminimization

9．3．1thegradientprojectionmethod

9．3．2aprojectednewtonmethod

9．3．3agradientprojection-reducednewtonmethod

9．3．4agradientprojection-cgmethod

9．3．5othermethods

9．4numericaltestresults

9．4．1resultsforone-dimensionaltestproblems

9．4．2resultsforatwo-dimensionaltestproblem

9．5iterativenonnegativeregularizationmethods

9．5．1richardson-lucyiteration

9．5．2amodifiedsteepestdescentalgorithm

exercises

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