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Introduction to Inverse Problems - University of Chicago

Introduction to Inverse ProblemsGuillaume Bal1 July 2, 20191 University of Chicago , Chicago , IL 60637; What is an Inverse Elements of anInverse problem .. and stability of the Measurement Operator .. model and Prior model .. algorithms .. Examples of Measurement Operator .. IP and Modeling. Application to MRI .. Inverse Problems , Smoothing, and ill-posedness .. Problems and Lipschitz Stability.. Problems and Unbounded Operators.. and Sobolev scale of hilbert spaces.. 152 Integral Geometry. Radon The Radon Transform .. Tomography .. dimensional X-ray (Radon) transform .. dimensional Radon transform .. Attenuated Radon Transform .. Photon Emission Computed Tomography.

We introduce a Hilbert scale of spaces in section1.4to quantify such an ampli cation for a restricted but pedagogically useful class of inverse problems. These preliminary notations set the stage for the introductory analysis in later chap-

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Transcription of Introduction to Inverse Problems - University of Chicago

1 Introduction to Inverse ProblemsGuillaume Bal1 July 2, 20191 University of Chicago , Chicago , IL 60637; What is an Inverse Elements of anInverse problem .. and stability of the Measurement Operator .. model and Prior model .. algorithms .. Examples of Measurement Operator .. IP and Modeling. Application to MRI .. Inverse Problems , Smoothing, and ill-posedness .. Problems and Lipschitz Stability.. Problems and Unbounded Operators.. and Sobolev scale of hilbert spaces.. 152 Integral Geometry. Radon The Radon Transform .. Tomography .. dimensional X-ray (Radon) transform .. dimensional Radon transform .. Attenuated Radon Transform .. Photon Emission Computed Tomography.

2 hilbert problem .. of the Attenuated Radon Transform .. (i): The problem , an elliptic equation .. (ii): jump conditions .. (iii): reconstruction formulas .. 373 Integral geometry and Radon transform and FIO .. Oscillatory integrals, symbols, and phases .. Schwartz kernels and Fourier integral operators .. Critical set and propagation of singular support .. Applications to DO and Radon transform .. Propagation of singularities for FIO.. Front Set and Distributions.. of singularities in linear transforms .. Critical sets and Lagrangian manifolds .. Local theory of FIO .. Elliptic FIO and Elliptic DO.. Local tomography and partial data.

3 594 Integral Geometry. Generalized Ray Generalized Ray Transform in two dimensions.. of curves.. Ray Transform.. operator and rescaled Normal operator .. Pseudo-differential operators and GRT .. of phase (x y) .. of a parametrix.. of symbol, smoothing and composition .. of DO in the spacesHs(X) .. and injectivity .. Kinematic Inverse Source problem .. equation .. form and energy estimates .. result .. Summary on GRT.. Application to generalized Radon transforms .. Composition and continuity of FIO .. 79 Appendices81A Remarks on composition of FIOs and Stationary Remarks on composition of FIOs .. Method of stationary phase.

4 885 Inverse wave One dimensional Inverse scattering problem .. Linearized Inverse Scattering problem .. and linearization .. field data and reconstruction .. toX-ray tomography .. Inverse source problem in PAT .. explicit reconstruction formula for the unit sphere .. explicit reconstruction for detectors on a plane .. One dimensional Inverse coefficient problem .. 1036 Inverse Kinematic and Inverse Transport Inverse Kinematic problem .. symmetry .. integral and Abel transform .. velocity Inverse problem .. Forward transport problem .. Inverse transport problem .. of the albedo operator and uniqueness result .. in Inverse transport .. 1227 Inverse Cauchy problem and Electrocardiac potential.

5 Half Space problem .. well posed problem .. electrocardiac application .. bounds and stability estimates .. continuation .. General two dimensional case .. equation on an annulus .. mapping theorem .. Backward Heat Equation .. Carleman Estimates, UCP and Cauchy problem .. 1348 Calder on Introduction .. Uniqueness and Stability .. to a Sch odinger equation .. of injectivity result .. of the stability result .. Complex Geometric Optics Solutions .. The Optical Tomography setting .. Anisotropy and non-uniqueness .. 1529 Coupled-physics IP I: PAT and Introduction to PAT and TE .. of photoacoustic tomography .. step: Inverse wave source problem .

6 Step: Inverse Problems with internal functionals .. of one coefficient.. to Transient Elastography .. Theory of quantitative PAT and TE .. and stability results in QPAT .. to Quantitative Transient Elastography .. in PAT and TE .. two dimensional case .. case .. The case of anisotropic coefficients .. algebra problem .. reconstructions .. problem .. 17310 Coupled-physics IP II: Ultrasound Modulation Tomography .. Inverse Problems in ultrasound modulation.. Eliminations and redundant systems of ODEs .. Elimination ofF.. System of ODEs forSj.. ODE solution and stability estimates .. Well-chosen illuminations .. The casen= 2 .. The casen 3 .. Remarks on hybrid Inverse Problems .

7 18611 Priors and Smoothness Regularization .. Ill-posed Problems and compact operators .. Regularity assumptions and error bound .. Regularization methods .. Sparsity and other Regularization Priors .. Smoothness Prior and Minimizations .. Sparsity Prior and Minimizations .. Bayesian framework and regularization .. Penalization methods and Bayesian framework .. Computational and psychological costs of the Bayesian framework 20512 Geometric Priors and Reconstructing the domain of inclusions .. Forward problem .. Factorization method .. Reconstruction of .. Reconstructing small inclusions .. First-order effects .. Stability of the reconstruction .. 219A Notation and Fourier transform.

8 221 Bibliography223 Index229 Chapter 1 What is an Inverse ProblemThree essential ingredients define an Inverse problem in this book. The central elementis theMeasurement Operator(MO), which maps objects of interest, calledparameters,to information collected about these objects, calledmeasurementsordata. The mainobjective of Inverse problem theory is to analyze such a MO, primarily its injectivity andstability properties. Injectivity of the MO means that acquired datauniquelycharacter-ize the parameters. Often, the inversion of the MO amplifieserrorsin the measurements,which we refer to estimatescharacterize this the amplification is considered too large by the user, which is a subjectivenotion, then the Inverse problem needs to be modified. How this should be done dependson the structure ofnoise.

9 The second essential ingredient of an Inverse problem is thusanoisemodel, for instance a statement about its size in a given metric, or, if available,its statistical a MO and a noise model are available, the too large effect of noise onthe reconstruction is mitigated by imposing additional constraints on theparametersthat render the inversionwell-posed. These constraints take the form of aprior model,for instance assuming that the parameters live in a finite dimensional space, or thatparameters are sparsely represented in an appropriate three elements (MO, noise model, prior model) are described in more detail insection Examples of MO are given in section role of modeling is often crucial in the derivation of the triplet (MO, noisemodel, prior model).

10 In section , three different models of MO are obtained in asimplified setting of Magnetic Resonance Imaging, one of the most successful medicalimaging a MO has been selected and proved to be injective, we need to understand howits inversion amplifiesnoise. Typically, difficulties in Inverse Problems arise because suchan amplification becomes larger for higher frequencies. This is related to thesmoothingproperties of the MO. We introduce a hilbert scale of spaces in section to quantifysuch an amplification for a restricted but pedagogically useful class of Inverse preliminary notations set the stage for the introductory analysis in later chap-ters of several (MO, noise model, prior model) that find applications in many inverseproblems12 CHAPTER 1.


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