Sequential Estimation Of Discretization Errors In Inverse-Books Pdf

Sequential Estimation of Discretization Errors in Inverse
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Sequential Estimation of Discretization Errors in Inverse Problems. Oliver K Ernst, Submitted in partial fulfillment of the requirements of a. Bachelor of Science in Physics with a Mathematical Physics Concentration. Department of Physics,Case Western Reserve University. 2076 Adelbert Road Cleveland OH 44106,2 May 2014, This thesis was carried out under the supervision of. Daniela Calvetti and Erkki Somersalo, Department of Mathematics Applied Mathematics and Statistics. Case Western Reserve University,Committee Members,Kenneth Kowalski.
Harsh Mathur, Inverse problems are by nature computationally intensive and a key challenge in practical. applications is to reduce the computing time without sacrificing the accuracy of the solution. When using Finite Element FE or Finite Difference FD methods the computational bur. den of using a fine discretization is often unrealizable in practical applications Conversely. coarse discretizations introduce a modeling error which maybe become the predominant. part of the noise particularly when the data is collected with high accuracy In the Bayesian. framework for solving inverse problems it is possible to attain a super resolution in a coarse. discretization by treating the modeling error as an unknown that is estimated as part of the. inverse problem It has been previously proposed to estimate the probability density of the. modeling error in an off line calculation that is performed prior to solving the inverse prob. lem In this thesis a dynamic method to obtain these estimates is proposed derived from. the idea of Ensemble Kalman Filtering EnKF The modeling error statistics are updated. sequentially based on the current ensemble estimate of the unknown quantity and concomi. tantly these estimates update the likelihood function reducing the variance of the posterior. distribution A small ensemble size allows for rapid convergence of the estimates and the. need for any prior calculations is eliminated The viability of the method is demonstrated in. application to Electrical Impedance Tomography EIT an imaging method that measures. spatial variations in the conductivity distribution inside a body. Thinking is more interesting than knowing but less interesting than looking. Denken ist interessanter als Wissen aber nicht als Anschauen. Johann Wolfgang von Goethe,Acknowledgements, I wish to thank my advisors Prof Erkki Somersalo and Prof Daniela Calvetti for their. guidance and for countless enlightening discussions Gau wrote I have had my results for. a long time but I do not yet know how I am to arrive at them I am grateful that they. always took the time to explain what I was looking at and why. I also wish to express my gratitude to Prof Kenneth Kowalsi and Prof Kenneth Singer. for organizing the senior project class and furthermore for the countless hours spent lec. turing viewing presentations and reading reports without which this work would not have. been possible, Last but not least I wish to thank my parents for everything. 1 Introduction 1,2 Inverse Problems in the Bayesian Framework 3. 2 1 Tikhonov Regularization 3,2 2 Bayesian Inverse Problems 6.
2 3 Discretization Error 7,3 Estimation of the Modeling Error 10. 3 1 Off Line Modeling Error Approximation 10,3 2 Dynamic Modeling Error Approximation 11. 4 Application to EIT 14,4 1 Forward Model 14,4 1 1 Boundary Conditions 16. 4 2 Discretized Model 17,5 Computed Examples in EIT 22. 5 1 Prior Density 22,5 2 Conductivity Reconstructions 22.
5 3 Convergence and Discussion 24,6 Conclusions 30. Appendices 32, A Derivations of EIT Equation from Maxwells Equations 33. B Gau Newton Algorithm 37,B 1 Wiener Filtering 39,C Implementation details in EIDORS 40. Introduction, The tradeoff between computational efficiency and the accuracy of the solution is a key. challenge in solving inverse problems In practical applications the discretization of the. inverse problem for example using FE or FD methods leads to a difference between the. model and the reality Using a finer discretization decreases this discrepancy but may. increase the computational burden beyond what is feasible In particular when the forward. map is non linear a coarse discretization is used that differs significantly from the model. used to produce the data synthetic or experimental. The insufficiency of a model to describe reality is referred to as model discrepancy and. accounting for it is a topic of active study 14 18 The model discrepancy due to numerical. discretization error has been addressed in the literature in the context of Bayesian inverse. problems 1 17 together with other types of model mismatches such as poorly known. geometry 21 22 boundary clutter in the data 5 or insufficient modeling of the underlying. physics 12 13 27 Particularly when experimental measurements are precise the modeling. error may predominate the noise and will significantly decrease the quality of the solution. if not compensated for, This thesis addresses the problem of accounting for the discretization error in the Bayesian.
framework for solving inverse problems The error is treated as a realization of a random. variable with some unknown probability density Since the actual value of the modeling. error depends on the unknown parameter of interest the estimates of the error depend on. the information available about the unknown Approximating the probability density of the. modeling error therefore becomes part of the inverse problem. Previously the error s probability density has been estimated in an off line calculation. that is performed prior to solving the inverse problem A sample of draws from the prior. density is generated and the modeling error is estimated by computing the model predictions. from two meshes one representing an accurate approximation of the continuous model the. other representing the discretized mesh used in the inverse solver Using the same mesh to. both generate the data and solve the inverse problem is known as an inverse crime and. leads to results that are overly optimistic, In this thesis we investigate the possibility of estimating the modeling error in a dynamic. fashion by updating it while we iteratively improve the solution of the inverse problem In. particular unlike in the earlier works this estimate is not based on draws from the prior. but rather on forward simulations using the current ensemble of estimates analogous to the. classical estimation of the error of numerical solvers for differential equations The proposed. Chapter 1 Introduction 2, algorithm is a modification of an EnKF algorithm with an artificial time parameter that. refers to the updating round A similar idea of interpreting iterations as an artificial time step. in the EnKF framework for time invariant inverse problems was proposed recently in 15. The method is applied to the discrete ill posed inverse problem in EIT an imaging. method which measures spatial variations in the electric conductivity distribution inside a. body Currents are applied through electrodes on the surface of the body and resulting. relative voltages are measured Benefits of the method include its non invasive nature and. rapid measurement time with applications ranging from biomedical imaging to industrial. process tomography The partial differential equation model that describes the forward. model in EIT is assumed to provide a good approximation of the physical measurement. setting while the discretization of the equation generates an approximation error. The thesis is divided into six chapters Following the Introduction in Chapter 1 Chapter. 2 presents an introduction to solving inverse problems in the Bayesian framework and the. connection of this method to Tikhonov regularization The origin of the discretization error. follows and its connection to the inverse problem solution is made explicit Chapter 3 reviews. the previously proposed off line algorithm for estimating the modeling errors probability. density and presents the dynamic method that is the subject of this thesis Chapter 4 derives. the partial differential equation model that describes EIT and discretizes the model using. the FE method Chapter 5 presents a number of computed examples that demonstrate the. convergence and efficiency of the method in comparison to the off line calculation Finally. Chapter 6 discusses the significance of the work presented in this thesis and suggestions for. future research directions are given,Inverse Problems in the Bayesian. In this chapter regularization methods for ill posed inverse problems are reviewed 11. followed by regularization in the Bayesian framework 16 Finally the origin of discretization. errors is discussed 1 16 17 arriving at a formulation of the forward model. 2 1 Tikhonov Regularization, We begin by considering the discrete inverse problem of recovering an unknown x from a. linear measurement model,b Ax e x Rn b e Rm A Rm n 2 1.
where b is a given data vector and the additive measurement noise e is some zero mean. We consider the general case where m 6 n and furthermore where A is not invertible A. first approach to solve 2 1 in this case is to find the solution that minimizes the residual. x arg min kb Ax0 k2 2 2,where k k2 denotes the two norm. However the solution with the minimum residual is in general not the solution that most. closely resembles the true unknown To see this consider the singular value decomposition. SVD of the matrix A,A U V U Rm m Rm n V Rn n 2 3, where the matrices satisfy the following properties. diag 1 2 r, Chapter 2 Inverse Problems in the Bayesian Framework 4. where Im is the unit matrix of dimension m m r is the rank of A and 1 2. r 0 are the singular values of A,Using the SVD the solution to 2 1 is. xSVD A b V U b vi 2 4, where ui and vi are the ith columns of U and V respectively In practice the noiseless.
data vector b0 b e is not known However if it were the effect of the noise on the exact. solution can be written as,xSVD vi 2 5, In this form it is apparent that despite minimizing the residual the solution will tend. to infinity if the singular values either,1 Span several orders of magnitude or. 2 Are close to zero, Inverse problems of this type are called linear discrete ill posed problems If the singu. lar values span several orders of magnitude the unknown error components e in the data. dominate the behavior of the solution Even when there is no measurement error present. in the data but the singular values are close to zero problems of this type do not admit a. numerically stable solution, A simple modification of 2 4 to overcome this difficulty is the Truncated SVD TSVD. solution in which the sum is truncated at some index 1 k r The choice of k is deter. mined both by the noise level in the data and by examining the singular values specific to the. problem While effective at estimating the general behavior of the solution this approach. is not particularly elegant as there is only a single parameter to adjust The truncation. discards a portion of the information about the solution limiting further refinement. These challenges motivate the introduction of some additional parameters into the prob. lem to finely control the behavior of the solution known as regularization Certainly the. most widespread form of regularization is Tikhonov regularization where the minimization. problem 2 2 is replaced by,x L arg min kb Axk22 2 kLxk22 2 6.
where R is referred to as the regularization parameter and L Rp n is the regular. ization matrix, Tikhonov regularization alters the original problem statement 2 1 The minimization. problem s equivalent augmented matrix is, Chapter 2 Inverse Problems in the Bayesian Framework 5. where 0p Rp is a column vector with all zero entries The effect of this change can be. seen by manipulating the analogous form of the normal equations A Ax A b to attain. x L A A 2 L L A b 2 7, It is worth mentioning that this Tikhonov regularized solution closely resembles the form. of Wiener Filtering In general it can be shown by singular value decomposition that the. Tikhonov and Wiener solutions are equivalent given as. xTikhonov A C 1 A 2 1 A C 1 b 2 8,xWiener A A A 2 C b 2 9. From the normal equations 2 7 the effect of the regularization parameter and matrix. can be elucidated by considering the general singular value decomposition GSVD of the. matrices A L defined as,A UCX U Rm m C Rm n X Rn n 2 10.
L VSX V Rp p S Rp n 2 11,where the following conditions are satisfied. C C S S In,diag 1 2 p 1 p 1 0,M diag 1 2 p 1 1 p 0. where we have assumed for simplicity that p n m,The solution to the normal equations is. X i2 u i b, where xi is the ith column of X The coefficients fi i2 i2 2 2i are known as filter. factors For a common choice of the regularization matrix L Ip the singular values are. simply i 1, The solution 2 12 shows that through a careful choice of the regularization parameter.
and matrix it is possible to compensate for the behavior of the singular values i of A This. makes it possible balance minimizing the residual and minimizing the norm of the solution. A small regularization parameter recovers the previous minimization problem 2 2 A. large regularization parameter effectively minimizes the norm of the solution Lx 2. Clearly the choice of the regularization parameters has a significant effect on. The method is applied to the discrete ill posed inverse regularization methods for ill posed inverse L is the Cholesky factorization of the inverse of

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