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Approximation of Inverses of BTTB Matrices for Preconditioning Applications

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Approximation of Inverses of BTTB Matrices for Preconditioning Applications Eindhoven University of Technology MASTER Approximation of inverses of BTTB matrices for preconditioning applications Schneider, F.S. Award date: 2017 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain APPROXIMATION OF INVERSES OF BTTB MATRICES for Preconditioning Applications MASTERTHESIS by Frank Schneider December 2016 Dr. Maxim Pisarenco Department of Research ASML Netherlands B.V., Veldhoven Dr. Michiel Hochstenbach Department of Mathematics and Computer Science Technische Universiteit Eindhoven (TU/e) Prof. Dr. Bernard Haasdonk Institute of Applied Analysis and Numerical Simulation Universität Stuttgart APPROXIMATION OF INVERSES of BTTB MATRICES m for Preconditioning Applications Frank Schneider December 2016 Submitted in partial fulfillment of the requirements for the degree of Master of Science (M.Sc) in Industrial and Applied Mathematics (IAM) to the Department of Mathematics and Computer Science Technische Universiteit Eindhoven (TU/e) as well as for the degree of Master of Science (M.Sc) in Simulation Technology to the Institute of Applied Analysis and Numerical Simulation Universität Stuttgart The work described in this thesis has been carried out under the auspices of - Veldhoven, The Netherlands. ABSTRACT The metrology of integrated circuits (ICs) requires multiple solutions of a large-scale linear system. The time needed for solving this sys- tem, greatly determines the number of chips that can be processed per time unit. Since the coefficient matrix is partly composed of block-Toeplitz- Toeplitz-block (BTTB) matrices, approximations of its inverse are interesting candidates for a preconditioner. In this work, different approximation techniques such as an ap- proximation by sums of Kronecker products or an approximation by inverting the corresponding generating function are examined and where necessary generalized for BTTB and BTTB-block matrices. The computational complexity of each approach is assessed and their uti- lization as a preconditioner evaluated. The performance of the discussed preconditioners is investigated for a number of test cases stemming from real life applications. v ACKNOWLEDGEMENT First and foremost I wish to thank my supervisor from ASML Maxim Pisarenco. Maxim has supported me not only by providing valuable feedback over the course of the thesis, but by being always there to answer all my questions. He guided the thesis while allowing me freedom to explore the areas that tempted me the most. I also want to thank my supervisor from the TU/e Michiel Hochsten- bach who was an excellent resource of knowledge, academically and emotionally. Thank you for all the helpful feedback not only regard- ing the work and the thesis, but also regarding future plans. I owe thanks to the members of my thesis committee, professor Barry Koren and Martijn van Beurden from the TU/e and professor Bernard Haasdonk from the university of Stuttgart. Thank you for your valuable guidance and insightful comments. Thank you very much, everyone! Frank Schneider Eindhoven, December 28, 2016. vii CONTENTS i introduction1 1 motivation3 1.1 Photolithography . 4 1.1.1 Metrology . 5 1.2 Other Applications . 8 1.2.1 Deblurring Images . 9 1.2.2 Further Applications . 11 2 linear systems 13 2.1 Iterative Solvers . 13 2.1.1 CG Method . 13 2.1.2 Other Methods . 16 2.2 Preconditioning . 17 3 toeplitz systems 19 3.1 Multi-level Toeplitz Matrices . 21 3.2 Circulant Matrices . 23 3.3 Hankel . 23 4 problem description 25 4.1 Full Problem . 25 4.2 BTTB-Block System . 28 4.3 BTTB System . 29 5 thesis overview 31 ii preconditioners 33 6 overview over the preconditioning techniques 35 7 full c preconditioner 37 7.1 Application to Full Problem . 37 7.1.1 Inversion . 37 7.1.2 MVP . 37 8 circulant approximation 39 8.1 Circulant Approximation for Toeplitz Matrices . 39 8.1.1 Circulant Preconditioners . 40 8.2 Circulant Approximation for BTTB Matrices . 42 8.2.1 Toeplitz-block Matrices . 43 8.2.2 Block-Toeplitz Matrices . 43 8.3 Application to BTTB-block Matrices . 45 8.3.1 Inversion . 45 8.3.2 MVP . 46 9 inverse generating function approach 47 9.1 Inverse Generating Function for Toeplitz and BTTB Matrices . 47 ix 9.1.1 Unknown Generating Function . 48 9.1.2 Numerical Integration for Computing the Fourier Coefficients . 49 9.1.3 Numerical Inversion of the Generating Function 50 9.1.4 Example . 51 9.1.5 Efficient Inversion andMVP........... 51 9.2 Inverse Generating Function for BTTB-block Matrices . 52 9.2.1 General Approach . 53 9.2.2 Preliminaries . 53 9.2.3 Proof of Clustering of the Eigenvalues . 57 9.2.4 Example . 63 9.3 Regularizing Functions . 63 9.4 Numerical Experiments . 65 9.4.1 Convergence of theIGF.............. 65 9.4.2 IGF for aBTTB-block Matrix . 66 10 kronecker product approximation 69 10.1 Optimal Approximation for BTTB Matrices . 69 10.1.1 Algorithm . 71 10.1.2 Inverse and MVP . 71 10.2 BTTB-block Matrices . 73 10.2.1 One Term Approximation . 75 10.2.2 Multiple Terms Approximation . 77 10.3 Numerical Experiments . 79 10.3.1 Convergence of the Kronecker Product Approx- imation . 79 10.3.2 Decay of Singular Values . 80 10.3.3 Relation to the Generating Function . 80 11 more ideas 83 11.1 Transformation Based Preconditioners . 83 11.1.1 Discrete Sine and Cosine Transform . 83 11.1.2 Hartley Transform . 84 11.2 Banded Approximations . 85 11.3 Koyuncu Factorization . 86 11.4 Low-Rank Update . 88 iii benchmarks 91 12 benchmarks 93 12.1 Transformation-based Preconditioner . 99 12.2 Kronecker Product Approximation . 100 12.3 Inverse Generating Function . 101 12.4 Banded Approximation . 103 iv conclusion 105 13 future work 107 13.1 Inverse Generating Function . 107 13.1.1 Regularization . 107 13.1.2 Other Kernels . 107 x 13.2 Kronecker Product Approximation . 108 13.2.1 Using a Common Basis . 108 13.3 Preconditioner Selection . 108 14 conclusion 111 v appendix 113 a inversion formulas for kronecker product ap- proximation 115 a.1 One Term Approximation . 115 a.1.1 Sum Approximation . 115 a.2 Multiple Terms Approximation . 119 a.2.1 Sum Approximation . 119 bibliography 123 xi LISTOFFIGURES Figure 1.1 Moore’s law. 3 Figure 1.2 Photolithographic process. 4 Figure 1.3 Close-up of a wafer. 5 Figure 1.4 Effect of focus on the gratings. 6 Figure 1.5 Indirect grating measurement. 6 Figure 1.6 Shape parameters for a trapezoidal grating. 7 Figure 1.7 Example for aPSF................. 10 Figure 1.8 Blurring problem. 10 Figure 2.1 Minimization function. 14 Figure 2.2 Convergence of gradient descent and conjugate gradient (CG) method for different functions φ. 15 Figure 2.3 Preconditioner trade off. 18 Figure 4.1 Sparsity patterns of the matrices C, G and G as well as the resulting matrix A......... 25 Figure 4.2 Sparsity pattern of C................ 26 Figure 4.3 Color plots of all levels of C........... 27 Figure 8.1 Color plots for a Toeplitz-block matrix and its circulant-block approximation. 43 Figure 8.2 Color plots for a block-Toeplitz matrix and its block-circulant approximation. 44 Figure 9.1 Illustration of the inverse generating function approach (marked in red). 48 Figure 9.2 Illustration of the inverse generating function approach for unknown generating functions, with the changes marked in red. 49 Figure 9.3 Illustration of the inverse generating function approach with numerical Integration (highlighted in red). 50 Figure 9.4 Illustration of the inverse generating function approach using a sampled generating function. 51 Figure 9.5 Color plots for the inverse of the originalBTTB matrix, T[f]-1, the result of the inverse gener- ating function method T[1=f] and the difference between those two. 52 Figure 9.6 Illustration of the inverse generating function for Toeplitz-block matrices. 53 Figure 9.7 Color plots for the inverse of the original 2 × 2 BTTB-block matrix, T[F (x, y)]-1, the result of the inverse generating function method T[1=F (x,y)] and the difference of those two. 64 xii Figure 9.8 Degrees of regularization. 65 Figure 9.9 Convergence of the inverse generating func- tion (IGF) method towards the exact inverse. 66 Figure 9.10 Distribution of eigenvalues for theIGF..... 68 Figure 10.1 Relative difference of the Kronecker product approximation (using all terms) and the orig- inalBTTB matrix, for 500 randomly created test cases. 80 Figure 10.2 Decay of the singular value of a sample test case. 81 Figure 10.3 Relation of the Kronecker product approx- imation and the generating functions (taken from the test case 1b). 82 Figure 10.4 Convergence of the Generating Function. 82 Figure 11.1 Color plots for aBTTB matrix and the ap- proximation resulting from discrete sine trans- form (DST)II...................
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