Structured Matrix Computations and ApplicationsApplications to PDEs An elliptic problem on the...
Transcript of Structured Matrix Computations and ApplicationsApplications to PDEs An elliptic problem on the...
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Structured Matrix Computations and Applications
Michael K. NgDepartment of MathematicsHong Kong Baptist University
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Outline
Structured matrices have been around for a long time and are encountered in various fields of application.Toeplitz matrices, circulant matrices, Hankel matrices, semiseparablematrices, Kronecker product matrices, 2-by-2 block matrices …
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Outline
Toeplitz MatricesOverviewTheoryApplicationsResearch Problems
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Toeplitz Matrices
A matrix is said to be a Toeplitz matrix if it is constant along its diagonals
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Background
The name Toeplitz originates from the work of Otto Toeplitz (1911) on blinearforms related to Laurent seriesTime series: Yule-Walker Equations (1927) Levinson’s work (1947) in formulating the Wiener filtering problem
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An example of Toeplitz system
Linear prediction is a particularly important topic in digital signal processingThe determination of the optimal linear filter for prediction requires the solution of a set of linear equations having a Toeplitz structureStationary time series
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Direct Methods
Schur algorithm (1917) – a test for determining the positive definiteness of a Toeplitz matrixLevinson (1947)Durbin (1960)Trench (1964)O(n2) algorithmsSmall Large systems (Recursive)
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Direct Methods
The Gohberg-Semencul Formula
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Superfast Direct ToeplitzSolvers
Brent et al. (1980)Bitmead and Anderson (1980)Morf (1980)de Hong (1986)Ammar Gragg (1988)O(nlog2n) algorithmsRecursive from n n/2 n/4 n/8 …
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Look-ahead algorithms
Singular or ill-conditioned principal submatricesAvoid breakdowns or near-breakdowns by skipping such submatricesGueguen (1981), Delsarte et al (1985), Chan and Hansen (1992), Sweet (1993)Worst case: O(n3) algorithms
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Stability
The stability properties of symmetric positive definite Toepltiz matrices: Sweet (1984), Bunch (1985), Cybenko(1987), Bojanczyk et al (1995)Weakly stable (residual is small for well-conditioned matrices)Look-ahead methods are stable
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Stability
Toeplitz matrices Cauchy matricesPartial pivoting stable ? Gohberg et al (1995)Displacement representation error growth Gu (1995), Chandrasekaran and Sayed(1996), Park and Elden (1996): QR-type algorithm on displacement representation stable
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Iterative Methods
Rino (1970) and Ekstrom (1974): a decomposition of Toeplitz matrix into a circulant matrices and iterative methods Strang (1986), Olkin (1986): the use of preconditioned conjugate gradient method with circulant matrices as preconditioners for Toeplitz systems
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Circulant Preconditioners
Circulant matrices: Toeplitz matrices where each column is a circular shift of its preceding column
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Circulant Preconditioners
Design of circulant matrices, Strang’spreconditioners (1986), R. Chan’s preconditioners (1988), Ku and Kuo’spreconditioners (1992) …
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Circulant Preconditioners
T. Chan preconditioners (1988) optimal preconditionersmin || C – T ||F
Tyrtyshnikov preconditioners (1992) superoptimal preconditionersmin || I – C-1T ||F
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Results
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Results
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Circulant Preconditioners
The eigenvalues of Strang’spreconditioner is the values of the convolution product of the DirichletkernelThe eigenvalues of T. Chan’s preconditioner is the values of the convolution product of the Fejer kernelConvergence results/different conditions
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Transform-based Preconditioners
Circulant matrices are precisely those matrices that can diagonalized by the discrete Fourier transformSine transformCosine transformHartley transformA effective basis: shift matrices
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Ill-conditioned systems
Zeros of f
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Ill-conditioned Systems
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Ill-conditioned Systems
The generalized Jackson kernel forms an approximate convolution identity match the zeros automatically
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Multigrid Methods
Use projection/restriction operators to generate a sequence of sub-systemsThe zeros can be matched (zeros of f)
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Recursive Preconditioners
Use the principal submatrices as preconditionersMatch the zeros automaticallySolve the subsystems recursivelyIdea of direct methodsUse the Gohberg-Semencul formula to represent the inverses of submatrices
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Block-Toeplitz-Toeplitz-blockSystems
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ResultsBlock-circulant-circulant-block preconditioners
can be defined based on the block structure
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Toeplitz Least Squares Problems
Min || T x – b ||22
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Applications to PDEs
An elliptic problem on the unit-square with Dirichlet boundary conditionsCirculant preconditioners are not optimal condition number O(n)Sine transform based preconditionersare optimal condition number O(1)Boundary conditions are matched
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Example
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Sinc-Galerkin Methods for BVPs
Toeplitz-plus-diagonal systemsToeplitz known f / banded prec.
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PDEs
Hyperbolic and parabolic equations
Block-circulant preconditioners by Holmgren and Otto (1992), Jin and Chan (1992), Hemmingsson (1996)
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Applications to ODEs
Boundary value methods
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Results
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Applications to Integral Equations
Displacement kernel k(s,t)=k(s-t)Circulant integral operatorDiscretization schemes (modified prec.)
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Boundary Integral Equations
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Applications to QueueingNetworks
The previous talk given by W. K. Ching
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Applications to Signal Processing
Linear prediction filter Circulant preconditioners can be appliedProbabilistic convergence result
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Applications to Image Processing
Deconvolution problemPoint spread functions Toeplitzmatrices subject to boundary conditions
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Deconvolution Problems
RegularizationVery ill-conditioned Toeplitz matricesDirect inversion noises amplificationMany possible solutionsRegularization restricts the set of admissible solutionsTikhonov regularization: L2 or H1 norm
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Deconvolution Problems
Periodic boundary conditionZero boundary conditionReflective boundary condition
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Example
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Example
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Example
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Image Restoration Problems
Other deblurring matrices: spatial variant matricesOther measures in the fitting term: L1 norm (non-Gaussian noises)Other regularization methods: TV norm, edge-preserving methods (convex, nonconvex), Lipschitz regularization methodsOther constraints: nonnegativity
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Data-fitting term
Data-fitting term is L1 norm|| A f – g ||1 + regularization Non-Guassian noisesNonlinear problemsNonsmoothnonnegativity
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Spatial-variant Matrices
Example: Superresolution imagingSeveral low-resolution imagesDownsampling, missing pixels, motions, zooming, etcTransformed based preconditioners are not effective
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TV-norm
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Results
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Exact TV ComputationChambolle (JMIV 2004) studied the discrete version of the exact total variation denoisingmodelChambolle has shown that the solution can be given by using the orthogonal projection of the noisy image onto the convex set with the magnitude of the discrete divergence being less than or equal to 1.Constrained minimization problems projected gradient method (slow convergence)Semismooth Newton method (fast convergence)
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Exact TV with Blur
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Exact TV with Blur
Alternating minimization algorithmPreconditioning techniques
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Example
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Other Regularization Methods
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Blind Deconvolution Example
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Research Problems
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Research Problems
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Research Problems
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Research Problems
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Thank you very much !
Q/A