Combining First- and Second-Order Continuity Constraints in Ultrasound Elastography (original) (raw)
2013
Abstract. The condition number of a Gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the least squares polynomial approximation. Given a polynomial basis, we consider the problem of finding a set of points and/or weights which minimizes the condition number of the Gram matrix. The objective function f in the minimization problem is nonconvex and nonsmooth. We present an expression of the Clarke generalized gradient of f and show that f is Clarke regular and strongly semismooth. Moreover, we develop a globally convergent smoothing method to solve the minimization problem by using the exponential smoothing function. To illustrate applications of minimizing the condition number, we report numerical results for the Gram matrix defined by the weighted Vandermonde-like matrix for least squares approximation on an interval and for the Gram matrix defined by an orthonormal set of real spherical harmonics for least squares approximatio...
Iterative methods for singular linear equations and least-squares problems
2006
CG, MINRES, and SYMMLQ are Krylov subspace methods for solving large symmetric systems of linear equations. CG (the conjugate-gradient method) is reliable on positive-definite systems, while MINRES and SYMMLQ are designed for indefinite systems. When these methods are applied to an inconsistent system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a leastsquares solution but not necessarily the minimum-length solution (often called the pseudoinverse solution). This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems.
On the Regularization in J-Matrix Methods
The J-Matrix Method, 2008
We investigate the effects of the regularization procedure used in the J-Matrix method. We show that it influences the convergence, and propose an alternative regularization approach.We explicitly perform some model calculations to demonstrate the improvement.
1983
Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the ab tract emtered to Block 20, Ii dit.rent from Report) NA ,W. SUPPLEMENTARY NOTES The findings in this report are not to be construed as an official Department of the Army position, unless so designated by other authorized documents. I. KEY WOROS (Continue on overoe side it ncesar and Identify by block number) HYPERBOLIC SYSTEMS OF PDE, GEODETIC ADJUSTMENTS, M-MATRICES, LEAST SQUARES COMPUTATIONS, SPARSE MATRICES, STATIONARY DISTRIBUTIONS OF MARKOV CHAINS, STRUCTURAL OPTIMIZATION. 40. AUSTRACT (ConSime m revwe side It necesarr aid idefltfy by block masb*r) This report contains a summary of the major accomplishments under the ARO sponsored projects, "Research in Numerical Linear Algebra" and "Numerical Methods for Large-Scale Least Squares and Other Problems", over the six year period 1977-1983. A list of 28 research publications resulting from this work is also included.
Structured Matrices in Numerical Linear Algebra, 2018
Structured Matrices in Numerical Linear Algebra Analysis Algorithms and Applications-Springer Interna Dario Andrea Bini • Fabio Di Benedetto • Eugene Tyrtyshnikov • Marc Van Barel Editors Structured Matrices in Numerical Linear Algebra Analysis, Algorithms and Applications Editors Dario Andrea Bini Department of Mathematics University of Pisa Pisa, Italy Fabio Di Benedetto Department of Mathematics University of Genoa Genoa, Italy Eugene Tyrtyshnikov Institute of Numerical Mathematics Russian Academy of Sciences Moscow, Russia Marc Van Barel Department of Computer Science KU Leuven Heverlee, Belgium
Hessian matrix-free lagrange-newton-krylov-schur-schwarz methods for elliptic inverse problems
2006
Convergence history of the solution error norm for 7 *, = (0.12)fc x 10~4 (top left), for 7 = 1. 0 x 1 0~4 (top right), for 7 = 1. 0 x 1 0-6 (bottom left), and for 7 = 1.0 x 1CT7 (bottom right) for 1% noise d a ta w ith P C I, the exact Schur complements and Tikhonov regularization.. .. 23 Convergence history of the residual norm for 7 * = (0.12)& x 10-4 (top left), for 7 = 1. 0 x 1 0~4 (top right), for 7 =. 0 x 1 0-6 (bottom left), and for 7 = 1. 0 x 1 0~7 (bottom right) for 1 % noise d a ta w ith P C I, the exact Schur complements and the Tikhonov regularization 24 Convergence history of the state error for 7 *, = (0.12)fc x 10~4 (top left), for 7 = 1. 0 x 1 (T4 (top right), for 7 = 1. 0 x 1 0-6 (bottom left), and for 7 = 1.0 x 1 0-7 (bottom right) for 1 % noise d a ta w ith P C I, the exact Schur complements and the Tikhonov regularization............. 25 M atching param eter for 7 *, = (O.l)* x 10_1 (top left), the m atching param eter for 7 *, = (0 .8)fc x 1 0-8 (top right), the m atching param eter for 7 k = (0.9131)fc x 10~9 (bottom left), and the m atching param eter for 7 ;, = (0.21 l) fe x 1CT5 (bottom right) for 1% error d a ta w ith P C I and the iterative Tikhonov regularization.