A standard way to solve linear algebraic systems $Au=f,\,\,(*)$ with ill-conditioned matrices $A$ is to use variational regularization. This leads to solving the equation $(A^*A+aI)u=A^*f_\d$, where $a$ is a regularization parameter, and $f_\d$ are noisy data, $||f-f_\d||\leq \d$. Numerically it requires to calculate products of matrices $A^*A$ and inversion of the matrix $A^*A+aI$ which is also ill-conditioned if $a>0$ is small. We propose a new method for solving (*) stably, given noisy data $f_\d$. This method, the DSM (Dynamical Systems Method) is developed in this paper for selfadjoint $A$. It consists in solving a Cauchy problem for systems of ordinary differential equations.
This work investigates the electrical impedance tomography (EIT) problem when only limited boundary measurements are available, which is known to be challenging due to the extreme ill-posedness. Based on the direct sampling method (DSM), we propose deep direct sampling methods (DDSMs) to locate inhomogeneous inclusions in which two types of deep neural networks (DNNs) are constructed to approximate the index function(functional): fully connected neural network(FNN) and convolutional neural network (CNN). The proposed DDSMs are easy to be implemented, capable of incorporating multiple Cauchy da…
There are some types of ill-conditioned algebraic equations that have difficulty in obtaining accurate roots and coefficients that must be expressed with a multiple precision floating-point number. When all their roots are simple, the problem solved via eigensolver (eigenvalue method) is well-conditioned if the corresponding companion matrix has its small condition number. However, directly solving them with Newton or simultaneous iteration methods (direct iterative method for short) should be considered as ill-conditioned because of increasing density of its root distribution. Although a grea…
Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standa…
Given a diagonalizable matrix $A$, we study the stability of its invariant subspaces when its matrix of eigenvectors is ill-conditioned. Let $\mathcal{X}_1$ be some invariant subspace of $A$ and $X_1$ be the matrix storing the right eigenvectors that spanned $\mathcal{X}_1$. It is generally believed that when the condition number $κ_2(X_1)$ gets large, the corresponding invariant subspace $\mathcal{X}_1$ will become unstable to perturbation. This paper proves that this is not always the case. Specifically, we show that the growth of $κ_2(X_1)$ alone is not enough to destroy the stability. As a…
Solving a singular linear system for an individual vector solution is an ill-posed problem with a condition number infinity. From an alternative perspective, however, the general solution of a singular system is of a bounded sensitivity as a unique element in an affine Grassmannian. If a singular linear system is given through empirical data that are sufficiently accurate with a tight error bound, a properly formulated general numerical solution uniquely exists in the same affine Grassmannian, enjoys Lipschitz continuity and approximates the underlying exact solution with an accuracy in the sa…
This report shows on real data that the direct methods such as LDL decomposition and Gaussian elimination for solving linear systems with ill-conditioned matrices provide inaccurate results due to divisions by very small numbers, which in turn results in peaking phenomena and large estimation errors. Richardson iteration provides accurate results without peaking phenomena since division by small numbers is absent in the Richardson approach. In addition, two preconditioners are considered and compared in the Richardson iteration: 1) the simplest and robust preconditioner based on the maximum ro…
We introduce a software package called HIFIR for preconditioning sparse, unsymmetric, ill-conditioned, and potentially singular systems. HIFIR computes a hybrid incomplete factorization, which combines multilevel incomplete LU factorization with a truncated, rank-revealing QR factorization on the final Schur complement. This novel hybridization is based on the new theory of approximate generalized inverse and $ε$-accuracy. It enables near-optimal preconditioners for consistent systems and enables flexible GMRES to solve inconsistent systems when coupled with iterative refinement. In this paper…
We propose an iterative algorithm for low-rank matrix completion that can be interpreted as an iteratively reweighted least squares (IRLS) algorithm, a saddle-escaping smoothing Newton method or a variable metric proximal gradient method applied to a non-convex rank surrogate. It combines the favorable data-efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. We establish the first local convergence guarantee from a minimal number of samples for that class of algorithms, showing that the method attains a local quadratic convergence rate. Furthermo…
The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well known that it can be very effective assuming regularity of the domain and boundary conditions. The main drawback of the MFS is that the matrices involved typically are ill-conditioned and this may prevent to achieve high accuracy. In this work, we propose a new algorithm to remove the ill conditioning of the classical MFS in the context of Laplace equation defined in planar domains. The main idea is to expand the MFS basis functions in …
Shifted CholeskyQR for computing the QR factorization of ill-conditioned matrices
Hokkaido University（北海道大学）；Arup（奥雅纳）；National Institute of Informatics（国立情报学研究所）；The University of Electro-Communications（电气通信大学）；JST CREST（日本科学技术振兴机构战略创造事业）；Waseda university（早稻田大学）
Analytic continuation is ill-posed, but becomes merely ill-conditioned (although with an infinite condition number) if it is known that the function in question is bounded in a given region of the complex plane. In an annulus, the Hadamard three-circles theorem implies that the ill-conditioning is not too severe, and we show how this explains the effectiveness of Chebfun and related numerical methods in evaluating analytic functions off the interval of definition. By contrast, we show that analytic continuation is far more ill-conditioned in a strip or a channel, with exponential loss of digit…
We show that the condition number of any cyclically contiguous $p\times q$ submatrix of the $N\times N$ discrete Fourier transform (DFT) matrix is at least $$ \exp \left( \fracπ{2} \left[\min(p,q)- \frac{pq}{N}\right] \right)~, $$ up to algebraic prefactors. That is, fixing any shape parameters $(α,β):=(p/N,q/N)\in(0,1)^2$, the growth is $e^{ρN}$ as $N\to\infty$ with rate $ρ= \fracπ{2}[\min(α,β)- αβ]$. Such Vandermonde system matrices arise in many applications, such as Fourier continuation, super-resolution, and diffraction imaging. Our proof uses the Kaiser-Bessel transform pair (of which we…
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University（厦门大学数学科学学院与福建省数学建模与高性能科学计算重点实验室）
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University（厦门大学数学科学学院、福建省数学建模与高性能科学计算重点实验室）
Widely employed for the accurate solution of the electroencephalography forward problem, the symmetric formulation gives rise to a first kind, ill-conditioned operator ill-suited for complex modelling scenarios. This work presents a novel preconditioning strategy based on an accurate spectral analysis of the operators involved which, differently from other Calderón-based approaches, does not necessitate the barycentric refinement of the primal mesh (i.e., no dual matrix is required). The discretization of the new formulation gives rise to a well-conditioned, symmetric, positive-definite system…
Vandermonde matrices are exponentially ill-conditioned, rendering the familiar "polyval(polyfit)" algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes the problem.
Brown University（布朗大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
In this paper we study the numerical instabilities of the 5- and 7-point problems for essential and fundamental matrix estimation in multiview geometry. In both cases we characterize the ill-posed world scenes where the condition number for epipolar estimation is infinite. We also characterize the ill-posed instances in terms of the given image data. To arrive at these results, we present a general framework for analyzing the conditioning of minimal problems in multiview geometry, based on Riemannian manifolds. Experiments with synthetic and real-world data then reveal a striking conclusion: t…
