A proximal approach to the inversion of ill-conditioned matrices
نویسندگان
چکیده
We propose a general proximal algorithm for the inversion of ill-conditioned matrices. This algorithm is based on a variational characterization of pseudo-inverses. We show that a particular instance of it (with constant regularization parameter) belongs to the class of fixed point methods. Convergence of the algorithm is also discussed.
منابع مشابه
1 2 Ja n 20 06 DSM for solving ill - conditioned linear algebraic systems ∗ †
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 δ , where a is a regularization parameter, and f δ are noisy data, ||f − f δ || ≤ δ. Numerically it requires to calculate products of matrices A * A and inversion of the matrix A * A + aI which is also ill-con...
متن کاملIll-Conditioned Matrices Are Componentwise Near to Singularity
For a square matrix normed to 1, the normwise distance to singularity is well known to be equal to the reciprocal of the condition number. In this paper we give an elementary and selfcontained proof for the fact that an ill-conditioned matrix is also not far from a singular matrix in a componentwise sense. This is shown to be true for any weighting of the componentwise distance. In words: Ill-c...
متن کاملNovel Algorithms Based on the Conjugate Gradient Method for Inverting Ill-Conditioned Matrices, and a New Regularization Method to Solve Ill-Posed Linear Systems
We propose novel algorithms to calculate the inverses of ill-conditioned matrices, which have broad engineering applications. The vector-form of the conjugate gradient method (CGM) is recast into a matrix-form, which is named as the matrix conjugate gradient method (MCGM). The MCGM is better than the CGM for finding the inverses of matrices. To treat the problems of inverting illconditioned mat...
متن کاملA class of arbitrarily ill-conditioned floating-point matrices∗
Let IF be a floating-point number system with basis β ≥ 2 and an exponent range consisting at least of the exponents 1 and 2. A class of arbitrarily ill-conditioned matrices is described the coefficients of which are elements of IF. Due to the very rapidly increasing sensitivity of those matrices they might be regarded als “almost” ill-posed problems. The condition of those matrices and their s...
متن کاملOn Block Matrices Associated with Discrete Trigonometric Transforms and Their Use in the Theory of Wave Propagation
Block matrices associated with discrete Trigonometric transforms (DTT’s) arise in the mathematical modelling of several applications of wave propagation theory including discretizations of scatterers and radiators with the Method of Moments, the Boundary Element Method, and the Method of Auxiliary Sources. The DTT’s are represented by the Fourier, Hartley, Cosine, and Sine matrices, which are u...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009