Solution Of Linear Operator Equations Research Articles

Estimates of matrix solutions of operator equations with random parameters under uncertainties

We investigate problems of estimating solutions of linear operator equations with random parameters under conditions of uncertainty. We establish that the guaranteed rms estimates of the matrices are found as solutions of special optimization problems under certain observations of the system state. As the output signals of the system, we have observations that are described by linear functions from the solutions of such equations with random right-hand sides, which have unknown second moments. Under the condition that the observation second moments of the right-hand parts and errors belong to certain sets, it is proved that the guaranteed estimates are expressed through solutions of operator equation systems. When the linear operator is given by the scalar product of rectangular matrices, a quasi-minimax estimate and its error are constructed. It is shown that the quasi-minimax estimation error tends to zero when the number of observations tends to infinity. An example of calculating the guaranteed rms estimate of the matrix's trace, which is a solution of a matrix equation with a random parameter, is given.

Tiresia: A code for molecular electronic continuum states and photoionization

The Tiresia program [1] provides access to numerically accurate solutions of the one-particle Schrödinger equation for highly excited states of complex polyatomic molecules, both bound and continuum, that cannot be described by conventional Quantum Chemistry approaches. It is based on an expansion of the required solution in a local multicentric basis set, with primitive functions built as products of a radial B-spline times a real spherical harmonic. In conjunction with Density Functional Theory (DFT), it has been extensively employed in a large variety of photoionization studies, also for rather large systems. Highly excited bound states as well as wavepacket propagation can also be accurately described. In fact, the flexibility of the basis essentially allows accurate solutions of linear operator equations, like Poisson or inhomogeneous perturbative equations, which are employed in the code. The program is parallelized with standard MPI-I instructions and makes extensive use of the Scalapack linear algebra library. Ancillary programs are available for the evaluation of photoionization cross sections and angular distributions from randomly to fully oriented molecules. Program summaryProgram Title: TiresiaCPC Library link to program files:https://doi.org/10.17632/fcrjxwgjxh.1Licensing provisions: GPLv3Programming language: Fortran77, Fortran90, MPISupplementary material: Program manual documentNature of problem: Accurate solutions for highly excited and continuum electronic states in complex polyatomic molecules. Molecular photoionization cross sections and angular distributions under high energy, high-intensity radiation pulses from Synchrotron radiation and laser sources, photoelectron imaging in pump-probe experiments, basis for electronic wavepackets under ultrafast or nonperturbative excitation.Solution method: Solution of the Schrödinger and similar linear operator equations in a finite domain is obtained via basis set expansion. Flexible basis set, that may approach practical completeness within the domain, is obtained as a multicenter set of B-spline radial functions times spherical harmonics. Accurate numerical integration is employed for the evaluation of matrix elements, and conventional diagonalization for bound states, or Galerkin approach for the full multichannel solution in the continuum. Full hamiltonian and dipole matrices in the spectral basis are available for time propagation. DFT many-body description is available, and strong correlations in the bound states may be incorporated via Dyson orbitals.Additional comments including restrictions and unusual features: The code is noted for computational efficiency, which allows fast yet reasonably accurate photoionization calculations for medium-sized molecules, allowing, e.g., calculations at many molecular geometries as required to describe time-resolved photoelectron spectra in pump-probe experiments.

Минимаксные среднеквадратические оценки матричных параметров в задачах линейной регрессии в условиях неопределенности

Исследована проблема оценки параметров в задачах линейной регрессии со случайными матричными коэффициентами. При условии, что наблюдаются случайные линейные функции от неизвестных матриц со случайными погрешностями, имеющими неизвестные корреляционные матрицы, исследованы задачи гарантированного среднеквадратичного оценивания линейных функций от матриц. Получены оценки сверху и снизу гарантированных среднеквадратических погрешностей линейных оценок по наблюдениям линейных функций от матриц в том случае, когда известны множества, которым принадлежат неизвестные матрицы и корреляционные матрицы погрешностей наблюдений. Установлено, что в некотором частном случае такие оценки являются точными. При предположении, что множества ограничены, выпуклы и замкнуты, получены более точные двусторонние оценки для гарантированных погрешностей. Найдены условия, когда гарантированные среднеквадратические погрешности приближаются к нулю при увеличении количества наблюдений. Приведены необходимые и достаточные условия несовмещенности линейных оценок линейных функций от матриц. Введено понятие квазиоптимальных оценок для линейных функций от матриц и доказано, что в классе несмещенных оценок квазиоптимальные оценки существуют и единые. Для таких оценок получены условия сходимости к нулю гарантированных среднеквадратических погрешностей. Также для линейных оценок неизвестных матриц введено понятие квазимимимаксных оценок и доказано, что они не смещены. Для специальных множеств, которым принадлежат неизвестная матрица и корреляционные матрицы погрешностей наблюдений, такие оценки выражены из-за решения линейных операторных уравнений в конечномерном пространстве. Для квазимимимаксных оценок при определенных предположениях определен вид гарантированной среднеквадратичной погрешности оценки неизвестной матрицы. Показано, что такие погрешности ограничиваются сверху суммой следов известных матриц. Приведен пример нахождения минимаксной несмещенной линейной оценки для специального вида случайных матриц, входящих в уравнение наблюдения.

Source Conditions for Non-Quadratic Tikhonov Regularization

In this paper, we consider convex Tikhonov regularization for the solution of linear operator equations on Hilbert spaces. We show that standard fractional source conditions can be employed in order to derive convergence rates in terms of the Bregman distance, assuming some stronger convexity properties of either the regularization term or its convex conjugate. In the special case of quadratic regularization, we are able to reproduce the whole range of Hölder type convergence rates known from classical theory.

Discretization Methods for Three-Dimensional Singular Integral Equations of Electromagnetism

Theorems providing the convergence of approximate solutions of linear operator equations to the solution of the original equation are proved. The obtained theorems are used to rigorously mathematically justify the possibility of numerical solution of the 3D singular integral equations of electromagnetism by the Galerkin method and the collocation method.

Two-sided estimates and positivity conditions for solutions of linear operator equations in a Banach lattice

Two-sided estimates and positivity conditions for solutions of linear operator equations in a Banach lattice

The residual method for regularizing ill-posed problems

Although the residual method, or constrained regularization, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces. In addition, we prove convergence rates in terms of (generalized) Bregman distances, which can also be applied to non-convex regularization functionals.We provide three examples that show the applicability of our theory. The first example is the regularized solution of linear operator equations on Lp-spaces, where we show that the results of Tikhonov regularization generalize unchanged to the residual method. As a second example, we consider the problem of density estimation from a finite number of sampling points, using the Wasserstein distance as a fidelity term and an entropy measure as regularization term. It is shown that the densities obtained in this way depend continuously on the location of the sampled points and that the underlying density can be recovered as the number of sampling points tends to infinity. Finally, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions.

Hypergeometric series solutions of linear operator equations

Let K be a field and L : K [ x ] → K [ x ] be a linear operator acting on the ring of polynomials in x over the field K. We provide a method to find a suitable basis { b k ( x ) } of K [ x ] and a hypergeometric term c k such that y ( x ) = ∑ k = 0 ∞ c k b k ( x ) is a formal series solution to the equation L ( y ( x ) ) = 0 . This method is applied to construct hypergeometric representations of orthogonal polynomials from the differential/difference equations or recurrence relations they satisfied. Both the ordinary cases and the q-cases are considered.

Acceleration of the generalized Landweber method in Banach spaces via sequential subspace optimization

The Landweber method is a well-known iterative procedure to compute regularized solutions of linear operator equations in Hilbert spaces. Unfortunately it is also known to be very slow. Likewise its generalization to Banach spaces has good regularizing properties but slow convergence. This article intends to give a short survey about the use of sequential subspace optimization to accelerate this method while preserving its regularizing properties.

Fast regularizing sequential subspace optimization in Banach spaces

We are concerned with fast computations of regularized solutions of linear operator equations in Banach spaces in case only noisy data are available. To this end we modify recently developed sequential subspace optimization methods in such a way that the therein employed Bregman projections onto hyperplanes are replaced by Bregman projections onto stripes whose width is in the order of the noise level.

Linear Convergence of Iterative Soft-Thresholding

In this article a unified approach to iterative soft-thresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the so-called finite basis injectivity property or the minimizer possesses a so-called strict sparsity pattern. Moreover it is shown that the constants can be calculated explicitly in special cases (i.e. for compact operators). Furthermore, the techniques also can be used to establish linear convergence for related methods such as the iterative thresholding algorithm for joint sparsity and the accelerated gradient projection method.

Metric and Bregman projections onto affine subspaces and their computation via sequential subspace optimization methods

Abstract In this article we investigate and prove relationships between metric and Bregman projections induced by powers of the norm of a Banach space. We consider Bregman projections onto affine subspaces of Banach spaces and deduce some interesting analogies to results which are well known for Hilbert spaces. Using these concepts as well as ideas from sequential subspace optimization techniques we construct efficient iterative methods to compute Bregman projections onto affine subspaces that are connected to linear, bounded operators between Banach spaces. Especially these methods can be used to compute minimum-norm solutions of linear operator equations or best approximations in the range of a linear operator. Numerical experiments illuminate the performance of our iterative algorithms and demonstrate a significant acceleration compared to the Landweber method.

Minimization of Tikhonov Functionals in Banach Spaces

Tikhonov functionals are known to be well suited for obtaining regularized solutions of linear operator equations. We analyze two iterative methods for finding the minimizer of norm‐based Tikhonov functionals in Banach spaces. One is the steepest descent method, whereby the iterations are directly carried out in the underlying space, and the other one performs iterations in the dual space. We prove strong convergence of both methods.

Certain approximate methods for solving linear operator equations

A solution of linear operator equations in the Hilbert space is constructed by using the best polynomial approximation of the inverse operator. This approach gives rise to certain iteration processes. Error estimates manifest that the suggested schemes may be fairly efficient.

Bounds for perturbed solutions of linear operator equations in Hilbert space

Some perturbation results on relative errors of solutions to singular linear operator equations in Hilbert space are given for consistent systems and for weighted least-squares problems with the help of weighted generalized inverses.

Special formal series solutions of linear operator equations

The transformation which assigns to a linear operator L the recurrence satisfied by coefficient sequences of the polynomial series in its kernel, is shown to be an isomorphism of the corresponding operator algebras. We use this fact to help factoring q-difference and recurrence operators, and to find ‘nice’ power series solutions of linear differential equations. In particular, we characterize generalized hypergeometric series that solve a linear differential equation with polynomial coefficients at an ordinary point of the equation, and show that these solutions remain hypergeometric at any other ordinary point. Therefore, to find all generalized hypergeometric series solutions, it suffices to look at a finite number of points: all the singular points, and a single, arbitrarily chosen ordinary point. We also show that at a point x= a we can have power series solutions with: • polynomial coefficient sequence — only if the equation is singular at a+1, • non-polynomial rational coefficient sequence — only if the equation is singular at a.

Direct methods for the approximate solution of operator equations with nonzero kernel

We justify direct methods for the approximate solution of linear operator equations with nonzero kernels and apply these methods to the justification of projective methods for the approximate solution of standard singular integral equations with Cauchy kernels and positive index on the unit disk.

Iterated solutions of linear operator equations with the Tau method

The Tau Method produces polynomial approximations of solutions of differential equations. The purpose of this paper is (i) to extend the recursive formulation of this method to general linear operator equations defined in a separable Hilbert space, and (ii) to develop an iterative refinement procedure which improves on the accuracy of Tau approximations. Applications to Fredholm integral equations demonstrate the effectiveness of this technique.

Convergence rates for moment collocation solutions of linear operator equations

Given a linear operator equation Kf=g with data g(x i )i=1,…,n, we consider the general moment collocation solution defined as the function f n that minimizes ||Pf n ||2 over a Hilbert space, subject to Kf n (x i )=g(x i )i=1,…,n. Here P is an orthogonal projection with a finite dimensional null space. In the case of P=I, the identity, it is known that if a certain kernel depending on K is continuous, then f n → f 0 , the true solution, as the maximum subinterval width → 0. Moreover, if the kernel satisfies a smoothness condition, then rates of convergence are known. In this paper we extend these results to the case with general P.

The Expected–Projection Method: Its Behavior and Applications to Linear Operator Equations and Convex Optimization

It was shown by Butnariu and Flåm [J. Nnmer. Funct. Anal. Optim. 15: 601–636, 1995] that, under some conditions, sequences generated by the expected projection method (EPM) in Hilbert spaces approximate almost common points of measurable families of closed convex subsets provided that such points exist. In this work we study the behavior of the EPM in the more general situation when the involved sets may or may not have almost common points and we give necessary and sufficient conditions for weak and strong convergence. Also, we show how the EPM can be applied to finding solutions of linear operator equations and to solving convex optimization problems.