Fredholm Equation Of First Kind Research Articles

Overlapping Domain Decomposition Preconditioner for Integral Equations

The discretization of certain integral equations, e.g., the first-kind Fredholm equation of Laplace's equation, leads to symmetric positive-definite linear systems, where the coefficient matrix is dense and often ill-conditioned. We introduce a new preconditioner based on a novel overlapping domain decomposition that can be combined efficiently with fast direct solvers. Empirically, we observe that the condition number of the preconditioned system is $O(1)$, independent of the problem size. Our domain decomposition is designed so that we can construct approximate factorizations of the subproblems efficiently. In particular, we apply the recursive skeletonization algorithm to subproblems associated with every subdomain. We present numerical results on problem sizes up to $16\,384^2$ in two dimensions (2D) and $256^3$ in three dimensions (3D), which were solved in less than 16 hours and three hours, respectively, on an Intel Xeon Platinum 8280M.

Peculiarities of data processing for optical measurements of disperse parameters of bimodal media

This study is focused on enhancing the informativity of optical measurement techniques for particulate matter. The problem is that the description of particulate matter with bimodal and multimodal distributions by an a priori defined analytical function of particle size distribution (for example, a log-normal distribution) is not accurate enough. Here, we explore if experimental data can be approximated by a multivariable function of particle size distribution instead of using the a priori defined log-normal distribution. For the comparison of the approximation results, experiments are conducted on standard samples with granulometric compositions OGS-01LM and OGS-08LM separately and jointly in a mix. The experimental data are recorded by a high-selectivity turbidimetric technique in water suspensions of these samples. The purpose of this study is to present the measurement results as a distribution function that enables one to identify more accurately the particle-size distribution profile and the corresponding disperse characteristics of the aerosol in question when measuring parameters of disperse media by optical techniques. The main objective of this work is to develop, implement and verify a search algorithm for the particle-size distribution function by way of a multi-parameter function. We show that the solution to the problem proposed herein is more universal because it allows slow and fast processes in suspensions and aerosols to be examined with a lower error. The algorithm can be applied to the problems which are based on solving first-kind Fredholm equations.

Integral identity for a class of ill-posed problems generated by a parabolic equation

Abstract The goal of the present study is to derive an integral identity for a class of ill-posed problems generated by a parabolic equation. The obtained result enables us to reduce the original ill-posed problem directly to the first-kind Fredholm equation with translation kernel. Various real world applications to a different fields of knowledge, including ecology and finances, are presented.

SIGEST

The SIGEST paper in this issue, “Nonlocal Aggregation Models: A Primer of Swarm Equilibria” by Andrew J. Bernoff and Chad M. Topaz, is from the SIAM Journal on Applied Dynamical Systems. One of the many remarkable things in this paper is the authors' clear account of the very direct connections between first-kind Fredholm integral equations and apocalyptic plagues of locusts. Those of you who solve integral equations for a living will not be surprised that a first-kind Fredholm equation can be apocalyptic. While the results in the paper apply to many types of swarms, the authors focus on the particular behavior of Schistocerca gregaria, the swarming desert locust in Africa. These locusts self-organize into bubble-like swarms having a dense group of locusts on the ground, a flying group overhead, and an empty gap between them. The swarm moves with the wind with a rolling motion (see Figure 1 in the paper). This type of swarming is observed in nature and captured by the model in the paper. The paper looks at the relation between two models of swarming behavior. The discrete model is a system of ODEs for the time-dependent path of each individual bug. The forces on the locust are both exogenous (gravity, air resistance, food, predators) and endogenous (the preference to avoid collisions and stay with the group). If the swarm is at all large, the system of ODEs can become intractable, hence motivating a continuum approach. The continuum model, which the authors derive from the discrete model, is a variational principle for the density of locusts. The necessary conditions for stationarity of the energy functional can be expressed as a first-kind Fredholm equation. The analysis in the paper is for the continuous model. The paper looks deeply into the stability of the swarm configurations and the types of forces on the locusts, finds exact solutions for swarm equilibria in many parameter regimes, and carefully examines the regularity of the solutions of the continuous model. They find parameters for which the bubble-like swarms are stable solutions for both models. This is a satisfying paper with deep analysis, a great application, and a discrete formulation that is simple enough for a reader to play with individually or share with a class.

A hybrid inverse method for the thermal design of radiative heating systems

This work applies the inverse analysis for the thermal design of a three-dimensional radiative enclosure formed with diffuse-gray surfaces. The objective is to determine the powers and locations of the heaters to attain prescribed uniform temperature and radiative heat flux on the design surface. The solution is based on a hybrid solution that couples two different approaches to solve the inverse problem, optimization and regularization. The determination of the locations of the heaters is treated via optimization with the generalized extremal optimization (GEO), whereas the evaluation of the required heat inputs in the heaters, a problem that embodies the Fredholm equation of first kind, is achieved through the truncated singular value decomposition (TSVD) regularization of the ill-conditioned system of equations. The results illustrate that the hybrid solution is a more flexible and accurate approach than the explicit solution based on regularization, and less time consuming than the implicit solution based solely on optimization.

A multilevel algorithm for inverse problems with elliptic PDE constraints

We present a multilevel algorithm for the solution of a source identification problem in which the forward problem is an elliptic partial differential equation on the 2D unit box. The Hessian corresponds to a Tikhonov-regularized first-kind Fredholm equation. Our method uses an approximate Hessian operator for which, first, the spectral decomposition is known, and second, there exists a fast algorithm that can perform the spectral transform. Based on this decomposition we propose a conjugate gradients solver which we precondition with a multilevel subspace projection scheme. The coarse-level preconditioner is an exact solve and the finer-levels preconditioner is one step of the scaled Richardson iteration. As a model problem, we consider the 2D-Neumann Poisson problem with variable coefficients and partial observations. The approximate Hessian for this case is the Hessian related to a problem with constant coefficients and full observations. We can use a fast cosine transform to compute the spectral transforms. We examine the effect of using Galerkin or level-discretized Hessian operators and we provide results from numerical experiments that indicate the effectiveness of the method for full and partial observations.

A wavelet‐based method for solving discrete first‐kind Fredholm equations with piecewise constant solutions

AbstractThe inverse problem of finding piecewise constant solutions to discrete Fredholm integral equations of the first kind arises in many applied fields, e.g. in geophysics. This equation is usually an ill‐posed problem when it is considered in a Hilbert space framework, requiring regularization techniques to control arbitrary error amplifications and to get adequate solutions. In this work, we describe an iterative regularizing method for computing piecewise constant solutions to first‐kind discrete Fredholm integral equations. The algorithm involves two main steps at each iteration: (1) approximating the solution using a new signal reconstruction algorithm from its wavelet maxima which involves a previous step of detecting discontinuities by estimation of its local Hölder exponents; and (2) obtaining a regularized solution of the original equation using the a priori knowledge and the above approximation. In order to check the behaviour of the proposed technique, we have carried out a statistical study from a high number of simulations obtaining excellent results. Their comparisons with the results coming from using classical Tikhonov regularization by the multiresolution support, total variation (TV) regularization and piecewise polynomial truncated singular value decomposition (PP‐TSVD) algorithm, serve to illustrate the advantages of the new method. Copyright © 2003 John Wiley & Sons, Ltd.

An adaptive LSQR algorithm for computing discontinuous solutions in deconvolution problems

Ill-posed problems described by first-kind Fredholm equations appear in many interesting practical cases in engineering or mathematical physics, such as the inverse problem of signal deconvolution, and require regularization techniques to get adequate solutions ([C. Sánchez-Ávila, A.R. Figueiras-Vidal, J. Comp. Appl. Math. 72 (1996) 21–39] and [A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov, A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer Academic Publishers, Dordrecht, 1995]). In this work we consider the noisy discrete version of this equation which describes a typical problem in signal processing (e.g., in geophysics): recovering the discontinuities of a solution showing sharp edges. Such solutions are suited to describe models, e.g., geological layers, where the coarse structure is more important than the fine structure. We present a new adaptive algorithm which is capable of computing solutions that are piecewise constant, without having to specify a priori the positions of the break points between the constant pieces. This iterative algorithm is built on the base of regularizing LSQR method [P.C. Hansen, Numerical Aspects of Linear Inversion, SIAM, USA, 1997] and consists of two steps at each iteration: 1. detecting the discontinuities by an adaptive procedure, and 2. solving the original equation by regularizing LSQR iteration. We have carried out a high number of simulations to check the performance of proposed technique considering different signal-to-noise ratios in order to study its capacity of recovering edges in very noisy environments. Here we show some representative examples.

Analysis of Scattering from Perfectly Conducting Plates by the Use of AMMM

In this article, an adaptive multiscale algorithm is presented for the analysis of scattering by a thin, perfectly conducting plate. This algorithm employs the conventional moment method and a special matrix transformation, which is derived from solving the first-kind Fredholm equation of the first-kind by the multiscale technique. Pulse functions and Dirac delta functions are used for expansion and testing functions, respectively, utilizing the conventional moment method. The impedance matrix and the source terms of the matrix equation are computed using a five-point average scheme. By use of the matrix transformation, the currents, source terms and impedance matrix can be arranged in the form of different scales resulting in an adaptive multiscale method. By going from one scale to another scale, the initial estimate for the solution can be predicted according to the properties of the multiscale technique. AMMM can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional moment method. Several numerical results are presented, which demonstrate that AMMM is a useful method to analyze the scattering from perfectly conducting plates.

Analysis of Scattering From Perfectly Conducting Plates By the Use of Ammm - Abstract

In this article, an adaptive multiscale algorithm is presented for the analysis of scattering by a thin, perfectly conducting plate. This algorithm employs the conventional moment method and a special matrix transformation, which is derived from solving the first-kind Fredholm equation of the first-kind by the multiscale technique. Pulse functions and Dirac delta functions are used for expansion and testing functions, respectively, utilizing the conventional moment method. The impedance matrix and the source terms of the matrix equation are computed using a five-point average scheme. By use of the matrix transformation, the currents, source terms and impedance matrix can be arranged in the form of different scales resulting in an adaptive multiscale method. By going from one scale to another scale, the initial estimate for the solution can be predicted according to the properties of the multiscale technique. AMMM can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional moment method. Several numerical results are presented, which demonstrate that AMMM is a useful method to analyze the scattering from perfectly conducting plates.

New POCS algorithms for regularization of inverse problems

Ill-posed problems described by first-kind Fredholm equations appear in many interesting practical cases in engineering or mathematical physics, such as deconvolution, and require regularization techniques to get adequate solutions. This paper presents, under a projection operators onto convex sets (POCS) framework to introduce the needed regularization operations, a series of non-adaptive and adaptive POCS regularization algorithms which offer as main advantages (besides including previously proposed methods) a lower computational load and the possibility of including any kind of constraints, linear and nonlinear. A series of simulation examples serve to appreciate the high performance of the proposed techniques in a typical problem: recovering the discontinuites of a sequence showing sharp edges.

A novel technique for the calculation of site energy densities from gas adsorption isotherm data

A new method of obtaining the site energy distribution for adsorption of gas molecules on a heterogeneous substrate is presented. The method exploits the Fourier convolution nature of the first-kind Fredholm equation which connects the amount of gas adsorbed (the data function) to the site energy density (the object function)via an assumed form for the local isotherm type. The method is capable of handling a wide variety of local isotherm types including both patchwise and random distributions and the effects of lateral interactions. The ill-posed nature of this inverse problem is clearly demonstrated in this method and the effect of noise in the data function on the accuracy of the inversion is easily understood, although a frequency extrapolation technique is used to minimize the effect of noise on the deconvolution.The method is designed to operate on modern adsorption data which can be obtained over a wide pressure regime and very closely spaced (essentially continuously) using the continuous addition technique.The effect of a wrong choice of local isotherm is discussed briefly.

Peculiarities of the inverse problems of vertical radio sounding of the ionosphere

At the interpretation of ionograms (obtained by ground-based or satellite soundings) in terms of electron density, or at that of absorption measurements in terms of an effective electron collision frequency or at that of Doppler measurements in terms of plasma motions, problems arise which might be seen as comparable in character. One can, in fact, resolve these problems by inversion of the first-kind Volterra and Fredholm integral equations. The fact that the first-kind Fredholm equations belong to the class of ambiguous problems necessitates the use of additional information and data which were measured with great precision. Therefore, the main problems in groundbased and satellite diagnostics of the ionosphere is an improvement of the precision in measuring parameters of radio signals and the development of controlling algorithms for the solution of inverse problems.

Generalised information theory for inverse problems in signal processing

A powerful technique for the solution of a number of experimental in verse problems, described by an underlying first-kind Fredholm equation is presented. Such problems include, for example, diffraction-limited imaging and the analysis of laser light scattering data. The technique requires the construction of the ‘singular system’ of the propblem which then provides exact orthonormal bases both for the description of sampled and truncated measured data and for the reconstructed continuous object ‘solution’ of the inversion. The singular-system approach may ber regarded as a theory of information which generalises in several directions the well known classical concepts of Shannon and Nyquist.

Bounds on Solutions of Linear Systems with Inaccurate Data

Previous work has dealt with the problem of finding the solution set of a system of m equations in n unknowns, Ax = b, where A and b are known only to some limited tolerance, A/sup 0/ - ..delta..A less than or equal to A less than or equal to A/sup 0/ + ..delta..A,b/sup 0/ - ..delta..b less than or equal to b less than or equal to b/sup 0/ + ..delta..b with ..delta..A and ..delta..b being arrays consisting of positive elements. For a given n-vector x, necessary and sufficient conditions have been established for x to be an exact solution for some system anti Ax = anti b, where A/sup 0/ - ..delta..A less than or equal to anti A less than or equal to A/sup 0/ + ..delta..A and b/sup 0/ reverse arrow - ..delta..A and b/sup 0/ - ..delta..b less than or equal to anti b less than or equal to b/sup 0/ + ..delta..b. These results are extended and the emphasis is changed. If the orthant in which the true solution vector x lies is known from some a priori consideration, and if A and b are as above, it is possible to compute bounds formore » x by linear programing. The tableau for the problem is developed. The results are extended to finding confidence intervals for x in the case where A and b are random samples from distributions with known variances. A discussion of the application of the technique to solving first-kind Fredholm equations is also given. The technique is applied to three example problems. 4 figures.« less