Error bounds for maximum entropy approximate solutions to Fredholm integral equations (original) (raw)

Properties of maximum entropy approximate solutions to Fredholm integral equations

Journal of Mathematical Physics, 1991

The properties of the maximum entropy approximate solutions to Fredholm integral equations of the second kind are discussed. It is proved that the approximate solutions exist and converge to the exact one for a wider class of problems than previously determined. The correspondence between the maximum entropy technique and the Galerkin method is established and on its basis an upper a posteriori bound for a solution is found.

Fredholm Integral Equations of the Second Kind (General Kernel

In Chap. 1, we conducted a thorough examination of the Fredholm integral equation of the second kind for an arbitrary complex parameter λ , assuming that the free term f (x) is complex-valued and continuous on the interval [a, b] and that the kernel K(x,t) is complex-valued, continuous, and separable on the square Q(a, b) = {(x,t) : [a, b] × [a, b]}. We stated the four Fredholm theorems and the Fredholm Alternative Theorem which provide for the construction of all possible solutions to the equation under these assumptions. A question naturally arises: What, if anything, can be proven if K(x,t) is a general kernel, i.e., an arbitrary kernel that is only assumed to be complex-valued and continuous? In this chapter, we will answer this question completely by proving that all of the Fredholm theorems continue to hold in this eventuality. In Sect. 2.1, we present several tools of the trade which are indispensible for the comprehension of the material in this chapter. In Sects. 2.2 and 2.3, we use these tools to show that the Fredholm integral equation of the second kind with a general kernel has a unique solution if the product of the parameter λ and the " size " of the kernel is small. In Sect. 2.4, we prove the validity of the Fredholm theorems for unrestricted λ and a general kernel. In Sect. 2.5, we show how to construct the resolvent kernel that appears in the solution to the integral equation recursively. In Sect. 2.6, we introduce numerical methods for producing an approximation to the solution of a Fredholm integral equation. These methods are necessary due to the inherent computational difficulties in constructing the resolvent kernel.

Hilbert Space Analysis of Integral Equations

This paper examines the theory of solving certain classes of integral equations using functional analysis methods of Hilbert Space. Included are equations with symmetric kernels and their associated self-adjoint linear operators, and in particular the Fredholm equa- tion. The Hilbert Space methods, while taking time and effort to establish, prove rewarding in that they then simplify approaches to solving such equations, on the way providing more focused and powerful results. Examples serve to illustrate the principles.

Linear Fredholm Integral Equations and the Integral of Kurzweil

Journal of Applied Analysis, 2002

We apply the Kurzweil-Henstock integral setting to prove a Fredholm Alternative-type result for the integral equation x (t) − K [a,b] α (t, s) x (s) ds = f (t) , t ∈ [a, b] , where x and f are Kurzweil integrable functions (possibly highly oscillating) defined on a compact interval [a, b] of the real line with values on Banach spaces. An application is given.

Optimal methods for specifying information in the solution of integral equations with analytic kernels

Ukrainian Mathematical Journal, 1996

We determine the exact order of the minimum radius of information in the logarithmic scale for Fredholm integral equations of the second kind with periodic analytic kernels and free terms. We show that the information complexity of the solution of Fredholm equations with analytic kernels is ~eater in order than the complexity of the approximation of analytic functions. This distinguishes the analytic case from the case of finite smoothness.

Metric and Probabilistic Information Associated with Fredholm Integral Equations of the First Kind

Journal of Integral Equations and Applications, 2002

The problem of evaluating the information associated with Fredholm integral equations of the first kind, when the integral operator is selfadjoint and compact, is considered here. The data function is assumed to be perturbed gently by an additive noise so that it still belongs to the range of the operator. First we estimate upper and lower bounds for the ε-capacity (and then for the metric information), and explicit computations in some specific cases are given; then the problem is reformulated from a probabilistic viewpoint and use is made of the probabilistic information theory. The results obtained by these two approaches are then compared.