Recursive equations for the predictive distributions of some determinantal processes

This paper provides recursive equations for the predictive distributions of one-dependent and two-dependent determinantal processes. Fixed order recursive equations can be applied both to efficiently simulate trajectories and to explore properties of the process.     

Some integral equations related to random Gaussian processes

To calculate the Laplace transform of the integral of the square of a random Gaussian process, we consider a nonlinear Volterra-type integral equation. This equation is a Ward identity for the generating correlation function. It turns out that for an important class of correlation functions, this identity reduces to a linear ordinary differential equation. We present sufficient conditions for this equation to be integrable (the equation coefficients are constant). We calculate the Laplace transform exactly for some concrete random Gaussian processes such as the “Brownian bridge” model and the Ornstein-Uhlenbeck model.     

On the integral equations of optimal signal detection and estimation theory

The analysis of many optimal signal detection and estimation processes involves the solution of Fredholm integral equations. A New approach to these equations is presented. It consists of reducing the Fredholm integral equation to a Cauchy system which is well suited to numerical solution.This research was supported by the National Institutes of Health, Grant No. GM-16437-03.     

Boundary arcs for integral equations

Behavior of boundary arcs for control systems is investigated when the systems are governed by integral equations of the Volterra type. The main result is in the form of a maximum principle. This result is then used to obtain necessary conditions for a minimum control problem.     

Variational-iterative method for integral equations

We study applications of the variational-iterative method to nonlinear integral equations with potential strongly monotone and Lipschitz-continuous operators and investigate its rate of convergence for special systems of coordinate functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1626–1635, December, 1990.     

Optimal fields for integral equations

The theory of optimal fields is developed for optimal control problems in which the state variables are determined by integral equations. The Hilbert integral is considered and the Hamilton-Jacobi equations are derived. The results obtained contain two maximum principles as special cases; one reflecting the special character of field theory and one corresponding to the results previously obtained by use of variations.     

On the eigenproblem for convolution integral equations

Summary This paper concerns the eigenproblem for convolution integral equations whose kernels can be expressed as finite or infinite Fourier transforms of integrable functions. A procedure which closely parallels previous work on displacement integral equations is derived and the problem of existence is treated. Approximations are obtained for both the eigenvalues and the eigenfunctions.The results of this paper are taken from the author's doctoral dissertation at the University of New Mexico. The research was supported by the United States Atomic Energy Commission.     

On the eigenproblem for displacement integral equations

Summary This paper is concerned with the eigenproblem for displacement integral equations which have kernels that are finite cosine transforms. Previous work is extended to a broader class of kernels and the eigenfunctions are treated for the first time.This work was supported by the United States Atomic Energy Commission.     

Analytical discussion for the mixed integral equations

This paper presents a numerical method for the solution of a Volterra–Fredholm integral equation in a Banach space. Banachs fixed point theorem is used to prove the existence and uniqueness of the solution. To find the numerical solution, the integral equation is reduced to a system of linear Fredholm integral equations, which is then solved numerically using the degenerate kernel method. Normality and continuity of the integral operator are also discussed. The numerical examples in Sect. 5 illustrate the applicability of the theoretical results.     

Liouville type theorems for integral equations and integral systems

In this paper, we establish some Liouville type theorems for positive solutions of some integral equations and integral systems in R ^N . The main technique we use is the method of moving planes in an integral form.     

Volterra integral equations and evolution equations with an integral condition

We suggest a simple method for reducing problems with an integral condition for evolution equations to a Volterra integral equation of the first kind. For Volterra equations of the convolution type, we indicate necessary and sufficient solvability conditions for the case in which the right-hand side lies in some classes of functions of finite smoothness. We use these conditions to construct examples of nonexistence of a local solution for the heat equation with an integral condition.     

Asymptotic distinction of counting processes

A canonical representation is obtained for the logarithm of the likelihood ratio. Limit theorems describing its asymptotic behavior are proved. Using these theorems, we study the rate of decrease of the probability of an error of the second-kind in the Neyman-Pearson test.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 972–979, July, 1993.     

Volterra integral equations

One presents a survey of the investigations in the theory of Volterra integral equations, reviewed in Ref. Zh. Mat. between 1966–1976.Translated from Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 15, pp. 131–198, 1977.     

Wavelet-based preconditioners for boundary integral equations

We study simple preconditioners for the conjugate gradient method when used to solve matrix systems arising from some hypersingular and weakly singular integral equations. The preconditioners, which are of the type of hierarchical basis preconditioners, are based on the decomposition of the piecewise-linear (respectively piecewise-constant) functions as the sum of prewavelets (respectively derivatives of prewavelets). We prove that with these preconditioners the preconditioned systems have condition numbers uniformly bounded with respect to the degrees of freedom. Numerical experiments support our analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multigrid method for nonlinear integral equations

In this paper, we introduce a multigrid method for solving the nonliear Urysohn integral equation. The algorithm is derived from a discrete resolvent equation which approximates the continuous resolvent equation of the nonlinear Urysohn integral equation. The algorithm is mathematically equivalent to Atkinson’s adaptive twogrid iteration. But the two are different computationally. We show the convergence of the algorithm and its equivalence to Atkinson’s adaptive twogrid iteration. In our numerical example, we compare our algorithm to other multigrid methods for solving the nonliear Urysohn integral equation including the nonlinear multigrid method introduced by Hackbush.     

Multigrid methods for boundary integral equations

Summary Multigrid methods are applied for solving algebraic systems of equations that occur to the numerical treatment of boundary integral equations of the first and second kind. These methods, originally formulated for partial differential equations of elliptic type, combine relaxation schemes and coarse grid corrections. The choice of the relaxation scheme is found to be essential to attain a fast convergent iterative process. Theoretical investigations show that the presented relaxation scheme provides a multigrid algorithm of which the rate of convergence increases with the dimension of the finest grid. This is illustrated for the calculation of potential flow around an aerofoil.     

Sparse grids for boundary integral equations

Summary. The potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in . Theoretical and numerical results on approximation rates, preconditioning, adaptivity and compression for piecewise constant and linear sparse grid spaces are obtained. Received March 17, 1998 / Revised version received September 10, 1998