Numerical Methods for Integral Equations

We first propose a multiscale Galerkin method for solving the Volterra integral equations of the second kind with a weakly singular kernel. Due to the special structure of Volterra integral equations and the ``shrinking support property of multiscale basis functions, a large number of entries of the coefficient matrix appearing in the resulting discrete linear system are zeros. This result, combined with a truncation scheme of the coefficient matrix, leads to a fast numerical solution of the integral equation. A quadrature method is designed especially for the weakly singular kernel involved inside the integral operator to compute the nonzero entries of the compressed matrix so that the quadrature errors will not ruin the overall convergence order of the approximate solution of the integral equation. We estimate the computational cost of this numerical method and its approximate accuracy. Numerical experiments are presented to demonstrate the performance of the proposed method. We also exploit two methods based on neural network models and the collocation method in solving the linear Fredholm integral equations of the second kind. For the first neural network (NN) model, we cast the problem of solving an integral equation as a data fitting problem on a finite set, which gives rise to an optimization problem. In the second method, which is referred to as the NN-Collocation model, we first choose the polynomial space as the projection space of the Collocation method, then approximate the solution of the integral equation by a linear combination of polynomials in that space. The coefficients of the linear combination are served as the weights between the hidden layer and the output layer of the neural network. We train both neural network models using gradient descent with Adam optimizer. Finally, we compare the performances of the two methods and find that the NN-Collocation model offers a more stable, accurate, and efficient solution

Full tomographic reconstruction of 2D vector fields using discrete integral data

Vector field tomography is a field that has received considerable attention in recent decades. It deals with the problem of the determination of a vector field from non-invasive integral data. These data are modelled by the vectorial Radon transform. Previous attempts at solving this reconstruction problem showed that tomographic data alone are insufficient for determining a 2D band-limited vector field completely and uniquely. This paper describes a method that allows one to recover both components of a 2D vector field based only on integral data, by solving a system of linear equations. We carry out the analysis in the digital domain and we take advantage of the redundancy in the projection data, since these may be viewed as weighted sums of the local vector field's Cartesian components. The potential of the introduced method is demonstrated by presenting examples of vector field reconstruction

ON OPTIMAL PROJECTION METHODS FOR SOLVING PERIODIC FRACTIONAL-INTEGRAL EQUATIONS

В статье решается задача оптимизации полиномиальных проекционных методов ре- шения периодических уравнений с дробно-интегральным оператором Вейля в главной части. Для класса регуляризованных интегральных уравнений дробного порядка, за- даваемых принадлежностью коэффициентов фиксированному классу Гельдера, в паре пространств гельдеровых функций доказана оптимальность по порядку точности из- вестных методов: Галеркина по тригонометрической системе функций, коллокации и подобластей по равноотстоящим узлам. Отсюда, как следствие, вытекает опти- мальность указанных методов и в соответствующем классе интегральных уравнений с дробно-интегральным оператором Вейля в главной части.We consider the solution for the problem of polynomial projection methods for solving periodic equations with a fractional-integral Weyl operator in the principal part. The optimality is proved in the order of accuracy for a class of regularized integral equations of fractional order. In this case the coefficients belong to a fixed Hoelder class in a pair of spaces of Hoelder functions. The optimality is proved on the trigonometric system of functions for the Galerkin method and on equidistant nodes for the methods of collocation and subdomains. Hence, as a consequence, the optimality of these methods also follows in the corresponding class of integral equations with a fractional-integral Weyl operator in the principal part.29-3

Fast integral equation methods for the Laplace-Beltrami equation on the sphere

Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere in the presence of multiple "islands" are presented. The surface of the sphere is first mapped to a multiply-connected region in the complex plane via a stereographic projection. After discretizing the integral equation, the resulting dense linear system is solved iteratively using the fast multipole method for the 2D Coulomb potential in order to calculate the matrix-vector products. This numerical scheme requires only O(N) operations, where $N$ is the number of nodes in the discretization of the boundary. The performance of the method is demonstrated on several examples

Adaptive Energy Preserving Methods for Partial Differential Equations

A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches, thereby also including finite element approaches. The schemes are then extended to accommodate $r$-, $h$- and $p$-adaptivity. The method is applied to the Korteweg-de Vries equation and the Sine-Gordon equation and results from numerical experiments are presented.Comment: 27 pages; some changes to notation and figure