Spectral collocation method for compact integral operators

We propose and analyze a spectral collocation method for integral equations with compact kernels, e.g. piecewise smooth kernels and weakly singular kernels of the form $\frac{1}{|t-s|^\mu}, \; 0\u3c\mu\u3c1.$ We prove that 1) for integral equations, the convergence rate depends on the smoothness of true solutions $y(t)$. If $y(t)$ satisfies condition (R): $\|y^{(k)}\|_{L^\infty[0,T]}\leq ck!R^{-k}$}, we obtain a geometric rate of convergence; if $y(t)$ satisfies condition (M): $\|y^{(k)}\|_{L^{\infty}[0,T]}\leq cM^k$, we obtain supergeometric rate of convergence for both Volterra equations and Fredholm equations and related integro differential equations; 2) for eigenvalue problems, the convergence rate depends on the smoothness of eigenfunctions. The same convergence rate for the largest modulus eigenvalue approximation can be obtained. Moreover, the convergence rate doubles for positive compact operators. Our numerical experiments confirm our theoretical results

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

Error analysis of numerical methods on graded meshes for stochastic Volterra equations

This paper presents the error analysis of numerical methods on graded meshes for stochastic Volterra equations with weakly singular kernels. We first prove a novel regularity estimate for the exact solution via analyzing the associated convolution structure. This reveals that the exact solution exhibits an initial singularity in the sense that its H\"older continuous exponent on any neighborhood of $t=0$ is lower than that on every compact subset of $(0,T]$. Motivated by the initial singularity, we then construct the Euler--Maruyama method, fast Euler--Maruyama method, and Milstein method based on graded meshes. By establishing their pointwise-in-time error estimates, we give the grading exponents of meshes to attain the optimal uniform-in-time convergence orders, where the convergence orders improve those of the uniform mesh case. Numerical experiments are finally reported to confirm the sharpness of theoretical findings

High Accuracy Combination Method For Solving the Systems of Nonlinear Volterra Integral and Integro-differential Equations with Weakly Singular Kernels of the Second Kind

This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples show that the algorithm is effective and saves storage capacity and computational cost