A stochastic simulation algorithm with iterative refinement for solving integral equations related to an inverse problem for the wave equation

Abstract A new stochastic simulation algorithm for solving integral equations arising in an inverse problem for the wave equation, known as the inverse coefficient recovery problem, is proposed. The method is capable of handling high-dimensional equations and achieves high accuracy due to the use of a hybrid randomized algorithm combined with Wilkinson-s iterative refinement technique. We consider the case of an integral equation whose kernel is represented by the autocorrelation function of a stationary random process, in particular Bessel functions of the zeroth and first order. Based on the Gelfand–Levitan approach, which reduces the nonlinear inverse problem to a family of linear integral equations, we focus on the numerical solution of the system of linear algebraic equations approximating the integral equation by combining a stochastic projection algorithm with randomized iterative refinement methods. To accelerate the convergence of the proposed algorithm, a circulant matrix preconditioner is employed. As demonstrated by theoretical estimates and numerical experiments, the developed hybrid algorithm enables the solution of large-scale systems of equations with high accuracy.