GENERALIZATION OF THE NYSTRÖM METHOD FOR SOLVING HIGHER-DIMENSIONAL FREDHOLM INTEGRAL EQUATIONS AND ITS COMPUTATIONAL PERFORMANCE

Abstract

This paper extends the Nyström method to the multidimensional domain of second kind Fredholm integral equations. The continuous integral operator over a domain Ω⊂ℝᵈ is replaced by a multidimensional quadrature rule and collocation on the quadrature nodes yields a dense weighted-kernel system. Although low dimensional nodes can be constructed directly and accurately by Tensor-product quadrature, the number of nodes increases exponentially with d. The resulting matrix will be required to have quadratic size and must be solved using cubic time for a direct solution, creating a severe dimensional bottleneck. In order to overcome this shortcoming, the paper considers the problem of sparse-grid quadrature, dimension-adaptive rules, quasi-Monte Carlo sampling, matrix-free iterative solution and low-rank kernel compression. The error analysis method is developed to separate the quadrature consistency error and the amplification by the inverse discrete Fredholm operator, which are multidimensional. The computational study is addressed theoretically in terms of node numbers, memory usage, arithmetic complexity, convergence measures and through a reproducible benchmark design. The analysis indicates that tensor products are still competitive for smooth two-dimensional problems at moderate resolution; on the other hand, sparse grids and low-rank iterative methods become more and more relevant as the dimension and the accuracy requirements increase. The generalized framework is simple and easily implementable, and maintains the simplicity of the Nyström method while offering practical ways to control the curse of dimensionality.