Multilevel approximation of Gaussian random fields: Fast simulation

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [Formula: see text] or, more generally, by a compact metric space [Formula: see text] such as a compact [Formula: see text]-manifold [Formula: see text]. A colored GRF [Formula: see text], admissible for our algorithms, solves the stochastic fractional-order equation [Formula: see text] for some [Formula: see text], where [Formula: see text] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [Formula: see text] is white noise on [Formula: see text]. We thus consider GRFs on [Formula: see text] with covariance operators of the form [Formula: see text]. The proposed algorithms numerically approximate samples of [Formula: see text] on nested sequences [Formula: see text] of regular, simplicial partitions [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. Work and memory to compute one approximate realization of the GRF [Formula: see text] on the triangulation [Formula: see text] of [Formula: see text] with consistency [Formula: see text], for some consistency order [Formula: see text], scale essentially linearly in [Formula: see text], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [Formula: see text] to white noise [Formula: see text]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

On sinc quadrature approximations of fractional powers of regularly accretive operators

We consider the finite element approximation of fractional powers of regularly accretive operators via the Dunford–Taylor integral approach. We use a sinc quadrature scheme to approximate the Balakrishnan representation of the negative powers of the operator as well as its finite element approximation. We improve the exponentially convergent error estimates from [A. Bonito and J. E. Pasciak, IMA J. Numer. Anal., 37 (2016), No. 3, 1245–1273] by reducing the regularity required on the data. Numerical experiments illustrating the new theory are provided.

Solving fractional diffusion equations by Sinc and radial basis functions

In this study, we approximate the solution of the fractional diffusion equations based on Gaussian radial basis function (GRBF). Our approach is based on the Caputo fractional derivative and the combination of GRBF and Sinc function, here the GRBF direct and GRBF-QR methods are developed. The Sinc quadrature rule combined with double exponential transformation (DE) has been used to approximate the fractional integral. Three examples have been considered to test the presented methods, we compared the results to verify the effectiveness of the presented methods against recent existing methods diven by [A radial basis functions method for fractional diffusion equations, J. Comput. Phys. 238 (2013) 71–81; An efficient Chebyshev-tau method for solving the space fractional diffusion equations, Appl. Math. Comput. 224 (2013) 259–267; A tau approach for solution of the space fractional diffusion equations, Comput. Math. Appl. 62 (2011) 1135–1142], and also conclude that GRBF-QR-Sinc method demonstrates better accuracy.

Numerical Approximation of Space-Time Fractional Parabolic Equations

In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e. an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator {E(t)} for the initial value problem can be written as a Dunford–Taylor integral involving the Mittag-Leffler function {e_{\alpha,1}} and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here α denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator {W(t)} and the forcing function {F(t)}. We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford–Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of {\frac{1}{h}}. The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying a sinc quadrature technique to approximate the Dunford–Taylor integral of the discrete operator and is free of any time stepping. The sinc quadrature of step size k involves {k^{-2}} nodes and results in an additional {O(\exp(-\frac{c}{k}))} error. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo-midpoint quadrature. This involves the average of {W_{h}(s)}, (the semi-discrete approximation to {W(s)}) over the quadrature interval. This average can also be written as a Dunford–Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford–Taylor integrals for computing the averages. We show that for a refined grid in time with a mesh of {O({\mathcal{N}}\log({\mathcal{N}}))} intervals, the error between the semi-discrete and fully discrete approximation is {O({\mathcal{N}}^{-2}+\log({\mathcal{N}})\exp(-\frac{c}{k}))}. We also report the results of numerical experiments that are in agreement with the theoretical error estimates.

FFT-based Kronecker product approximation to micromagnetic long-range interactions

We derive a Kronecker product approximation for the micromagnetic long-range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format, Tensor Trains) scales below linear in the volume size. Based on efficient usage of FFT for structured tensors, we are able to accelerate computations to quasi-linear complexity in the number of collocation points used in one dimension. Quadratic convergence of the underlying collocation scheme as well as exponential convergence in the separation rank of the approximations is proved. Numerical experiments on accuracy and complexity confirm the theoretical results.

A New Estimate of the Sing Method for Linear Parabolic Problems Including the Initial Point

For two new effcient methods for solving initial value problems in a Hilbert or Banach spaces based on a Sinc quadrature for an improper Dunford-Cauchy integrals over a path enveloping the spectrum of the operator we give a new unified estimate in the case of a Hilbert space.

Exponentially Convergent Parallel Discretization Methods for the First Order Evolution Equations

We propose a new discretization of an initial value problem for differen- tial equations of the first order in a Banach space with a strongly P-positive operator coefficient. Using the strong positiveness, we represent the solution as a Dunford- Cauchy integral along a parabola in the right half of the complex plane, then transform it into real integrals over (−∞,∞), and finally apply an exponentially convergent Sinc quadrature formula to this integral. The integrand values are the solutions of a finite set of elliptic problems with complex coefficients, which are independent and may be solved in parallel.