Horizontal crustal movement in China fitted by adaptive collocation with embedded Euler vector

The crustal movements of the Chinese mainland include an average regional movement trend of the mainland and complex local deformations. Thus, both trends in the crustal movement of the mainland and local distortions should be simultaneously taken into consideration in crustal movement estimations. A combined collocation model based on Euler vector (taken as trend parameters) and local distortions (taken as signals) is proposed in this paper. We assume that prior covariance matrices between signals and observations should be consistent with their uncertainties. Otherwise, the station movement estimates provided by the collocation will be distorted. Thus, an adaptive collocation estimator based on simplified Helmert variance components is applied. This means that the contributions of signals and observations to estimates of crustal movements are balanced and reasonable, and consistent covariance matrices of the signals and observations are achieved through the adjustment of the adaptive factor. The calculation of actual horizontal movements of the Chinese crust shows that the estimates of horizontal crustal movement velocities are made more accurate by the adaptive collocation model.

The Adaptive Wavelet Collocation Method and Its Application in Front Simulation

The adaptive wavelet collocation method （AWCM） is a variable grid technology for solving partial differential equations （PDEs） with high singularities. Based on interpolating wavelets, the AWCM adapts the grid so that a higher resolution is automatically attributed to domain regions with high singularities. Accuracy problems with the AWCM have been reported in the literature, and in this paper problems of efficiency with the AWCM are discussed in detail through a simple one-dimensional （1D） nonlinear advection equation whose analytic solution is easily obtained. A simple and efficient implementation of the AWCM is investigated. Through studying the maximum errors at the moment of frontogenesis of the 1D nonlinear advection equation with different initial values and a comparison with the finite difference method （FDM） on a uniform grid, the AWCM shows good potential for modeling the front efficiently. The AWCM is also applied to a two-dimensional （2D） unbalanced frontogenesis model in its first attempt at numerical simulation of a meteorological front. Some important characteristics about the model are revealed by the new scheme.

Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems

We present a method for solving partial differential equations using artificial neural networks and an adaptive collocation strategy.In this procedure,a coarse grid of training points is used at the initial training stages,while more points are added at later stages based on the value of the residual at a larger set of evaluation points.This method increases the robustness of the neural network approximation and can result in significant computational savings,particularly when the solution is non-smooth.Numerical results are presented for benchmark problems for scalar-valued PDEs,namely Poisson and Helmholtz equations,as well as for an inverse acoustics problem.

Adaptive Stroud Stochastic Collocation Method for Flow in Random Porous Media via Karhunen-Loeve Expansion

In this paper we develop a Stochastic Collocation Method(SCM)for flow in randomly heterogeneous porous media.At first,the Karhunen-Lo`eve expansion is taken to decompose the log transformed hydraulic conductivity field,which leads to a stochastic PDE that only depends on a finite number of i.i.d.Gaussian random variables.Based on the eigenvalue decay property and a rough error estimate of Stroud cubature in SCM,we propose to subdivide the leading dimensions in the integration space for random variables to increase the accuracy.We refer to this approach as adaptive Stroud SCM.One-and two-dimensional steady-state single phase flow examples are simulated with the new method,and comparisons are made with other stochastic methods,namely,the Monte Carlo method,the tensor product SCM,and the quasiMonte Carlo SCM.The results indicate that the adaptive Stroud SCM is more efficient and the statistical moments of the hydraulic head can be more accurately estimated.

Solving Delay Differential Equations through RBF Collocation

A general and easy-to-code numerical method based on radial basis functions(RBFs)collocation is proposed for the solution of delay differential equations(DDEs).It relies on the interpolation properties of infinitely smooth RBFs,which allow for a large accuracy over a scattered and relatively small discretization support.Hardy’s multiquadric is chosen as RBF and combined with the Residual Subsampling Algorithm of Driscoll and Heryudono for support adaptivity.The performance of the method is very satisfactory,as demonstrated over a cross-section of benchmark DDEs,and by comparison with existing general-purpose and specialized numerical schemes for DDEs.