High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.     

Regularized Lardy Scheme for Solving Singularly Perturbed Elliptic and Parabolic PDEs

Mediterranean Journal of Mathematics - In the present paper, we introduce screen generic lightlike submanifold of golden semi-Riemannian manifolds and give non-trivial examples of such...     

An ALE ESFEM for Solving PDEs on Evolving Surfaces

Numerical methods for approximating the solution of partial differential equations on evolving hypersurfaces using surface finite elements on evolving triangulated surfaces are presented. In the ALE ESFEM the vertices of the triangles evolve with a velocity which is normal to the hypersurface whilst having a tangential velocity which is arbitrary. This is in contrast to the original evolving surface finite element method in which the nodes move with a material velocity. Numerical experiments are presented which illustrate the value of choosing the arbitrary tangential velocity to improve mesh quality. Simulations of two applications arising in material science and biology are presented which couple the evolution of the surface to the solution of the surface partial differential equation.     

Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and its local kernel variants to approximate second-order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the ghost point diffusion maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points, specified along the estimated tangential direction at the sampled points on the boundary. The resulting GPDM estimator restricts the standard DM matrix to a set of extrapolation equations that estimates the function values at the ghost points. This adjustment is analogous to the classical ghost point method in a finite-difference scheme for solving PDEs on flat domains. As opposed to the classical DM, which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary. Applying the consistent GPDM estimator to solve well-posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions. We numerically validate the proposed mesh-free PDE solver on various problems defined on simple submanifolds embedded in Euclidean spaces as well as on an unknown manifold. Numerically, we also found that the GPDM is more accurate compared to DM in solving elliptic eigenvalue problems on bounded smooth manifolds. © 2021 Wiley Periodicals LLC.     

Adaptive Wavelet Methods for Elliptic Operator Equations with Nonlinear Terms

We present an adaptive wavelet method for the numerical solution of elliptic operator equations with nonlinear terms. This method is developed based on tree approximations for the solution of the equations and adaptive fast reconstruction of nonlinear functionals of wavelet expansions. We introduce a constructive greedy scheme for the construction of such tree approximations. Adaptive strategies of both continuous and discrete versions are proposed. We prove that these adaptive methods generate approximate solutions with optimal order in both of convergence and computational complexity when the solutions have certain degree of Besov regularity.     

Circulant Wavelet Preconditioners for Solving Elliptic Differential Equations and Boundary Integral Equations

In this paper is discussed solving an elliptic equation and a boundary integral equation of the second kind by representation of compactly supported wavelets. By using wavelet bases and the Galerkin method for these equations, we obtain a stiff sparse matrix that can be ill-conditioned. Therefore, we have to introduce an operator which maps every sparse matrix to a circulant sparse matrix. This class of circulant matrices is a class of preconditioners in a Banach space. Based on having some properties in the spectral theory for this class of matrices, we conclude that the circulant matrices are a good class of preconditioners for solving these equations. We called them circulant wavelet preconditioners (CWP). Therefore, a class of algorithms is introduced for rapid numerical application.     

Elliptic PDEs with Fibered Nonlinearities

In ℝ ^m ×ℝ ^n−m , endowed with coordinates x=(x′,x″), we consider bounded solutions of the PDE

用径向函数法求解偏微分方程

In this paper. Kansa′s method and Hermite collocation method with Radial Basis Func-tions is applied to solve partial differential equation. The resultant matrix generated from the Her-mite method is positive definite, which guarantees the reversibility of the matrix. The numerical re-sults indicate that the methods provides reversibility of the matrix. The numerical results indicatethat the method provieds an efficient algorithm for solving partial differential equations.     

The Jacobi Elliptic Function Method for Solving Zakharov Equation

The Zakharov equation to describe the laser plasma interaction process has very important sense, this paper gives the solitary wave solutions for Zakharov equation by using Jacobi elliptic function method.     

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

We propose a multiscale multilevel Monte Carlo(MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage,we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallestscale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.     

A Semi-Analytical Method of Solving the Fokker-Planck-Equation for High-Dimensional Nonlinear Mechanical Systems

Stochastic processes are a common way of describing systems that are subjected to random influences. Technical systems may be excited by road roughness or wind gusts, for example, as well as fluctuating system parameters, which can all be described by stochastic differential equations. In previous works by the author and others (see [1], for example) it has been demonstrated how a Galerkin-method can be used to obtain global numerical solutions of the Fokker-Planck-Equation (FPE) for nonlinear random systems. Computational efforts are reduced by orthogonal polynomial expansion of approximate solutions so that probability density functions (pdfs) for comparably high-dimensional problems have been computed successfully. Stationary mechanical systems with dimensions up to d = 10 have been investigated, including polynomial as well as non-smooth nonlinearities. This article presents results for different energy-harvester-systems under stochastic excitation, a field of research that has become the subject of increasing attention in the last years. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive Wavelet Methods II—Beyond the Elliptic Case

This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in [17] for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [26]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a significant modification of the ideas from [17]. The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.     

求解对流扩散方程的Haar小波方法

本文用Haar小波求解对流扩散方程，将满足初始和边界条件的常系数偏微分方程简化为较简单的代数方程组进行求解．实例说明了这种方法具有收敛速度快和计算容易的特点，同时又避免了用Daubechies小波求解微分方程需要计算相关系数的麻烦．本文所使用的方法可以求解一般的微（积）分方程．     

Adaptive multiscale methods for elliptic PDEs

This paper deals with the numerical solution of elliptic partial differential equations with an adaptive multiscale algorithm, which uses symmetric collocation and radial basis functions. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Iterative Tikhonov Regularization for Solving Singularly Perturbed Elliptic PDE

In this paper, we consider a class of singularly perturbed elliptical problems with homogeneous boundary conditions. We consider a regularized iterative method for solving such problems. Convergence analysis and error estimate are derived. The regularization parameter is chosen according to an a priori strategy. We give numerical results to illustrate that the method is implementable compared with numerical methods such as Shishkin and finite element schemes. The study demonstrates that the iterated regularized scheme can be considered as an alternate method for solving singularly perturbed elliptical problems.     

Haar小波方法求解分数阶Poisson方程的数值解

借助Haar小波正交函数的分数阶积分算子矩阵,通过离散未知变量,将待求Poisson方程转化为大型的线性代数方程组,然后利用Matlab软件进行编程求解,即可求得原问题的未知系数矩阵,代入原方程,从而求得数值解.数值结果表明,当Haar小波采取很小的级数项展开时,即可获得满意的数值精度,而且算法比较稳定,有很强的实际应用价值.     

平面中含Hardy位势临界参数的非线性椭圆型方程

本文首先证明了平面中含Hardy位势和临界参数的非线性椭圆方程在一个新Hilbert空间中无穷多个解的存在性,然后在该空间中还讨论了一类含Hardy位势的变分问题极小解的存在性．     

Adaptive first integral preserving methods for PDEs

We present a method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids, using a finite difference approach for spatial discretization and discrete gradients for time stepping. The method is extended to accommodate spatial adaptivity. A numerical experiment is carried out where the method is applied to the Sine-Gordon equation with moving mesh adaptivity. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)