arXiv subjects

Changtao Sheng

Publications and source records attributed to Changtao Sheng.

16 recordsLinked to original sources

arXiv2026

An hp-version time stepping spectral Monte Carlo method for semi-linear parabolic equations

In this paper, we present an $hp$-version time-stepping spectral Monte Carlo method for solving semi-linear parabolic equations. The key innovation lies in constructing an exponentially accurate stochastic algorithm that integrates a residual iteration scheme on Gauss-type nodes in both temporal and spatial directions with a reconstruction strategy rooted in spectral methods. To address the long-time simulations and initial singularities that are often challenging for traditional stochastic algorithms (e.g., walk-on-spheres method), we further develop an $hp$-version time-stepping framework that employs multiple time steps and, respectively, geometric time partitions with linearly increasing polynomial degrees to handle these difficulties. Notably, the proposed algorithm bypasses the need to solve linear systems required by traditional spectral methods and remarkably supports parallel computation at both temporal and spatial grid points. We rigorously establish exponential convergence rates for the multistep method within a finite number of iterations. Extensive numerical experiments are conducted to demonstrate the spectral accuracy and computational efficiency of the proposed method in long-time simulations, problems with initial singularities, and a five-dimensional problem, thereby validating the theoretical results.

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arXiv2026

A modified projected walk on spheres method for elliptic equations on high-dimensional embedded manifolds: algorithm and error estimates

In this paper, we propose a modified projected Walk on Spheres method (MPWoS) for screened Poisson equations on embedded manifolds. The method employs local extensions together with the Green representation in local Euclidean balls, coupled with a closest-point projection that maps the boundary samples back to the manifold. This formulation yields a meshfree and highly parallelizable stochastic recursion in the ambient Euclidean space, rather than a direct discretization of the Laplace-Beltrami operator on the manifold. The proposed approach can be viewed as a high-dimensional extension and modification of the projected Walk on Spheres method introduced for surface PDEs in [Sugimoto et al., SIGGRAPH Asia 2024 Conference Papers, pp. 1-10], with three main distinctions: a compensation term that corrects the discrepancy between the ambient Laplacian applied to the closest-point extension and the intrinsic Laplace-Beltrami operator on the manifold, an adaptive radius strategy determined by local geometric and boundary information, and a rigorous error analysis for the proposed algorithm. Under assumptions on the geometric projection and the prescribed compensation accuracy, we establish mean-square error estimates for the proposed Monte Carlo method in both the boundary and closed-manifold settings. Extensive numerical examples on parametrized, implicit, high-dimensional (up to 1000 dimensions), and point-cloud manifolds are presented to illustrate the convergence and efficiency of the proposed method across different geometries.

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arXiv2026

FNWoS: Fractional Neural Walk-on-Spheres Methods for High-Dimensional PDEs Driven by $α$-stable Lévy Process on Irregular Domains

In this paper, we develop a highly parallel and derivative-free fractional neural walk-on-spheres method (FNWoS) for solving high-dimensional fractional Poisson equations on irregular domains. We first propose a simplified fractional walk-on-spheres (FWoS) scheme that replaces the high-dimensional normalized weight integral with a constant weight and adopts a correspondingly simpler sampling density, substantially reducing per-trajectory cost. To mitigate the slow convergence of standard Monte Carlo sampling, FNWoS is then proposed via integrating this simplified FWoS estimator, derived from the Feynman-Kac representation, with a neural network surrogate. By amortizing sampling effort over the entire domain during training, FNWoS achieves more accurate evaluation at arbitrary query points with dramatically fewer trajectories than classical FWoS. To further enhance efficiency in regimes where the fractional order $α$ is close to 2 and trajectories become excessively long, we introduce a truncated path strategy with a prescribed maximum step count. Building on this, we propose a buffered supervision mechanism that caches training pairs and progressively refines their Monte Carlo targets during training, removing the need to precompute a highly accurate training set and yielding the buffered fractional neural walk-on-spheres method (BFNWoS). Extensive numerical experiments, including tests on irregular domains and problems with dimensions up to $1000$, demonstrate the accuracy, scalability, and computational efficiency of the proposed methods.

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arXiv2025

A derivative-free localized stochastic method for very high-dimensional semilinear parabolic PDEs

We develop a mesh-free, derivative-free, matrix-free, and highly parallel localized stochastic method for high-dimensional semilinear parabolic PDEs. The efficiency of the proposed method is built upon four essential components: (i) a martingale formulation of the forward backward stochastic differential equation (FBSDE); (ii) a small scale stochastic particle method for local linear regression (LLR); (iii) a decoupling strategy with a matrix-free solver for the weighted least-squares system used to compute $\nabla u$; (iv) a Newton iteration for solving the univariate nonlinear system in $u$. Unlike traditional deterministic methods that rely on global information, this localized computational scheme not only provides explicit pointwise evaluations of $u$ and $\nabla u$ but, more importantly, is naturally suited for parallelization across particles. In addition, the algorithm avoids the need for spatial meshes and global basis functions required by classical deterministic approaches, as well as the derivative-dependent and lengthy training procedures often encountered in machine learning. More importantly, we rigorously analyze the error bound of the proposed scheme, which is fully explicit in both the particle number $M$ and the time step size $Δt$. Numerical results conducted for problem dimensions ranging from $d=100$ to $d=10000$ consistently verify the efficiency and accuracy of the proposed method. Remarkably, all computations are carried out efficiently on a standard personal computer, without requiring any specialized hardware. These results confirm that the proposed method is built upon a principled design that not only extends the practically solvable range of ultra-high-dimensional PDEs but also maintains rigorous error control and ease of implementation.

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arXiv2025

An Efficient Finite Element Method for Multi-dimensional Nonlocal Laplacian on Uniform Grids

Computing the stiffness matrix for the finite element discretization of the nonlocal Laplacian on unstructured meshes is difficult, because the operator is nonlocal and can even be singular. In this paper, we focus on the $C^0$-piecewise linear finite element method (FEM) for the nonlocal Laplacian on uniform grids within a $d$-dimensional rectangular domain. By leveraging the connection between FE bases and B-splines (having attractive convolution properties), we can reduce the involved $2d$-dimensional integrals for the stiffness matrix entries into integrations over $d$-dimensional balls with explicit integrands involving cubic B-splines and the kernel functions, which allows for explicit study of the singularities and accurate evaluations of such integrals in spherical coordinates. We show the nonlocal stiffness matrix has a block-Toeplitz structure, so the matrix-vector multiplication can be implemented using fast Fourier transform (FFT). In addition, when the interaction radius $δ\to 0^+,$ the nonlocal stiffness matrix automatically reduces to the local one. Although our semi-analytic approach on uniform grids cannot be extended to general domains with unstructured meshes, the resulting solver can seamlessly integrate with the grid-overlay (Go) technique for the nonlocal Laplacian on arbitrary bounded domains.

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arXiv2025

Efficient implicit-explicit sparse stochastic method for high dimensional semi-linear nonlocal diffusion equations

In this paper, we present a sparse grid-based Monte Carlo method for solving high-dimensional semi-linear nonlocal diffusion equations with volume constraints. The nonlocal model is governed by a class of semi-linear partial integro-differential equations (PIDEs), in which the operator captures both local convection-diffusion and nonlocal diffusion effects, as revealed by its limiting behavior with respect to the interaction radius. To overcome the bottleneck of computational complexity caused by the curse of dimensionality and the dense linear systems arising from nonlocal operators, we propose a novel implicit-explicit scheme based on a direct approximation of the nonlinear Feynman-Kac representation. The incorporation of sparse grid interpolation significantly enhances the algorithm's scalability and enables its application to problems in high dimensions. To further address the challenges posed by hypersingular kernels, we design a sampling strategy tailored to their singular structure, which ensures accurate and stable treatment of the nonlocal operators within the probabilistic framework. Notably, the proposed method inherits unconditional stability from the underlying stochastic representation, without imposing constraints on the temporal and spatial discretization scales. A rigorous error analysis is provided to establish the convergence of the proposed scheme. Extensive numerical experiments, including some non-radial solutions in up to 100 dimensions, are presented to validate the robustness and accuracy of the proposed method.

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arXiv2025

Exponentially accurate spectral Monte Carlo method for linear PDEs and their error estimates

This paper introduces a spectral Monte Carlo iterative method (SMC) for solving linear Poisson and parabolic equations driven by $α$-stable Lévy process with $α\in (0,2)$, which was initially proposed and developed by Gobet and Maire in their pioneering works (Monte Carlo Methods Appl 10(3-4), 275--285, 2004, and SIAM J Numer Anal 43(3), 1256--1275, 2005) for the case $α=2$. The novel method effectively integrates multiple computational techniques, including the interpolation based on generalized Jacobi functions (GJFs), space-time spectral methods, control variates techniques, and a novel walk-on-sphere method (WOS). The exponential convergence of the error bounds is rigorously established through finite iterations for both Poisson and parabolic equations involving the integral fractional Laplacian operator. Remarkably, the proposed space-time spectral Monte Carlo method (ST-SMC) for the parabolic equation is unified for both $α\in(0,2)$ and $α=2$. Extensive numerical results are provided to demonstrate the spectral accuracy and efficiency of the proposed method, thereby validating the theoretical findings.

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arXiv2024

FEM on nonuniform meshes for nonlocal Laplacian: Semi-analytic Implementation in One Dimension

In this paper, we compute stiffness matrix of the nonlocal Laplacian discretized by the piecewise linear finite element on nonuniform meshes, and implement the FEM in the Fourier transformed domain. We derive useful integral expressions of the entries that allow us to explicitly or semi-analytically evaluate the entries for various interaction kernels. Moreover, the limiting cases of the nonlocal stiffness matrix when the interactional radius $δ\rightarrow0$ or $δ\rightarrow\infty$ automatically lead to integer and fractional FEM stiffness matrices, respectively, and the FEM discretisation is intrinsically compatible. We conduct ample numerical experiments to study and predict some of its properties and test on different types of nonlocal problems. To the best of our knowledge, such a semi-analytic approach has not been explored in literature even in the one-dimensional case.

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arXiv2023

A highly efficient and accurate divergence-free spectral method for curl-curl equation in two and three dimensions

In this paper, we present a fast divergence-free spectral algorithm (FDSA) for the curl-curl problem. Divergence-free bases in two and three dimensions are constructed by using the generalized Jacobi polynomials. An accurate spectral method with exact preservation of the divergence-free constraint point-wisely is then proposed, and its corresponding error estimate is established. We then present a highly efficient solution algorithm based on a combination of matrix-free preconditioned Krylov subspace iterative method and a fully diagonalizable auxiliary problem, which is derived from the spectral discretisations of generalized eigenvalue problems of Laplace and biharmonic operators. We rigorously prove that the dimensions of the invariant subspace of the preconditioned linear system resulting from the divergence-free spectral method with respect to the dominate eigenvalue $1$, are $(N-3)^2$ and $2(N-3)^3$ for two- and three-dimensional problems with $(N-1)^2$ and $2(N-1)^3$ unknowns, respectively. Thus, the proposed method usually takes only several iterations to converge, and astonishingly, as the problem size (polynomial order) increases, the number of iterations will decrease, even for highly indefinite system and oscillatory solutions. As a result, the computational cost of the solution algorithm is only a small multiple of $N^3$ and $N^4$ floating number operations for 2D and 3D problems, respectively. Plenty of numerical examples for solving the curl-curl problem with both constant and variable coefficients in two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.

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arXiv2023

Müntz ball polynomials and Müntz spectral-Galerkin methods for singular eigenvalue problems

In this paper, we introduce a new family of orthogonal systems, termed as the Müntz ball polynomials (MBPs), which are orthogonal with respect to the weight function: $\|x\|^{2θ+2μ-2} (1-\|x\|^{2θ})^α$ with the parameters $α>-1, μ>- 1/2$ and $θ>0$ in the $d$-dimensional unit ball $x\in {\mathbb B}^d=\big\{x\in\mathbb{R}^d: r=\|x\|\leq1\big\}$. We then develop efficient and spectrally accurate MBP spectral-Galerkin methods for singular eigenvalue problems including degenerating elliptic problems with perturbed ellipticity and Schrödinger's operators with fractional potentials. We demonstrate that the use of such non-standard basis functions can not only tailor to the singularity of the solutions but also lead to sparse linear systems which can be solved efficiently.

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arXiv2022

Efficient Monte Carlo Method for Integral Fractional Laplacian in Multiple Dimensions

In this paper, we develop a Monte Carlo method for solving PDEs involving an integral fractional Laplacian (IFL) in multiple dimensions. We first construct a new Feynman-Kac representation based on the Green function for the fractional Laplacian operator on the unit ball in arbitrary dimensions. Inspired by the "walk-on-spheres" algorithm proposed in [24], we extend our algorithm for solving fractional PDEs in the complex domain. Then, we can compute the expectation of a multi-dimensional random variable with a known density function to obtain the numerical solution efficiently. The proposed algorithm finds it remarkably efficient in solving fractional PDEs: it only needs to evaluate the integrals of expectation form over a series of inside ball tangent boundaries with the known Green function. Moreover, we carry out the error estimates of the proposed method for the $n$-dimensional unit ball. Finally, ample numerical results are presented to demonstrate the robustness and effectiveness of this approach for fractional PDEs in unit disk and complex domains, and even in ten-dimensional unit balls.

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arXiv2020

Generalised Hermite spectral methods for PDEs involving integral fractional Laplacian and Schrödinger operators

In this paper, we introduce two new families of generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate generalised Hermite spectral algorithms for PDEs with integral fractional Laplacian (IFL) and/or Schrödinger operators in $\mathbb R^d.$ As a generalisation of the G. Szegö's family in 1D (1939), the first family of GHPs (resp. GHFs) are orthogonal with respect to $|\bx|^{2μ} \e^{-|\bx|^2}$ (resp. $|\bx |^{2μ}$) in $\mathbb R^d$. We further define adjoint generalised Hermite functions (A-GHFs) which have an interwoven connection with the corresponding GHFs through the Fourier transform, and which are orthogonal with respect to the inner product $[u,v]_{H^s(\mathbb R^d)}=((-Δ)^{s/ 2}u, (-Δ)^{s/2} v )_{\mathbb R^d}$ associated with the IFL of order $s>0$. Thus, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds efficient and accurate in solving PDEs with the fractional Schrödinger operator: $(-Δ)^s +|\bs x|^{2μ}$ with $s\in (0,1]$ and $μ>-1/2.$ Following the same spirit, we construct the second family of GHFs, dubbed as Müntz-type generalised Hermite functions (M-GHFs), which are orthogonal with respect to an inner product associated with the underlying Schrödinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Müntz-type GHF spectral method leads to sparse matrices and spectrally accurate to some Schrödinger eigenvalue problems.

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arXiv2020

On explicit form of the FEM stiffness matrix for the integral fractional Laplacian on non-uniform meshes

We derive exact form of the piecewise-linear finite element stiffness matrix on general non-uniform meshes for the integral fractional Laplacian operator in one dimension, where the derivation is accomplished in the Fourier transformed space. With such an exact formulation at our disposal, we are able to numerically study some intrinsic properties of the fractional stiffness matrix on some commonly used non-uniform meshes (e.g., the graded mesh), in particular, to examine their seamless transition to those of the usual Laplacian.

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arXiv2020

On diagonal dominance of FEM stiffness matrix of fractional Laplacian and maximum principle preserving schemes for fractional Allen-Cahn equation

In this paper, we study diagonal dominance of the stiffness matrix resulted from the piecewise linear finite element discretisation of the integral fractional Laplacian under global homogeneous Dirichlet boundary condition in one spatial dimension. We first derive the exact form of this matrix in the frequency space which is extendable to multi-dimensional rectangular elements. Then we give the complete answer when the stiffness matrix can be strictly diagonally dominant. As one application, we apply this notion to the construction of maximum principle preserving schemes for the fractional-in-space Allen-Cahn equation, and provide ample numerical results to verify our findings.

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arXiv2019

Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., $O((N\log_2N)^d)$ with $N$ being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time-discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schr{ö}dinger equation by using the fourth-order time-splitting method together with the proposed MCF-spectral-Galerkin method.

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arXiv2018

Fundamental Gaps of the Fractional Schrödinger Operator

We study asymptotically and numerically the fundamental gap -- the difference between the first two smallest (and distinct) eigenvalues -- of the fractional Schrödinger operator (FSO) and formulate a gap conjecture on the fundamental gap of the FSO. We begin with an introduction of the FSO on bounded domains with homogeneous Dirichlet boundary conditions, while the fractional Laplacian operator defined either via the local fractional Laplacian (i.e. via the eigenfunctions decomposition of the Laplacian operator) or via the classical fractional Laplacian (i.e. zero extension of the eigenfunctions outside the bounded domains and then via the Fourier transform). For the FSO on bounded domains with either the local fractional Laplacian or the classical fractional Laplacian, we obtain the fundamental gap of the FSO analytically on simple geometry without potential and numerically on complicated geometries and/or with different convex potentials. Based on the asymptotic and extensive numerical results, a gap conjecture on the fundamental gap of the FSO is formulated. Surprisingly, for two and higher dimensions, the lower bound of the fundamental gap depends not only on the diameter of the domain, but also the diameter of the largest inscribed ball of the domain, which is completely different from the case of the Schrödinger operator. Extensions of these results for the FSO in the whole space and on bounded domains with periodic boundary conditions are presented.

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