Li-Lian Wang

Sheng Chen, Jie Shen, Li-Lian Wang

In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus, and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value problems (FBVPs) of general order, and show that with an appropriate choice of the parameters in GJFs, the resulted linear systems can be sparse and well-conditioned. Moreover, we derive error estimates with convergence rate only depending on the smoothness of data, so truly spectral accuracy can be attained if the data are smooth enough. The idea and results presented in this paper will be useful to deal with more general FDEs associated with Riemann-Liouville or Caputo fractional derivatives.

Xiaodan Zhao, Li-Lian Wang, Ziqing Xie

This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and valid for degree $n\ge 1$. We demonstrate the sharpness of the estimates by comparing with existing ones, in particular, the very recent results in [38, SIAM J. Numer. Anal., 2012]. We also extend this argument to estimate the Gegenbauer-Gauss quadrature remainder of analytic functions, which leads to some new tight bounds for quadrature errors.

Li-Lian Wang, Bo Wang, Xiaodan Zhao

This paper is concerned with fast and accurate computation of exterior wave equations truncated via exact circular or spherical nonreflecting boundary conditions (NRBCs, which are known to be nonlocal in both time and space). We first derive analytic expressions for the underlying convolution kernels, which allow for a rapid and accurate evaluation of the convolution with $O(N_t)$ operations over $N_t$ successive time steps. To handle the onlocality in space, we introduce the notion of boundary perturbation, which enables us to handle general bounded scatters by solving a sequence of wave equations in a regular domain. We propose an efficient spectral-Galerkin solver with Newmark's time integration for the truncated wave equation in the regular domain. We also provide ample numerical results to show high-order accuracy of NRBCs and efficiency of the proposed scheme.

Wenjie Liu, Li-Lian Wang, Huiyuan Li

In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville (RL) fractional integrals/derivatives, which is naturally arisen from exact representations of Chebyshev expansion coefficients, for optimal error estimates of Chebyshev approximations to functions with limited regularity. The essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions), and (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties. Under this framework, we are able to estimate the optimal decay rate of Chebyshev expansion coefficients for a large class of functions with interior and endpoint singularities, which are deemed suboptimal or complicated to characterize in existing literature. We can then derive optimal error estimates for spectral expansions and the related Chebyshev interpolation and quadrature measured in various norms, and also improve the available results in usual Sobolev spaces of integer regularity exponentials in several senses. As a by-product, this study results in some analytically perspicuous formulas particularly on GGF-Fs, which are potentially useful in spectral algorithms. The idea and analysis techniques can be extended to general Jacobi spectral approximations.

Ziqing Xie, Li-Lian Wang, Xiaodan Zhao

This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the Gegenbauer-Gauss and Gegenbauer-Gauss-Lobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates in the maximum norm are derived.

Jing Zhang, Huiyuan Li, Li-Lian Wang et al.

In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $α>-1$ on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both a weighted concentration integral operator, and a Sturm-Liouville differential operator. Different from existing works on multi-dimensional PSWFs, the ball PSWFs are defined as a generalisation of orthogonal {\em ball polynomials} in primitive variables with a tuning parameter $c>0$, through a "perturbation" of the Sturm-Liouville equation of the ball polynomials. From this perspective, we can explore some interesting intrinsic connections between the ball PSWFs and the finite Fourier and Hankel transforms. We provide an efficient and accurate algorithm for computing the ball PSWFs and the associated eigenvalues, and present various numerical results to illustrate the efficiency of the method. Under this uniform framework, we can recover the existing PSWFs by suitable variable substitutions.

Michael Daniel Samson, Huiyuan Li, Li-Lian Wang

This paper serves as our first effort to develop a new triangular spectral element method (TSEM) on unstructured meshes, using the rectangle-triangle mapping proposed in the conference note [21]. Here, we provide some new insights into the originality and distinctive features of the mapping, and show that this transform only induces a logarithmic singularity, which allows us to devise a fast, stable and accurate numerical algorithm for its removal. Consequently, any triangular element can be treated as efficiently as a quadrilateral element, which affords a great flexibility in handling complex computational domains. Benefited from the fact that the image of the mapping includes the polynomial space as a subset, we are able to obtain optimal $L^2$- and $H^1$-estimates of approximation by the proposed basis functions on triangle. The implementation details and some numerical examples are provided to validate the efficiency and accuracy of the proposed method. All these will pave the way for developing an unstructured TSEM based on, e.g., the hybridizable discontinuous Galerkin formulation.

Lunji Song, Zilan Cheng, Juan Diego Toscano et al.

We explicitly construct an approximate version of the Kolmogorov superpositions, which is composed of C2-inner and outer functions, and can approximate an arbitrary alpha Holder continuous function with accuracy of N to the power -alpha, where N denotes the number of outer summations. The inner functions are generated by applying suitable translations and dilations to a piecewise C2, strictly increasing function, while the outer functions are constructed rowwise through piecewise C2 interpolation using newly designed shape functions. This novel variant of Kolmogorov superpositions overcomes the wild and pathological behaviors of the inherent single variable functions, but retains the essence of Kolmogorov strategy of exact representation-an objective that Sprecher (Neural Netw. 144(2021)438-442) has actively pursued. We also discuss the implications of this new construction and demonstrate its applicability to related neural networks.

Wenjie Liu, Li-Lian Wang

The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(λ)}_ν(x)$ (right GGF-Fs) and ${}^{l}G^{(λ)}_ν(x)$ (left GGF-Fs) with $x\in (-1,1),$ $λ>-1/2$ and real $ν\ge 0,$ are special functions (usually non-polynomials), which are defined upon the hypergeometric representation of the classical Gegenbauer polynomial by allowing integer degree to be real fractional degree. Remarkably, the GGF-Fs become indispensable for optimal error estimates of polynomial approximation to singular functions, and have intimate relations with several families of nonstandard basis functions recently introduced for solving fractional differential equations. However, some properties of GGF-Fs, which are important pieces for the analysis and applications, are unknown or under explored. The purposes of this paper are twofold. The first is to show that for $λ,ν>0$ and $x=\cosθ$ with $θ\in (0,π),$ \begin{equation*}\label{IntRep-0N} (\sin φ)^λ\,{}^{r\!}G_ν^{(λ)}(\cos φ)= \frac{2^λΓ(λ+1/2)}{\sqrtπ {(ν+λ)^λ}} \, {\cos ((ν+λ)φ- λπ/2)} +{\mathcal R}_ν^{(λ)} (φ), \end{equation*} and derive the precise expression of the "residual" term ${\mathcal R}_ν^{(λ)} (φ).$ With this at our disposal, we obtain the bounds of GGF-Fs uniform in $ν.$ Under an appropriate weight function, the bounds are uniform for $θ\in [0,π]$ as well. Moreover, we can study the asymptotics of GGF-Fs with large fractional degree $ν.$ The second is to present miscellaneous properties of GGF-Fs for better understanding of this family of useful special functions.

Bo Wang, Li-Lian Wang, Ziqing Xie

We present in this paper a spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map.

Can Huang, Li-lian Wang

In this paper, we propose an accurate numerical means built upon a spectral-Galerkin method in spatial discretization and an enriched multi-step spectral-collocation approach in temporal direction, for Maxwell equations in Cole-Cole dispersive media in two-dimensional setting. Our starting point is to derive a new model involving only one unknown field from the original model with three unknown fields: electric, magnetic fields and the induced electric polarisation (described by a global temporal convolution of the electric field). This results in a second-order integral-differential equation with a weakly singular integral kernel expressed by the Mittag-Lefler (ML) function. The most interesting but challenging issue resides in how to efficiently deal with the singularity in time induced by the ML function which is an infinite series of singular power functions with different nature. With this in mind, we introduce a spectral-Galerkin method using Fourier-like basis functions for spatial discretization, leading to a sequence of decoupled temporal integral-differential equations (IDE) with the same weakly singular kernel involving the ML function as the original two-dimensional problem. With a careful study of the regularity of IDE, we incorporate several leading singular terms into the numerical scheme and approximate much regular part of the solution. Then we solve to IDE by a multi-step well-conditioned collocation scheme together with mapping technique to increase the accuracy and enhance the resolution. We show such an enriched collocation method is convergent and accurate. % analysis of the scheme is carried out.

Xiaodan Zhao, Li-Lian Wang

We report in this paper a fast and accurate algorithm for computing the exact spherical nonreflecting boundary condition (NRBC) for time-dependent Maxwell's equations. It is essentially based on a new formulation of the NRBC, which allows for the use of an analytic method for computing the involved inverse Laplace transform. This tool can be generically integrated with the interior solvers for challenging simulations of electromagnetic scattering problems. We provide some numerical examples to show that the algorithm leads to very accurate results.

Zilan Cheng, Li-Lian Wang, Zhongjian Wang

We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime based on one-step generative transport. Building on the Mean Flows, we learn a fully conditional amortized sampler with a neural-operator backbone that maps a reference Gaussian noise to approximate posterior samples. We show that while white-noise references may be admissible at fixed discretization, they become incompatible with the function-space limit, leading to instability in inference for Bayesian problems arising from PDEs. To address this issue, we adopt a prior-aligned anisotropic Gaussian reference distribution and establish the Lipschitz regularity of the resulting transport. Our method is not distilled from MCMC: training relies only on prior samples and simulated partial and noisy observations. Once trained, it generates a $64\times64$ posterior sample in $\sim 10^{-3}$s, avoiding the repeated PDE solves of MCMC while matching key posterior summaries.

Juan Diego Toscano, Li-Lian Wang, George Em Karniadakis

Inspired by the Kolmogorov-Arnold representation theorem and Kurkova's principle of using approximate representations, we propose the Kurkova-Kolmogorov-Arnold Network (KKAN), a new two-block architecture that combines robust multi-layer perceptron (MLP) based inner functions with flexible linear combinations of basis functions as outer functions. We first prove that KKAN is a universal approximator, and then we demonstrate its versatility across scientific machine-learning applications, including function regression, physics-informed machine learning (PIML), and operator-learning frameworks. The benchmark results show that KKANs outperform MLPs and the original Kolmogorov-Arnold Networks (KANs) in function approximation and operator learning tasks and achieve performance comparable to fully optimized MLPs for PIML. To better understand the behavior of the new representation models, we analyze their geometric complexity and learning dynamics using information bottleneck theory, identifying three universal learning stages, fitting, transition, and diffusion, across all types of architectures. We find a strong correlation between geometric complexity and signal-to-noise ratio (SNR), with optimal generalization achieved during the diffusion stage. Additionally, we propose self-scaled residual-based attention weights to maintain high SNR dynamically, ensuring uniform convergence and prolonged learning.

Zilan Cheng, Zhongjian Wang, Li-Lian Wang et al.

In this paper, we introduce Proper Orthogonal Decomposition Neural Operators (PODNO) for solving partial differential equations (PDEs) dominated by high-frequency components. Building on the structure of Fourier Neural Operators (FNO), PODNO replaces the Fourier transform with (inverse) orthonormal transforms derived from the Proper Orthogonal Decomposition (POD) method to construct the integral kernel. Due to the optimality of POD basis, the PODNO has potential to outperform FNO in both accuracy and computational efficiency for high-frequency problems. From analysis point of view, we established the universality of a generalization of PODNO, termed as Generalized Spectral Operator (GSO). In addition, we evaluate PODNO's performance numerically on dispersive equations such as the Nonlinear Schrodinger (NLS) equation and the Kadomtsev-Petviashvili (KP) equation.

Tao Tang, Li-Lian Wang, Huifang Yuan et al.

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach.

Yujian Jiao, Li-Lian Wang, Can Huang

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order $μ\in (0,1)$ to compute that of any order $k+μ$ with integer $k\ge 0,$ while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order $μ\in (0,1).$ Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at "interior" JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, are essential for our algorithm development.