arXiv subjects

Guanglian Li

Publications and source records attributed to Guanglian Li.

At least 19 recordsLinked to original sources

arXiv2026

Edge Multiscale Finite Element Methods

The objective of this paper is to review recent developments in Edge Multiscale Finite Element Methods (EMsFEM) for partial differential equations with heterogeneous coefficients or highly oscillatory solutions. Using elliptic equations with heterogeneous coefficients as an illustrative example, we present the key ideas of the method. We also analyze the approach while accounting for the discrete error in the multiscale basis functions. Extensive numerical tests are provided to validate the performance of the method.

math.NA↗

arXiv2026

Quasi-Monte Carlo finite element approximation for singularly perturbed convection-diffusion problems with random velocity

This paper studies the numerical approximation of a singularly perturbed convection-diffusion problem over a bounded polygonal domain in $\mathbb{R}^d$ ($d=2,3$), where the velocity field is modeled by a log-uniform random field, a setting typical in uncertainty quantification. We introduce a novel numerical framework for computing the expected value of the linear functionals of the solution. The approach combines a finite element discretization of the problem, a truncated Karhunen--Loève expansion to represent the stochastic velocity field, and a lattice-based quasi-Monte Carlo (QMC) method to estimate expectations over the parameter space. We provide a rigorous error analysis of the proposed scheme, establishing bounds on the mean squared error and demonstrating that the QMC method achieves a nearly linear optimal convergence rate, with a constant independent of the integration dimension. Furthermore, the convergence rate is shown to be independent of the singular perturbation parameter.

math.NA↗

arXiv2026

Linear Convergence of Parareal Algorithm for Semilinear Parabolic Equations

Long-time simulations of evolution equations present substantial computational challenges due to the inherently sequential nature of conventional time-stepping schemes. The parareal method, a leading parallel-in-time (PinT) algorithm, offers a promising approach to overcome the challenge by introducing concurrency in the time domain. While its convergence theory is well-established for linear problems, extending the theory to nonlinear problems, particularly when the problem data have only limited regularity, remains a significant challenge. In this work, we provide the convergence analysis of the parareal algorithm for solving semilinear parabolic equations with an $H^2$ initial data. We employ stable rational approximations and first-order linearization as coarse propagators, establish the linear convergence of the parareal algorithm and provide a sharp estimate for the convergence factor. The analysis combines the error-splitting technique from the superlinear convergence analysis of the parareal method, a refined linear convergence theory for linear parabolic equations, and \textsl{a priori} error estimates that are optimal with respect to the regularity of the problem data. The analysis shows the close connection between the convergence behavior of nonlinear models and their linear counterparts. Numerical experiments fully support the theoretical findings.

math.NA↗

arXiv2026

Dual Variational Neural Network for the $p$-Laplace Problem

The reliable and accurate numerical approximation of the $p$-Laplacian is particularly challenging in the extreme regimes $p \to 1^{+}$ and $p \gg 1$, where the operator becomes either highly singular or strongly degenerate, often causing severe instability in standard numerical methods. To address these difficulties, we propose a novel deep learning based framework, termed the dual variational neural network, for $p$-Laplace problems. The approach is based on a mixed formulation and an $L^q$-based Helmholtz decomposition, which decouples the original problem into two convex subproblems: a linear Poisson problem for the irrotational component and an unconstrained minimization problem over divergence-free fields for the solenoidal component. Following the decomposition, we employ two neural networks using a gradient--curl representation to approximate the flux, and further establish an error analysis of the neural approximation. The analysis relies on fundamental vector inequalities together with tools from statistical learning theory. Numerical experiments demonstrate robust convergence of the proposed method in challenging settings, including the extreme cases $p \to 1^{+}$ and $p \gg 1$, as well as the $p(x)$-Laplace equation.

math.NA↗

arXiv2026

Optimized Two-Step Coarse Propagators in Parareal Algorithms

In this work, we propose a novel framework for accelerating the parareal algorithm, in which the coarse propagator is formulated as a two-step method and optimized with respect to the convergence factor.} We derive a rigorous error estimate for the proposed two-step parareal algorithm, yielding an explicit bound on the linear convergence factor. This estimate is not only of theoretical interest: it provides a quantitative guideline for selecting and designing coarse propagators. Guided by this estimate, we {consider the linear parabolic equation as an illustrative example and }construct an optimized two-step coarse propagator~(O2CP) that delivers very fast convergence in practice. The resulting method attains an optimized convergence factor of approximately $0.0064$, substantially smaller than that of commonly used practical coarse propagators in the classical parareal setting, while keeping the computational cost moderate. Numerical experiments on linear and nonlinear parabolic equations fully support the theoretical analysis and demonstrate rapid convergence of the two-step parareal algorithm equipped with the O2CP.

math.NA↗

arXiv2026

Numerical homogenization for indefinite time-harmonic Maxwell equations

We propose a novel numerical homogenization method based on the edge multiscale approach for solving indefinite time-harmonic Maxwell equations in heterogeneous media with large wavenumber. Numerical methods for these equations in homogeneous media with high wavenumber are particularly challenging due to the so-called pollution effect: the mesh size must be significantly smaller than the reciprocal of the wavenumber to achieve a desired accuracy. This challenge is amplified in heterogeneous media, which frequently occur in practical applications such as metamaterial simulations, since resolving the heterogeneity is necessary for obtaining reliable solutions. Our approach overcomes this difficulty by avoiding explicit resolution of the heterogeneity, while employing a mesh size that depends almost linearly on the reciprocal of the wavenumber. The approximation properties and stability of the method rely critically on the development and rigorous analysis of a novel, nonstandard variational formulation, which constitutes the main innovation of this work. Extensive numerical experiments are provided to validate our theoretical findings.

math.NA↗

arXiv2026

Efficient Simulation and Calibration of the Rough Bergomi Model via Wasserstein Distance

Despite the empirical success of the rough Bergomi (rBergomi) model in modeling volatility dynamics, its practical use remains challenging due to high computational complexity in both pricing and calibration arising from its non-Markovian structure. To address these difficulties, we develop an efficient computational framework. First, we propose a modified-sum-of-exponentials (mSOE) Monte Carlo scheme within the class of hybrid multifactor approximations. The method combines an exact treatment of the singular kernel over the first time step with a sum-of-exponentials approximation over the remaining time interval, and exact Gaussian simulation of the resulting multifactor components. For a fixed number of exponential terms, the method maintains linear online complexity with respect to the number of time steps. It achieves high pricing accuracy in numerical experiments, particularly for out-of-the-money options. Second, building on this pricing engine, we formulate a calibration approach based on distributional matching of the terminal underlying asset via the Wasserstein-1 distance. Instead of fitting option prices only at selected strikes, this method compares model-generated and market-implied terminal distributions through the Kantorovich-Rubinstein dual representation. Numerical experiments indicate that the mSOE scheme exhibits stable convergence, and the Wasserstein-based calibration scheme improves parameter recovery, optimization stability, and out-of-sample performance relative to conventional MSE-based fitting in the rBergomi setting considered in this paper.

q-fin.CP↗

arXiv2026

Unsupervised Learning-based Calibration Scheme for Rough Volatility Models

Existing deep learning-based calibration scheme for rough volatility models predominantly rely on supervised learning frameworks, which incur significant computational costs due to the necessity of generating massive synthetic training datasets. In this work, we propose a novel unsupervised learning-based calibration scheme for rough volatility models that eliminates the data generation bottleneck. Our approach leverages the backward stochastic differential equation (BSDE) representation of the pricing function derived by Bayer et al. \cite{bayer2022pricing}. By treating model parameters as trainable variables, we simultaneously approximate the BSDE solution and optimize the parameters within a unified neural network training process, with the terminal misfit as the loss. We theoretically establish that the mean squared error between the model-implied prices and market data is bounded by the loss function. Furthermore, we prove that the loss can be minimized to an arbitary degree, depending on the model's market fitting capacity and the universal approximation capability of neural networks. Numerical experiments for both simulated and historical S\&P 500 data based on rough Bergomi (rBergomi) model demonstrate the efficiency and accuracy of the proposed scheme.

q-fin.CP↗

arXiv2026

A POD-DeepONet Framework for Forward and Inverse Design of 2D Photonic Crystals

We develop a reduced-order operator-learning framework for forward and inverse band-structure design of two-dimensional photonic crystals with binary, pixel-based $p4m$-symmetric unit cells. We construct a POD--DeepONet surrogate for the discrete band map along the standard high-symmetry path by coupling a POD trunk extracted from high-fidelity finite-element band snapshots with a neural branch network that predicts reduced coefficients. This architecture yields a compact and differentiable forward model that is tailored to the underlying Bloch eigenvalue discretization. We establish continuity of the discrete band map on the relaxed design space and prove a uniform approximation property of the POD--DeepONet surrogate, leading to a natural decomposition of the total surrogate error into POD truncation and network approximation contributions. Building on this forward surrogate, we formulate two end-to-end neural inverse design procedures, namely dispersion-to-structure and band-gap inverse design, with training objectives that combine data misfit, binarity promotion, and supervised regularization to address the intrinsic non-uniqueness of the inverse mapping and to enable stable gradient-based optimization in the relaxed space. Our numerical results show that the proposed framework achieves accurate forward predictions and produces effective inverse designs on practical high-contrast, pixel-based photonic layouts.

physics.optics↗

arXiv2025

Edge multiscale finite element methods for semilinear parabolic problems with heterogeneous coefficients

We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial discretization, which fails to resolve the spatial heterogeneity but maintains satisfactory accuracy independent of the heterogeneity. This is achieved by simultaneously constructing a steady-state multiscale ansatz space with certain approximation properties for the evolving solution and the initial data. The approximation properties of the multiscale ansatz space are derived using local-global splitting. A fully discrete scheme is analyzed using a first-order explicit exponential Euler scheme. We derive the error estimates in the $L^{2}$-norm and energy norm under the regularity assumptions for the semilinear term. The convergence rates depend on the coarse grid size and the level parameter. Finally, extensive numerical experiments are carried out to validate the efficiency of the proposed method.

math.NA↗

arXiv2025

Gradient-enhanced sparse Hermite polynomial expansions for pricing and hedging high-dimensional American options

We propose an efficient and easy-to-implement gradient-enhanced least squares Monte Carlo method for computing price and Greeks (i.e., derivatives of the price function) of high-dimensional American options. It employs the sparse Hermite polynomial expansion as a surrogate model for the continuation value function, and essentially exploits the fast evaluation of gradients. The expansion coefficients are computed by solving a linear least squares problem that is enhanced by gradient information of simulated paths. We analyze the convergence of the proposed method, and establish an error estimate in terms of the best approximation error in the weighted $H^1$ space, the statistical error of solving discrete least squares problems, and the time step size. We present comprehensive numerical experiments to illustrate the performance of the proposed method. The results show that it outperforms the state-of-the-art least squares Monte Carlo method with more accurate price, Greeks, and optimal exercise strategies in high dimensions but with nearly identical computational cost, and it can deliver comparable results with recent neural network-based methods up to dimension 100.

q-fin.CP↗

arXiv2025

Adaptive Crouzeix-Raviart finite elements for the first eigenpair of $p$-Laplacian

In this paper, we propose and analyze an adaptive Crouzeix-Raviart finite element method for computing the first Dirichlet eigenpair of the $p$-Laplacian problem. We prove that the sequence of error estimators produced by the adaptive algorithm has a vanishing limit and that, starting from a fine initial mesh, the relevant sequence of approximate eigenvalues converges to the first eigenvalue and the distance in a mesh-dependent broken norm between discrete eigenfunctions and the set composed of relevant continuous eigenfunctions also tends to zero. The analysis hinges on establishing a compactness property for Crouzeix-Raviart finite elements over a sequence of adaptively generated meshes, which represents key theoretical challenges and novelties. We present numerical results to illustrate the advantage of the proposed algorithm.

math.NA↗

arXiv2025

On the convergence analysis of MsFEM with oversampling: Interpolation error

In this paper, we investigate the approximation properties of two types of multiscale finite element methods with oversampling as proposed in [Hou \& Wu, {\textit{J. Comput. Phys.}}, 1997] and [Efendiev, Hou \& Wu, \textit{SIAM J. Numer. Anal.}, 2000] without scale separation. We develop a general interpolation error analysis for elliptic problems with highly oscillatory rough coefficients, under the assumption of the existence of a macroscopic problem with suitable $L^2$-accuracy. The distinct features of the analysis, in the setting of highly oscillatory periodic coefficients, include: (i) The analysis is independent of the first-order corrector or the solutions to the cell problems, and thus independent of their regularity properties; (ii) The analysis only involves the homogenized solution and its minimal regularity. We derive an interpolation error $\mathcal{O}\left(H+\fracε{H}\right)$ with $ε$ and $H$ being the period size and the coarse mesh size, respectively, when the oversampling domain includes one layer of elements from the target coarse element.

math.NA↗

arXiv2025

Adaptive Nonoverlapping Preconditioners for the Helmholtz Equation

The Helmholtz equation poses significant computational challenges due to its oscillatory solutions, particularly for large wavenumbers. Inspired by the Schur complement system for elliptic problems, this paper presents a novel substructuring approach to mitigate the potential ill-posedness of local Dirichlet problems for the Helmholtz equation. We propose two types of preconditioners within the framework of nonoverlapping spectral additive Schwarz (NOSAS) methods. The first type of preconditioner focuses on the real part of the Helmholtz problem, while the second type addresses both the real and imaginary components, providing a comprehensive strategy to enhance scalability and reduce computational cost. Our approach is purely algebraic, which allows for adaptability to various discretizations and heterogeneous Helmholtz coefficients while maintaining theoretical convergence for thresholds close to zero. Numerical experiments confirm the effectiveness of the proposed preconditioners, demonstrating robust convergence rates and scalability, even for large wavenumbers.

math.NA↗

arXiv2025

Quasi-Monte Carlo finite element approximation of the Navier-Stokes equations with initial data modeled by log-normal random fields

In this paper, we analyze the numerical approximation of the Navier-Stokes problem over a bounded polygonal domain in $\mathbb{R}^2$, where the initial condition is modeled by a log-normal random field. This problem usually arises in the area of uncertainty quantification. We aim to compute the expectation value of linear functionals of the solution to the Navier-Stokes equations and perform a rigorous error analysis for the problem. In particular, our method includes the finite element, fully-discrete discretizations, truncated Karhunen-Loéve expansion for the realizations of the initial condition, and lattice-based quasi-Monte Carlo (QMC) method to estimate the expected values over the parameter space. Our QMC analysis is based on randomly-shifted lattice rules for the integration over the domain in high-dimensional space, which guarantees the error decays with $\mathcal{O}(N^{-1+δ})$, where $N$ is the number of sampling points, $δ>0$ is an arbitrary small number, and the constant in the decay estimate is independent of the dimension of integration.

math.NA↗

arXiv2024

On the Numerical Approximation of the Karhunen-Loève Expansion for Random Fields with Random Discrete Data

In many applications, random fields reflect uncertain parameters, and often their moments are part of the modeling process and thus well known. However, there are practical situations where this is simply not the case. Therefore, we do not assume that we know moments or expansion terms of the random fields, but only have discretized samples of them. The main contribution of this paper concerns the approximation of the true covariance operator from these finite measurements. We derive explicit error estimates that include the finite-rank approximation error of the covariance operator, the Monte Carlo-type error for sampling in the stochastic domain, and the numerical discretization error in the physical domain. For this purpose, we use modern tapering covariance estimators adapted to high-dimensional applications, where the dimension is introduced by the resolution of the measurement process. This allows us to give sufficient conditions on the three discretization parameters to guarantee that the error is kept below a prescribed accuracy $\varepsilon$.

math.NA↗

arXiv2024

Adaptive finite element approximations of the first eigenpair associated with $p$-Laplacian

In this paper, we propose an adaptive finite element method for computing the first eigenpair of the $p$-Laplacian problem. We prove that starting from a fine initial mesh our proposed adaptive algorithm produces a sequence of discrete first eigenvalues that converges to the first eigenvalue of the continuous problem and the distance between discrete eigenfunctions and the normalized eigenfunction set corresponding to the first eigenvalue in $W^{1,p}$-norm also tends to zero. Extensive numerical examples are provided to show the effectiveness and efficiency.

math.NA↗

arXiv2024

Wavelet-based Edge Multiscale Finite Element Methods for Singularly Perturbed Convection-Diffusion Equations

We propose a novel efficient and robust Wavelet-based Edge Multiscale Finite Element Method (WEMsFEM) motivated by \cite{MR3980476,GL18} to solve the singularly perturbed convection-diffusion equations. The main idea is to first establish a local splitting of the solution over a local region by a local bubble part and local Harmonic extension part, and then derive a global splitting by means of Partition of Unity. This facilitates a representation of the solution as a summation of a global bubble part and a global Harmonic extension part, where the first part can be computed locally in parallel. To approximate the second part, we construct an edge multiscale ansatz space locally with hierarchical bases as the local boundary data that has a guaranteed approximation rate \noteLg{both inside and outside of the layers}. The key innovation of this proposed WEMsFEM lies in a provable convergence rate with little restriction on the mesh size. Its convergence rate with respect to the computational degree of freedom is rigorously analyzed, which is verified by extensive 2-d and 3-d numerical tests.

math.NA↗