arXiv subjects

Jianliang Qian

Publications and source records attributed to Jianliang Qian.

At least 19 recordsLinked to original sources

arXiv2026

Fast Operator-Splitting Methods for Nonlinear Elliptic Equations

Nonlinear elliptic problems arise in many fields, including plasma physics, astrophysics, and optimal transport. In this article, we propose a novel operator-splitting/finite element method for solving such problems. We begin by introducing an auxiliary function in a new way for a semilinear elliptic partial differential equation, leading to the development of a convergent operator-splitting/finite element scheme for this equation. The algorithm is then extended to fully nonlinear elliptic equations of the Monge-Ampère type, including the Dirichlet Monge-Ampère equation and Pucci's equation. This is achieved by reformulating the fully nonlinear equations into forms analogous to the semilinear case, enabling the application of the proposed splitting algorithm. In our implementation, a mixed finite element method is used to approximate both the solution and its Hessian matrix. Numerical experiments show that the proposed method outperforms existing approaches in efficiency and accuracy, and can be readily applied to problems defined on domains with curved boundaries.

math.NA↗

arXiv2026

High-order WENO-based semi-implicit Newton-type fast sweeping methods for static Hamilton-Jacobi equations

In this paper, we propose high-order weighted essentially non-oscillatory (WENO)-based semi-implicit Newton-type Gauss-Seidel Lax-Friedrichs fast sweeping methods for solving the generalized Eikonal equation arising in wave propagation through a moving fluid. Building upon the Newton-type framework of Li and Qian (2020), which updates the solution line-wise using Newton's method with a tridiagonal and strictly diagonally dominant Jacobian, we extend the local solver to fifth-, seventh-, and ninth-order accuracy by incorporating high-order WENO approximations of the spatial derivatives into the numerical Hamiltonian. Three alternating sweeping strategies, namely column-wise, row-wise, and column-row-wise, are considered within the Gauss-Seidel iteration framework. Numerical examples in both two and three spatial dimensions demonstrate the efficiency and accuracy of the proposed schemes.

math.NA↗

arXiv2026

Level-set physics-informed neural networks for domain inverse problems of gravimetry

We propose level-set physics-informed neural networks (PINNs) for domain inverse problems of gravimetry. The domain inverse problem establishes a correctness class for ill-posed inverse gravimetry, which we solve within the PINNs framework. Directly representing the domain inverse problem via neural networks is problematic due to the discontinuous nature of interfaces. We consider a level-set formulation where the neural network represents a continuous level-set function, and its zero level-set depicts sharp interfaces. To overcome the challenges of exploding and vanishing gradients caused by sharp interfaces during training, we propose an interface-aware backpropagation strategy. By redefining the derivative associated with interface evolution, this strategy enables a broader support region to drive the evolution process. Detailed analysis is provided to justify the efficacy of this strategy. Additionally, we introduce a simple procedure for adaptive refinement of collocation points near interfaces. The selection of network architecture is investigated by studying the solution spaces and the approximation properties of neural networks. Finally, extensive 2D and 3D numerical examples demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2025

Inverse problems for infinite-dimensional transport PDEs on Wasserstein space

We develop a foundational framework for inverse problems governed by evolutionary partial differential equations (PDEs) on the Wasserstein space of probability measures. While the forward problems for such transport-type PDEs have been extensively and intensively studied, their corresponding inverse problems--which aim to reconstruct unknown operators, cost functions, or interaction kernels from observed solution data--remain largely unexplored at this level of generality. The cornerstone of our theory is a systematic approach featuring high-order calculus on the Wasserstein space and a progressive variational scheme. This methodology is specifically designed to address the challenges inherent in inverse problems for infinite-dimensional, nonlinear, and nonlocal transport PDEs. We demonstrate the power and versatility of our theory through two canonical examples: inverse problems for both the Mean Field Control (MFC) Dynamic Programming Equation and the Mean Field Game (MFG) Master Equation. Our work provides, for the first time, a unified foundation for identifying cost functions and interaction kernels from value function data. This establishes a new and fertile field of mathematical research with significant implications for both theory and applications in stochastic control and mean field games.

math.OC↗

arXiv2025

A Linear-complexity Tensor Butterfly Algorithm for Compressing High-dimensional Oscillatory Integral Operators

This paper presents a multilevel tensor compression algorithm called tensor butterfly algorithm for efficiently representing large-scale and high-dimensional oscillatory integral operators, including Green's functions for wave equations and integral transforms such as Radon transforms and Fourier transforms. The proposed algorithm leverages a tensor extension of the so-called complementary low-rank property of existing matrix butterfly algorithms. The algorithm partitions the discretized integral operator tensor into subtensors of multiple levels, and factorizes each subtensor at the middle level as a Tucker-type interpolative decomposition, whose factor matrices are formed in a multilevel fashion. For a $d$-dimensional integral operator discretized into a $2d$-mode tensor with $n^{2d}$ entries, the overall CPU time and memory requirement scale as $O(n^d)$, in stark contrast to the $O(n^d\log n)$ requirement of existing matrix algorithms such as matrix butterfly algorithm and fast Fourier transforms (FFT), where $n$ is the number of points per direction. When comparing with other tensor algorithms such as quantized tensor train (QTT), the proposed algorithm also shows superior CPU and memory performance for tensor contraction. Remarkably, the tensor butterfly algorithm can efficiently model high-frequency Green's function interactions between two unit cubes, each spanning 512 wavelengths per direction, which represents over $512\times$ larger problem sizes than existing butterfly algorithms. On the other hand, for a problem representing 64 wavelengths per direction, which is the largest size existing algorithms can handle, our tensor butterfly algorithm exhibits 200x speedups and $30\times$ memory reduction comparing with existing ones. Moreover, the tensor butterfly algorithm also permits $O(n^d)$-complexity FFTs and Radon transforms up to $d=6$ dimensions.

math.NA↗

arXiv2024

Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms

Starting from the Kirchhoff-Huygens representation and Duhamel's principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagator. Second, using the Fourier transform in time, we derive the corresponding Eulerian short-time propagator in frequency domain; on top of this propagator, we further develop a time-frequency-time (TFT) method for the Cauchy problem of time-domain wave equations. Third, we further propose the time-frequency-time-frequency (TFTF) method for the corresponding point-source Helmholtz equation, which provides Green's functions of the Helmholtz equation for all angular frequencies within a given frequency band. Fourth, to implement TFT and TFTF methods efficiently, we introduce butterfly algorithms to compress oscillatory integral kernels at different frequencies. As a result, the proposed methods can construct wave field beyond caustics implicitly and advance spatially overturning waves in time naturally with quasi-optimal computational complexity and memory usage. Furthermore, once constructed the Hadamard integrators can be employed to solve both time-domain wave equations with various initial conditions and frequency-domain wave equations with different point sources. Numerical examples for two-dimensional wave equations illustrate the accuracy and efficiency of the proposed methods.

math.NA↗

arXiv2024

Optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations

We study the optimal rate of convergence in periodic homogenization of the viscous Hamilton-Jacobi equation $u^\varepsilon_t + H(\frac{x}{\varepsilon},Du^\varepsilon) = \varepsilon Δu^\varepsilon$ in $\mathbb R^n\times (0,\infty)$ subject to a given initial datum. We prove that $\|u^\varepsilon-u\|_{L^\infty(\mathbb R^n \times [0,T])} \leq C(1+T) \sqrt{\varepsilon}$ for any given $T>0$, where $u$ is the viscosity solution of the effective problem. Moreover, we show that the $O(\sqrt{\varepsilon})$ rate is optimal for a natural class of $H$ and a Lipschitz continuous initial datum, both theoretically and through numerical experiments. It remains an interesting question to investigate whether the convergence rate can be improved when $H$ is uniformly convex. Finally, we propose a numerical scheme for the approximation of the effective Hamiltonian based on a finite element approximation of approximate corrector problems.

math.AP↗

arXiv2023

Hadamard integrator for time-dependent wave equations: Lagrangian formulation via ray tracing

We propose a novel Hadamard integrator for the self-adjoint time-dependent wave equation in an inhomogeneous medium. First, we create a new asymptotic series based on the Gelfand-Shilov function, dubbed Hadamard's ansatz, to approximate the Green's function of the time-dependent wave equation. Second, incorporating the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation, we develop an original Hadamard integrator for the Cauchy problem of the time-dependent wave equation and derive the corresponding Lagrangian formulation in geodesic polar coordinates. Third, to construct the Hadamard integrator in the Lagrangian formulation efficiently, we use a short-time ray tracing method to obtain wavefront locations accurately, and we further develop fast algorithms to compute Chebyshev-polynomial based low-rank representations of both wavefront locations and variants of Hadamard coefficients. Fourth, equipped with these low-rank representations, we apply the Hadamard integrator to efficiently solve time-dependent wave equations with highly oscillatory initial conditions, where the time step size is independent of the initial conditions. By judiciously choosing the medium-dependent time step, our new Hadamard integrator can propagate wave field beyond caustics implicitly and advance spatially overturning waves in time naturally. Moreover, since the integrator is independent of initial conditions, the Hadamard integrator can be applied to many different initial conditions once it is constructed. Both two-dimensional and three-dimensional numerical examples illustrate the accuracy and performance of the proposed method.

math.NA↗

arXiv2023

Operator Splitting/Finite Element Methods for the Minkowski Problem

The classical Minkowski problem for convex bodies has deeply influenced the development of differential geometry. During the past several decades, abundant mathematical theories have been developed for studying the solutions of the Minkowski problem, however, the numerical solution of this problem has been largely left behind, with only few methods available to achieve that goal. In this article, focusing on the two-dimensional Minkowski problem with Dirichlet boundary conditions, we introduce two solution methods, both based on operator-splitting. One of these two methods deals directly with the Dirichlet condition, while the other method uses an approximation of this Dirichlet condition. This relaxation of the Dirichlet condition makes this second method better suited than the first one to treat those situations where the Minkowski and the Dirichlet condition are not compatible. Both methods are generalizations of the solution method for the canonical Monge-Ampère equation discussed by Glowinski et al. (Journal of Scientific Computing, 79(1), 1-47, 2019); as such they take advantage of a divergence formulation of the Minkowski problem, well-suited to a mixed finite element approximation, and to the the time-discretization via an operator-splitting scheme, of an associated initial value problem. Our methodology can be easily implemented on convex domains of rather general shape (with curved boundaries, possibly). The numerical experiments we performed validate both methods and show that if one uses continuous piecewise affine finite element approximations of the smooth solution of the Minkowski problem and of its three second order derivatives, these two methods provide nearly second order accuracy for the $L^2$ and $L^{\infty}$ error. One can extend easily the methods discussed in this article, to address the solution of three-dimensional Minkowski problem.

math.NA↗

arXiv2022

A Fast Butterfly-compressed Hadamard-Babich Integrator for High-Frequency Helmholtz Equations in Inhomogeneous Media with Arbitrary Sources

We present a butterfly-compressed representation of the Hadamard-Babich (HB) ansatz for the Green's function of the high-frequency Helmholtz equation in smooth inhomogeneous media. For a computational domain discretized with $N_v$ discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all $N_v$ cell centers to each other. The overall CPU time and memory requirement scale as $O(N_v\log^2N_v)$ for any bounded 2D domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields an $O(N_v^{4/3})$ CPU complexity, which can be further reduced to quasi-linear complexities with proposed remedies. The scheme can also efficiently handle scattering problems involving inclusions in inhomogeneous media. Although the current construction of our HB integrator does not accommodate caustics, the resulting HB integrator itself can be applied to certain sources, such as concave-shaped sources, to produce caustic effects. Compared to finite-difference frequency-domain (FDFD) methods, the proposed HB integrator is free of numerical dispersion and requires fewer discretization points per wavelength. As a result, it can solve wave-propagation problems well beyond the capability of existing solvers. Remarkably, the proposed scheme can accurately model wave propagation in 2D domains with 640 wavelengths per direction and in 3D domains with 54 wavelengths per direction on a state-the-art supercomputer at Lawrence Berkeley National Laboratory.

physics.comp-ph↗

arXiv2022

An Efficient Operator-Splitting Method for the Eigenvalue Problem of the Monge-Ampère Equation

We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Ampère operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by Abedin and Kitagawa (Inverse iteration for the {M}onge-{A}mp{è}re eigenvalue problem, {\it Proceedings of the American Mathematical Society}, 148 (2020), no. 11, 4975-4886). Modifying the theoretical formulation, we develop an efficient algorithm for computing the eigenvalue and eigenfunction of the Monge-Ampère operator by solving a constrained Monge-Ampère equation during each iteration. Our method consists of four essential steps: (i) Formulate the Monge-Ampère eigenvalue problem as an optimization problem with a constraint; (ii) Adopt an indicator function to treat the constraint; (iii) Introduce an auxiliary variable to decouple the original constrained optimization problem into simpler optimization subproblems and associate the resulting new optimization problem with an initial value problem; and (iv) Discretize the resulting initial-value problem by an operator-splitting method in time and a mixed finite element method in space. The performance of our method is demonstrated by several experiments. Compared to existing methods, the new method is more efficient in terms of computational cost and has a comparable rate of convergence in terms of accuracy.

math.NA↗

arXiv2021

Learning Rays via Deep Neural Network in a Ray-based IPDG Method for High-Frequency Helmholtz Equations in Inhomogeneous Media

We develop a deep learning approach to extract ray directions at discrete locations by analyzing highly oscillatory wave fields. A deep neural network is trained on a set of local plane-wave fields to predict ray directions at discrete locations. The resulting deep neural network is then applied to a reduced-frequency Helmholtz solution to extract the directions, which are further incorporated into a ray-based interior-penalty discontinuous Galerkin (IPDG) method to solve the Helmholtz equations at higher frequencies. In this way, we observe no apparent pollution effects in the resulting Helmholtz solutions in inhomogeneous media. Our 2D and 3D numerical results show that the proposed scheme is very efficient and yields highly accurate solutions.

math.NA↗

arXiv2020

On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Ampère Operator

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the following ingredients: (i) A divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step $h\rightarrow 0$. We considered also test problems with no known exact solutions.

math.NA↗

arXiv2017

A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

We propose a hybrid approach to solve the high-frequency Helmholtz equation with point source terms in smooth heterogeneous media. The method is based on the ray-based finite element method (ray-FEM), whose original version can not handle the singularity close to point sources accurately. This pitfall is addressed by combining the ray-FEM, which is used to compute the smooth far-field of the solution accurately, with a high-order asymptotic expansion close to the point source, which is used to properly capture the singularity of the solution in the near-field. The method requires a fixed number of grid points per wavelength to accurately represent the wave field with an asymptotic convergence rate of $\mathcal{O}(ω^{-1/2})$, where $ω$ is the frequency parameter in the Helmholtz equation. In addition, a fast sweeping-type preconditioner is used to solve the resulting linear system. We present numerical examples in 2D to show both accuracy and efficiency of our method as the frequency increases. In particular, we provide numerical evidence of the convergence rate, and we show empirically that the overall complexity is $\mathcal{O}(ω^2)$ up to a poly-logarithmic factor.

math.NA↗

arXiv2017

A ray-based IPDG method for high-frequency time-domain acoustic wave propagation in inhomogeneous media

The numerical approximation of high-frequency wave propagation in inhomogeneous media is a challenging problem. In particular, computing high-frequency solutions by direct simulations requires several points per wavelength for stability and usually requires many points per wavelength for a satisfactory accuracy. In this paper, we propose a new method for the acoustic wave equation in inhomogeneous media in the time domain to achieve superior accuracy and stability without using a large number of unknowns. The method is based on a discontinuous Galerkin discretization together with carefully chosen basis functions. To obtain the basis functions, we use the idea from geometrical optics and construct the basis functions by using the leading order term in the asymptotic expansion. Also, we use a wavefront tracking method and a dimension reduction procedure to obtain dominant rays in each cell. We show numerically that the accuracy of the numerical solutions computed by our method is significantly higher than that computed by the IPDG method using polynomials.Moreover, the relative errors of our method grow only moderately as the frequency increases.

math.NA↗

arXiv2017

Min-max formulas and other properties of certain classes of nonconvex effective Hamiltonians

This paper is the first attempt to systematically study properties of the effective Hamiltonian $\overline{H}$ arising in the periodic homogenization of some coercive but nonconvex Hamilton-Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min-max formulas for a class of nonconvex $\overline{H}$. Secondly, we analytically and numerically investigate other related interesting phenomena, such as "quasi-convexification" and breakdown of symmetry, of $\overline{H}$ from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for same class of nonconvex Hamilton-Jacobi equations. Some conjectures and problems are also proposed.

math.AP↗

arXiv2016

Perfectly-matched-layer boundary integral equation method for wave scattering in a layered medium

For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive, since they are formulated on lower-dimension boundaries or interfaces, and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on the Green's function of the background medium must evaluate the expensive Sommefeld integrals. Alternative BIE methods based on the free-space Green's function give rise to integral equations on unbounded interfaces which are not easy to truncate, since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. The PMLs are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space. Our PML-based BIE method uses the Green's function of the PML-transformed free space to define the boundary integral operators. The method is efficient, since the Green's function of the PML-transformed free space is easy to evaluate and the PMLs are very effective in truncating the unbounded interfaces. Numerical examples are presented to validate our method and demonstrate its accuracy.

math.NA↗

arXiv2016

Learning Dominant Wave Directions For Plane Wave Methods For High-Frequency Helmholtz Equations

We present a ray-based finite element method (ray-FEM) by learning basis adaptive to the underlying high-frequency Helmholtz equation in smooth media. Based on the geometric optics ansatz of the wave field, we learn local dominant ray directions by probing the medium using low-frequency waves with the same source. Once local ray directions are extracted, they are incorporated into the finite element basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve approximations for both local ray directions and the high frequency wave field iteratively. The method requires a fixed number of grid points per wavelength to represent the wave field and achieves an asymptotic convergence as the frequency $ω\rightarrow \infty$ without the pollution effect. A fast solver is developed for the resulting linear system with an empirical complexity $\mathcal{O}(ω^d)$ up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.

math.NA↗