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Kejia Pan

Publications and source records attributed to Kejia Pan.

At least 19 recordsLinked to original sources

A Neural Network Architecture Based on Attention Gate Mechanism for 3D Magnetotelluric Forward Modeling

Traditional three-dimensional magnetotelluric (MT) numerical forward modeling methods, such as the finite element method (FEM) and finite volume method (FVM), suffer from high computational costs and low efficiency due to limitations in mesh refinement and computational resources. We propose a novel neural network architecture named MTAGU-Net, which integrates an attention gating mechanism for 3D MT forward modeling. Specifically, a dual-path attention gating module is designed based on forward response data images and embedded in the skip connections between the encoder and decoder. This module enables the fusion of critical anomaly information from shallow feature maps during the decoding of deep feature maps, significantly enhancing the network's capability to extract features from anomalous regions. Furthermore, we introduce a synthetic model generation method utilizing 3D Gaussian random field (GRF), which accurately replicates the electrical structures of real-world geological scenarios with high fidelity. Numerical experiments demonstrate that MTAGU-Net outperforms conventional 3D U-Net in terms of convergence stability and prediction accuracy, with the structural similarity index (SSIM) of the forward response data consistently exceeding 0.98. Moreover, the network can accurately predict forward response data on previously unseen datasets models, demonstrating its strong generalization ability and validating the feasibility and effectiveness of this method in practical applications.

cs.LG

Analysis of harmonic average method for interface problems with discontinuous solutions and fluxes

Harmonic average method has been widely utilized to deal with heterogeneous coefficients in solving differential equations. One remarkable advantage of the harmonic averaging method is that no derivative of the coefficient is needed. Furthermore, the coefficient matrix of the finite difference equations is an M-matrix which guarantees the stability of the algorithm. It has been numerically observed but not theoretically proved that the method produces second order pointwise accuracy when the solution and flux are continuous even if the coefficient has finite discontinuities for which the method is inconsistent ($O(1)$ in the local truncation errors). It has been believed that there are some fortunate error cancellations. The harmonic average method does not converge when the solution or the flux has finite discontinuities. In this paper, not only we rigorously prove the second order convergence of the harmonic averaging method for one-dimensional interface problem when the coefficient has a finite discontinuities and the solution and the flux are continuous, but also proposed an {\em improved harmonic average method} that is also second order accurate (in the $L^{\infty}$ norm), which allows discontinuous solutions and fluxes along with the discontinuous coefficients. The key in the convergence proof is the construction of the Green's function. The proof shows how the error cancellations occur in a subtle way. Numerical experiments in both 1D and 2D confirmed the theoretical proof of the improved harmonic average method.

math.NA

Fourth-order compact finite difference schemes for solving biharmonic equations with Dirichlet boundary conditions

In this study, we propose a genuine fourth-order compact finite difference scheme for solving biharmonic equations with Dirichlet boundary conditions in both two and three dimensions. In the 2D case, we build upon the high-order compact (HOC) schemes for flux-type boundary conditions originally developed by Zhilin Li and Kejia Pan [SIAM J. Sci. Comput., 45 (2023), pp. A646-A674] to construct a high order compact discretization for coupled boundary conditions. When considering the 3D case, we modify carefully designed undetermined coefficient methods of Li and Pan to derive the finite difference approximations of coupled boundary conditions. The resultant FD discretization maintains the global fourth order convergence and compactness. Unlike the very popular Stephenson method, the number of unknows do not increase with dimensions. Besides, it is noteworthy that the condition number of the coefficient matrix increases at a rate of $O(h^{-2})$ in both 2D and 3D. We also validate the performance of the proposed genuine HOC methods through nontrivial examples.

math.NA

High order two-grid finite difference methods for interface and internal layer problems

Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal boundaries. In this paper, alternative approaches based on two different grids are developed for some interface and internal layer problems, which are different from adaptive mesh refinement (AMR) techniques. For one dimensional, or two-dimensional problems with straight interfaces or boundary layers that are parallel to one of the axes, the discussion is relatively easy. One of challenges is how to construct a fourth order compact finite difference scheme at boarder grid points that connect two meshes. A two-grid method that employs a second order discretization near the interface in the fine mesh and a fourth order discretization away from the interface in the coarse and boarder grid points is proposed. For two dimensional problems with a curved interface or an internal layer, a level set representation is utilized for which we can build a fine mesh within a tube $|φ({\bf x}) | \le δh$ of the interface. A new super-third seven-point discretization that can guarantee the discrete maximum principle has been developed at hanging nodes. The coefficient matrices of the finite difference equations developed in this paper are M-matrices, which leads to the convergence of the finite difference schemes. Non-trivial numerical examples presented in this paper have confirmed the desired accuracy and convergence.

math.NA

Convergence study of IB methods for Stokes equations with no-slip boundary conditions

Peskin's Immersed Boundary (IB) model and method are among one of the most important modeling tools and numerical methods. The IB method has been known to be first order accurate in the velocity. However, almost no rigorous theoretical proof can be found in the literature for Stokes equations with a prescribed velocity boundary condition. In this paper, it has been shown that the pressure of the Stokes equation has a convergence order $O(\sqrt{h} |\log h| )$ in the $L^2$ norm while the velocity has an $O(h |\log h| )$ convergence order in the infinity norm in two-space dimensions. The proofs are based on splitting the singular source terms, discrete Green functions on finite lattices with homogeneous and Neumann boundary conditions, a new discovered simplest $L^2$ discrete delta function, and the convergence proof of the IB method for elliptic interface problems \cite{li:mathcom}. The conclusion in this paper can apply to problems with different boundary conditions as long as the problems are wellposed. The proof process also provides an efficient way to decouple the system into three Helmholtz/Poisson equations without affecting the order of convergence. A non-trivial numerical example is also provided to confirm the theoretical analysis and the simple new discrete delta function.

math.NA

High order compact schemes for flux type BCs

In this paper new innovative fourth order compact schemes for Robin and Neumann boundary conditions have been developed for boundary value problems of elliptic PDEs in two and three dimensions. Different from traditional finite difference operator approach, which may not work for flux type of boundary conditions, carefully designed undetermined coefficient methods are utilized in developing high order compact (HOC) schemes. The new methods not only can be utilized to design HOC schemes for flux type of boundary conditions but can also be applied to general elliptic PDEs including Poisson, Helmholtz, diffusion-advection, and anisotropic equations with linear boundary conditions. In the new developed HOC methods, the coefficient matrices are generally M-matrices, which guarantee the discrete maximum principle for well-posed problems, so the convergence of the HOC methods. The developed HOC methods are versatile and can cover most of high order compact schemes in the literature. The HOC methods for Robin boundary conditions and for anisotropic diffusion and advection equations with Robin or even Dirichlet boundary conditions are likely the first ones that have ever been developed. With the help of pseudo-inverse, or SVD solutions, we have also observed that the developed HOC methods usually have smaller error constants compared with traditional HOC methods when applicable. Non-trivial examples with large wave numbers and oscillatory solutions are presented to confirm the performance of the new HOC methods.

math.NA

A conservative difference scheme with optimal pointwise error estimates for two-dimensional space fractional nonlinear Schrödinger equations

In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE).The scheme has the property of mass and energy conservation on the discrete level, with an unconditional stability and a second order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established and proved for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings.

math.NA

A multigrid-reduction-in-time solver with a new two-level convergence for unsteady fractional Laplacian problems

The multigrid-reduction-in-time (MGRIT) technique has proven to be successful in achieving higher run-time speedup by exploiting parallelism in time. The goal of this article is to develop and analyze a MGRIT algorithm, using FCF-relaxation with time-dependent time-grid propagators, to seek the finite element approximations of unsteady fractional Laplacian problems. The multigrid with line smoother proposed in [L. Chen, R. H. Nochetto, E. Ot{á}rola, A. J. Salgado, Math. Comp. 85 (2016) 2583--2607] is chosen to be the spatial solver. Motivated by [B. S. Southworth, SIAM J. Matrix Anal. Appl. 40 (2019) 564--608], we provide a new temporal eigenvalue approximation property and then deduce a generalized two-level convergence theory which removes the previous unitary diagonalization assumption on the fine and coarse time-grid propagators required in [X. Q. Yue, S. Shu, X. W. Xu, W. P. Bu, K. J. Pan, Comput. Math. Appl. 78 (2019) 3471--3484]. Numerical computations are included to confirm the theoretical predictions and demonstrate the sharpness of the derived convergence upper bound.

math.NA

A linearized energy--conservative finite element method for the nonlinear Schr\"{o}dinger equation with wave operator

In this paper, we propose a linearized finite element method (FEM) for solving the cubic nonlinear Schr\"{o}dinger equation with wave operator. In this method, a modified leap-frog scheme is applied for time discretization and a Galerkin finite element method is applied for spatial discretization. We prove that the proposed method keeps the energy conservation in the given discrete norm. Comparing with non-conservative schemes, our algorithm keeps higher stability. Meanwhile, an optimal error estimate for the proposed scheme is given by an error splitting technique. That is, we split the error into two parts, one from temporal discretization and the other from spatial discretization. First, by introducing a time-discrete system, we prove the uniform boundedness for the solution of this time-discrete system in some strong norms and obtain error estimates in temporal direction. With the help of the preliminary temporal estimates, we then prove the pointwise uniform boundedness of the finite element solution, and obtain the optimal $L^2$-norm error estimates in the sense that the time step size is not related to spatial mesh size. Finally, numerical examples are provided to validate the convergence-order, unconditional stability and energy conservation.

math.NA

An efficient multigrid solver for 3D biharmonic equation with a discretization by 25-point difference scheme

In this paper, we propose an efficient extrapolation cascadic multigrid (EXCMG) method combined with 25-point difference approximation to solve the three-dimensional biharmonic equation. First, through applying Richardson extrapolation and quadratic interpolation on numerical solutions on current and previous grids, a third-order approximation to the finite difference solution can be obtained and used as the iterative initial guess on the next finer grid. Then we adopt the bi-conjugate gradient (Bi-CG) method to solve the large linear system resulting from the 25-point difference approximation. In addition, an extrapolation method based on midpoint extrapolation formula is used to achieve higher-order accuracy on the entire finest grid. Finally, some numerical experiments are performed to show that the EXCMG method is an efficient solver for the 3D biharmonic equation.

math.NA

A three-level linearized difference scheme for the coupled nonlinear fractional Ginzburg-Landau equation

In this paper, the coupled fractional Ginzburg-Landau equations are first time investigated numerically. A linearized implicit finite difference scheme is proposed. The scheme involves three time levels, is unconditionally stable and second-order accurate in both time and space variables. The unique solvability, the unconditional stability and optimal pointwise error estimates are obtained by using the energy method and mathematical induction. Moreover, the proposed second-order method can be easily extended into the fourth-order method by using an average finite difference operator for spatial fractional derivatives and Richardson extrapolation for time variable. Finally, numerical results are presented to confirm the theoretical results.

math.NA

A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers' equation

In this paper, an efficient and high-order accuracy finite difference method is proposed for solving multidimensional nonlinear Burgers' equation. The third-order three stage Runge-Kutta total variation diminishing (TVD) scheme is employed for the time discretization, and the three-point combined compact difference (CCD) scheme is used for spatial discretization. Our method is third-order accurate in time and sixth-order accurate in space. The CCD-TVD method treats the nonlinear term explicitly thus it is very efficient and easy to implement. In addition, we prove the unique solvability of the CCD system under non-periodic boundary conditions. Numerical experiments including both two-dimensional and three-dimensional problems have been conducted to demonstrate the high efficiency and accuracy of the proposed method.

math.NA

Parallel-in-Time with Fully Finite Element Multigrid for 2-D Space-fractional Diffusion Equations

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions. We firstly obtain a fully discrete scheme via using the linear finite element method to discretize spatial and temporal derivatives to propagate solutions. Next, we present a non-intrusive time-parallelization and its two-level convergence analysis, where we algorithmically and theoretically generalize the MGRIT to time-dependent fine time-grid propagators. Finally, numerical illustrations show that the obtained numerical scheme possesses the saturation error order, theoretical results of the two-level variant deliver good predictions, and significant speedups can be achieved when compared to parareal and the sequential time-stepping approach.

math.NA

A fourth-order maximum principle preserving operator splitting scheme for three-dimensional fractional Allen-Cahn equations

In this paper, by using Strang's second-order splitting method, the numerical procedure for the three-dimensional (3D) space fractional Allen-Cahn equation can be divided into three steps. The first and third steps involve an ordinary differential equation, which can be solved analytically. The intermediate step involves a 3D linear fractional diffusion equation, which is solved by the Crank-Nicolson alternating directional implicit (ADI) method. The ADI technique can convert the multidimensional problem into a series of one-dimensional problems, which greatly reduces the computational cost. A fourth-order difference scheme is adopted for discretization of the space fractional derivatives. Finally, Richardson extrapolation is exploited to increase the temporal accuracy. The proposed method is shown to be unconditionally stable by Fourier analysis. Another contribution of this paper is to show that the numerical solutions satisfy the discrete maximum principle under reasonable time step constraint. For fabricated smooth solutions, numerical results show that the proposed method is unconditionally stable and fourth-order accurate in both time and space variables. In addition, the discrete maximum principle is also numerically verified.

math.NA

Nanostructures imaging via numerical solution of a 3-d inverse scattering problem without the phase information

Inverse scattering problems without the phase information arise in imaging of nanostructures whose sizes are hundreds of nanometers as well as in imaging of biological cells. The governing equation is the 3-d generalized Helmholtz equation with the unknown coefficient, which represents the spatially distributed dielectric constant. It is assumed that only the modulus of the complex valued wave field is measured on a frequency range. The phase is not measured. Two rigorous numerical methods are tested and their performances are compared for realistic ranges of parameters.

math.AP

A linearly implicit conservative difference scheme for the generalized Rosenau-Kawahara-RLW equation

This paper concerns the numerical study for the generalized Rosenau-Kawahara-RLW equation obtained by coupling the generalized Rosenau-RLW equation and the generalized Rosenau-Kawahara equation. We first derive the energy conservation law of the equation, and then develop a three-level linearly implicit difference scheme for solving the equation. We prove that the proposed scheme is energy-conserved, unconditionally stable and second-order accurate both in time and space variables. Finally, numerical experiments are carried out to confirm the energy conservation, the convergence rates of the scheme and effectiveness for long-time simulation.

math.NA

An extrapolation cascadic multigrid method combined with a fourth order compact scheme for 3D poisson equation

In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the $L^2$-norm and $L^{\infty}$-norm for the solution and its gradient when the exact solution belongs to $C^6$. Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.

math.NA

A new extrapolation cascadic multigrid method for 3D elliptic boundary value problems on rectangular domains

In this paper, we develop a new extrapolation cascadic multigrid (ECMG$_{jcg}$) method, which makes it possible to solve 3D elliptic boundary value problems on rectangular domains of over 100 million unknowns on a desktop computer in minutes. First, by combining Richardson extrapolation and tri-quadratic Serendipity interpolation techniques, we introduce a new extrapolation formula to provide a good initial guess for the iterative solution on the next finer grid, which is a third order approximation to the finite element (FE) solution. And the resulting large sparse linear system from the FE discretization is then solved by the Jacobi-preconditioned Conjugate Gradient (JCG) method. Additionally, instead of performing a fixed number of iterations as cascadic multigrid (CMG) methods, a relative residual stopping criterion is used in iterative solvers, which enables us to obtain conveniently the numerical solution with the desired accuracy. Moreover, a simple Richardson extrapolation is used to cheaply get a fourth order approximate solution on the entire fine grid. Test results are reported to show that ECMG$_{jcg}$ has much better efficiency compared to the classical MG methods. Since the initial guess for the iterative solution is a quite good approximation to the FE solution, numerical results show that only few number of iterations are required on the finest grid for ECMG$_{jcg}$ with an appropriate tolerance of the relative residual to achieve full second order accuracy, which is particularly important when solving large systems of equations and can greatly reduce the computational cost. It should be pointed out that when the tolerance becomes smaller, ECMG$_{jcg}$ still needs only few iterations to obtain fourth order extrapolated solution on each grid, except on the finest grid. Finally, we present the reason why our ECMG algorithms are so highly efficient for solving such problems.

math.NA