In this paper, we propose a numerical approach for the fractional reaction sub-diffusion equation with a Caputo-Hadamard derivative of fractional order α∈ (0,1) , whose solutions typically exhibit singular behavior at the initial time. The approach involves the use of an L2-type discrete operator of order 3-α to approximate the fractional derivative on nonuniform temporal meshes, while a standard second-order difference method is employed for uniform spatial meshes. Detailed discussions are presented on the truncation errors and coefficients of the proposed discrete fractional derivative operator. The stability as well as the accuracy of the resulting numerical scheme is rigorously analyzed on a unique nonuniform mesh. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.
We aim to propose a Riemann-problem-solver-free nonstaggered central difference scheme (NCDS), which is well-balanced and positivity-preserving for two-dimensional shallow water flows over nonflat bottom topography with wet-dry fronts. The well-balanced property of the NCDS for the two-dimensional shallow water equations at wet-dry fronts is still unclear. The NCDS is “generously multidimensional” and consists of three steps: a forward step, a corrector step, and a backward step. Each step needs to guarantee the water depth to be nonnegative and retain the lake at rest. The key ingredient of the NCDS is the backward step. It is very nontrivial and technical to obtain both the well-balanced and positivity-preserving properties when the computational domain contains wet-dry fronts for the NCDS. The traditional NCDS fails to retain the lake at rest when the computational domain contains wet-dry fronts. We introduce a novel backward step, which is well-balanced and positivity-preserving, to recover numerical solutions on nonstaggered cells. Finally, we show several numerical results of two-dimensional shallow water equations with wet-dry fronts to verify these properties.
We investigate the inverse scattering transform for the focusing nonlinear Schrödinger (NLS) equation with a particular class of nonvanishing boundary conditions (NVBCs), especially in the case of reflectionless potentials that give rise to a transmission coefficient with an arbitrary finite number N pairs of higher-order poles. The inverse problem is characterized in terms of a 2× 2 matrix Riemann-Hilbert (RH) problem equipped with several residue conditions at N pairs of higher-order poles. In the reflectionless case, we point out that the N-multipole soliton solutions including higher-order Kuznetsov-Ma breathers and Akhmediev breathers can be reconstructed by a linear algebraic system. Furthermore, we verify these special solutions by numerical simulations and display their density structures, also derive some Peregrine solitons by choosing appropriate parameters and taking limits.
Reaction-diffusion equations model various biological, physical, sociological, and environmental phenomena. Often, numerical simulations are used to understand and discover the dynamics of such systems. Following the extension of the nonlinear Lyapunov theory applied to some class of reaction-diffusion partial differential equations (PDEs), we develop the first fully discrete Lyapunov discretizations that are consistent with the stability properties of the continuous parabolic reaction-diffusion models. The proposed framework provides a systematic procedure to develop fully discrete schemes of arbitrary order in space and time for solving a broad class of equations equipped with a Lyapunov functional. The new schemes are applied to solve systems of PDEs, which arise in epidemiology and oncolytic M1 virotherapy. The new computational framework provides physically consistent and accurate results without exhibiting scheme-dependent instabilities and converging to unphysical solutions. The proposed approach represents a capstone for developing efficient, robust, and predictive technologies for simulating complex phenomena.
This paper presents a novel approach to preserving invariants in the time-implicit numerical discretization of high-order nonlinear wave equations. Many high-order nonlinear wave equations have an infinite number of conserved quantities. Designing time-implicit numerical schemes that preserve many conserved quantities simultaneously is challenging. The proposed method utilizes Lagrange multipliers to reformulate the local discontinuous Galerkin (LDG) discretization as a conservative discretization. Combining an implicit spectral deferred correction method for time discretization can achieve a high-order scheme in both temporal and spatial dimensions. We use the Korteweg-de Vries equations as an example to illustrate the implementation of the algorithm. The algorithm can be easily generalized to other nonlinear wave problems with conserved quantities. Numerical examples for various high-order nonlinear wave equations demonstrate the effectiveness of the proposed methods in preserving invariants while maintaining the high accuracy.
Solutions of convection-dominated convection-diffusion problems usually possess layers, which are regions where the solution has a steep gradient. It is well known that many classical numerical discretization techniques face difficulties when approximating the solution to these problems. In recent years, physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems ((I)BVPs) received a lot of interest. This paper studies various loss functionals for PINNs that are especially designed for convection-dominated convection-diffusion problems and that are novel in the context of PINNs. They are numerically compared to the vanilla and an hp-variational loss functional from the literature based on two steady-state benchmark problems whose solutions possess different types of layers. We observe that the best novel loss functionals reduce the L^2( ) error by 17.3
This paper examines an optimal control problem for mean-field systems under partial observation. The state system is described by a controlled mean-field forward-backward stochastic differential equation that features correlated noises between the system and the observation. Furthermore, the observation coefficients are allowed to depend not only on the control process but also on its probability distribution. Assuming a convex control domain and allowing all coefficients of the systems to be random, we derive directly a variational formula for the cost function in a given control process direction in terms of the Hamiltonian and the associated adjoint system without relying on variational systems under standard assumptions on the coefficients. As an application, we present the necessary and sufficient conditions for the optimality of our control problem using Pontryagin’s maximum principle in a unified manner. The research provides insights into the optimization of mean-field systems under partial observation, which has practical implications for various applications.
In this article, we develop a posteriori error analysis of a nonconforming finite element method for a linear quadratic elliptic distributed optimal control problem with two different sets of constraints, namely (i) integral state constraint and integral control constraint; (ii) integral state constraint and pointwise control constraints. In the analysis, we have taken the approach of reducing the state-control constrained minimization problem into a state minimization problem obtained by eliminating the control variable. The reliability and efficiency of a posteriori error estimator are discussed. Numerical results are reported to illustrate the behavior of the error estimator.
This paper proposes a finite element method for solving the periodic steady-state problem for the scalar-valued and vector-valued Poisson equations, a simple reduction model of the Maxwell equations under the Coulomb gauge. Introducing a new potential variable, we reformulate two systems composed of the scalar-valued and vector-valued Poisson problems to a single Hodge-Laplace problem for the 1-form in $\mathbb{R}^4$ using the standard de Rham complex. Consequently, we can directly apply the Finite Element Exterior Calculus (FEEC) theory in $\mathbb{R}^4$ to deduce the well-posedness, stability, and convergence. Numerical examples using the cubical element are reported to validate the theoretical results.
It is important to compute the Hilbert transform of a given function defined on a finite interval. In 2013, Micchelli and his collaborators proposed a fast algorithm, which is called the Hilbert spline transform, to calculate the Hilbert transform of a given function on a finite interval with the computational complexity O(nlog n) , where the spline knots were chosen to be the midpoints of sampling points. A natural question is that, whether or not the spline knots can be chosen to be the same as the sampling points. This paper gives a positive answer to this question. Besides, the analytic expression of the Hilbert transform of B-splines of any order is also established. Furthermore, the problem of how to choose spline coefficients, using the quasi-interpolation method or interpolation method, is also considered, although both make sure an optimal approximation order. Several interesting numerical examples are implemented and compared with most of the existing methods. Numerical results show that the proposed algorithm has a relatively high computational accuracy as well as a relatively low computational complexity.
A broad framework for time-fractional advection-diffusion equations is considered by incorporating a general class of Caputo-type fractional derivatives that includes many well-known fractional derivatives as particular cases. Within this general context, in this paper, we propose a numerical approach to provide approximate solutions to time-fractional advection-diffusion models that involve fractional derivatives of the generalized class. The developed approach is based on discretizing the studied models with respect to spatio-temporal domains using non-uniform meshes. Moreover, we discuss the stability of the proposed approach, which does not require solving large systems of linear equations. Numerical results and 3D graphics are presented for some time-fractional advection-diffusion models using several fractional derivatives. The feasibility and the reliability of the proposed approach are clearly demonstrated by comparing the numerical solutions of the studied models with the exact solutions in cases with a known solution.
In this paper, we consider solving a least-squares problem to the generalized Sylvester quaternion tensor equation. From the properties of quaternions and the Kronecker product, a tensor form of the LSQR algorithm is proposed for solving this problem, the convergence analysis of which is then established. Numerical results are reported to illustrate the feasibility and validity of the proposed algorithm compared with the tensor form of the CGLS method, including when the algorithm is tested with some randomly generated data and color video restoration problems.
A stable and high-order accurate solver for linear and nonlinear parabolic equations is presented. An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an explicit singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solution is performed by the recently developed Hierarchical Poincaré-Steklov (HPS) method. This is a fast direct solver for elliptic equations that decomposes the space domain into a hierarchical tree of subdomains and builds spectral collocation solvers locally on the subdomains. These ideas are naturally combined in the presented method since the singly diagonal coefficient in ESDIRK and a fixed time step ensures that the coefficient matrix in the implicit solution of HPS remains the same for all time stages. This means that the precomputed inverse can be efficiently reused, leading to a scheme with complexity (in two dimensions) 𝒪(N^1.5) for the precomputation where the solution operator to the elliptic problems is built, and then 𝒪(N log N) for the solution in each time step. The stability of the method is proved for first order in time and any order in space, and numerical evidence substantiates a claim of the stability for a much broader class of time discretization methods. Numerical experiments supporting the accuracy of the efficiency of the method in one and two dimensions are presented.
Very recently, Qi and Cui extended the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts and proved that they have a Perron eigenpair and a Perron-Frobenius eigenpair. The Collatz method was also extended to find the Perron eigenpair. Qi and Cui proposed two conjectures. One is the k-order power of a dual number matrix, which tends to zero if and only if the spectral radius of its standard part is less than one, and another is the linear convergence of the Collatz method. In this paper, we confirm these conjectures and provide the theoretical proof. The main contribution is to show that the Collatz method R-linearly converges with an explicit rate.
The randomized block Kaczmarz (RBK) method is a randomized orthogonal projection iterative approach, which plays an important role in solving large-scale linear systems. A key point of this type of method is to select working rows effectively during iterations. However, in most of the RBK-type methods, one has to scan all the rows of the coefficient matrix in advance to compute probabilities or paving, or to compute the residual vector of the linear system in each iteration to determine the working rows. These are unfavorable for big data problems. To cure these drawbacks, we propose a semi-randomized block Kaczmarz (SRBK) method with simple random sampling for large-scale linear systems in this paper. The convergence of the proposed method is established. Numerical experiments on some real-world and large-scale data sets show that the proposed method is often superior to many state-of-the-art RBK-type methods for large linear systems.
The photoacoustic tomography (PAT) is a new biomedical imaging modality. It has great advantages in early diagnosis of the human disease and accurate monitoring of disease progression. In photoacoustic imaging, when a beam of short-pulsed laser illuminates the biological tissue, the photoacoustic effect leads to the emergence of acoustic waves in the tissue. The initial acoustic pressure in the tissue reveals the structures of the tissue. The purpose of the PAT reconstruction problem is to obtain the initial acoustic pressure in the tissue from the collected photoacoustic signal information. In this paper, we propose a rank minimization-based regularization model for the sparse-view photoacoustic image reconstruction problem. We design a proximal alternating iterative algorithm to solve the model and the convergence of the algorithm is demonstrated by utilizing the Kudyka-Łojasiewicz theory. The experimental results show that the proposed method is competitive with the existing state-of-the-art PAT reconstruction methods in terms of both reconstructed quantities and visual effects for the sparse-view PAT reconstruction problem.
The electrocardiogram (ECG) segmentation needs to separate different waves from an ECG and cluster the waves simultaneously. Clusterwise regression is a useful approach that can segment and cluster the data simultaneously. In this paper, we apply the clusterwise regression method to segment the ECG. By modeling the ECG signal wave by the Gaussian mixture model (GMM) and introducing a weight function, we propose a minimization model that consists of the weighted sum of the negative log-likelihood and the total variation (TV) of the weight function. The TV of the weight function enforces the temporal consistency. A supervised algorithm is designed to solve the proposed model. Experimental results show the efficiency of the proposed method for the ECG segmentation.
This paper further extends the shift-splitting (SS) and local shift-splitting (LSS) preconditioners to solve the general block two-by-two linear systems. We demonstrate that the eigenvalues of the corresponding preconditioned matrices cluster tightly around 2 by detailed spectral property analysis. Numerical experiments not only validate the theoretical results but also show the effectiveness and superiority of the SS and LSS preconditioners by comparing them with some existing preconditioners applied to the generalized minimal residual (GMRES) method for solving the block two-by-two linear systems.
We demonstrate how different computational approaches affect the performance of solving the generalized eigenvalue problem (GEP). The layered piezo device is studied for resonance frequencies using different meshes, sparse matrix representations, and numerical methods in COMSOL Multiphysics and ACELAN-COMPOS packages. Specifically, the matrix-vector and matrix-matrix product implementation for large sparse matrices is discussed. The shift-and-invert Lanczos method is used to solve the partial symmetric GEP numerically. Different solvers are compared in terms of the efficiency. The results of numerical experiments are presented.
