Compact Approximation Research Articles

A new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation

The fractional Schrödinger equation is an important fractional nonlinear evolution equation, and the study of its numerical solution has profound scientific meaning and wide application prospects. This paper proposes a new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation (TFSE). The Caputo time-fractional derivative is discretized by high-order formula and the fourth-order compact difference approximation is applied for spatial discretization. A new nonlinear compact difference scheme with temporal second-order and spatial fourth-order accuracy is constructed, which is solved by the efficient linearized iterative algorithm. The unconditional stability and convergence are analysed by the energy method. The unique existence and maximum-norm estimate of new compact difference scheme solution are obtained. Theoretical analysis shows that the convergence accuracy of new compact difference scheme is with the strong regularity assumption. Numerical experiments verify theoretical results and indicate that the proposed method is an efficient numerical method for solving TFSE.

An alternating direction implicit compact finite difference scheme for the multi-term time-fractional mixed diffusion and diffusion–wave equation

An alternating direction implicit(ADI) compact finite difference scheme for the two-dimensional multi-term time-fractional mixed diffusion and diffusion–wave equation is given in this paper. By using the Riemann–Liouville fractional integral operator on both sides of the original equation, we obtain a time-fractional integro–differential equation with the order of the highest derivative being one. Using the weighted and shifted Grünwald formulas of Riemann–Liouville fractional derivative and fractional integral, and the Crank–Nicolson approximation, a temporal second-order approximation is obtained for the equivalent integrodifferential system. In the spatial direction, a fourth-order compact approximation is employed to give a fully discrete scheme. Using the splitting techniques for higher dimensional problems, we derive the fully discrete ADI Crank–Nicolson scheme. With the positive definiteness of the related time discrete coefficients and spatial operators, the stability and convergence (second-order in time and fourth-order in space) in the discrete L2-norm are proved by using energy method. Two numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.

Fractional‐order LC filter modeling, implementation, and analysis for distributed power generation system

SummaryThis paper proposes a comprehensive mathematical modeling and analysis considering fractional‐order LC filters (FOLCF) in the utility grid‐connected mode (UGCM) and the standalone mode (SM) of operation for the distributed power generation system. The fractional‐order behavior of LC filter elements helps to more properly depict a practical inductor or capacitor than the integer‐order LC filters (IOLCF) in the distributed power generation system. The fractional‐order model of UGCM and SM is obtained in the synchronously rotating reference frame using coordinate transformation. A detailed stability and time domain analysis is performed on the developed FO UGCM and SM mathematical models. The stability of fractional‐order LC filters increases for orders less than 1, based on the performance evaluation of the FO and IO mathematical models for UGCM and SM with the corresponding designed controller gains. The FOLCFs are approximated using the constant phase element method, which gives a compact and exact approximation of the fractional‐order inductor and capacitor elements. The implementation of FOLCF with UGCM and SM operations verifies the effectiveness of the proposed model in real‐time simulation. The results show the FOLCF model provides better dynamic tracking and disturbance rejection response than the IOLCF model. Hence, the FOLCF model imparts immunity against local load disturbances to regulate the point of common coupling voltage and frequency in standalone mode. Furthermore, a significant improvement in % total harmonic distortion is also observed in grid current and load voltage.

A note on numerical radius attaining mappings

We prove that if every bounded linear operator (or N N -homogeneous polynomials) on a Banach space X X with the compact approximation property attains its numerical radius, then X X is a finite dimensional space. Moreover, we present an improvement of the polynomial James’ theorem for numerical radius proved by Acosta, Becerra Guerrero and Galán [Q. J. Math. 54 (2003), pp. 1–10]. Finally, the denseness of weakly (uniformly) continuous 2 2 -homogeneous polynomials on a Banach space whose Aron-Berner extensions attain their numerical radii is obtained.

Quantitative Characterizations of Bounded Compact Approximation Property by Semi-Fredholm Operators

Quantitative Characterizations of Bounded Compact Approximation Property by Semi-Fredholm Operators

Model-Based Reinforcement Learning and Neural-Network-Based Policy Compression for Spacecraft Rendezvous on Resource-Constrained Embedded Systems

Autonomous spacecraft rendezvous is very challenging in increasingly complex space missions. In this article, we present our approach model-based reinforcement learning for spacecraft rendezvous guidance (MBRL4SRG). We build a Markov decision process model based on the Clohessy-Wiltshire equation of spacecraft dynamics and use dynamic programming to solve it and generate the decision table as the optimal agent policy. Since the onboard computing system of spacecraft is resource constrained in terms of both memory size and processing speed, we train a neural network (NN) as a compact and efficient function approximation to the tabular representation of the decision table. The NN outputs are formally verified using the verification tool ReluVal, and the verification results show that the robustness of the NN is maintained. Experimental results indicate that MBRL4SRG achieves lower computational overhead than the conventional proportional–integral–derivative algorithm and has higher trustworthiness and better computational efficiency during training than the model-free reinforcement learning algorithms.

A high-resolution fuzzy transform combined compact scheme for 2D nonlinear elliptic partial differential equations

This paper proposes a new high-resolution fuzzy transform algorithm for solving two-dimensional nonlinear elliptic partial differential equations (PDEs). The underlying new computational method implements the method of so-called approximating fuzzy components, which evaluate the solution values with fourth-order accuracy at internal mesh points. Triangular basic functions and fuzzy components are locally determined by linear combinations of solution values at nine points. Such a scheme connects the proposed method of approximating fuzzy components with the exact values of the solution using a linear system of equations. Compact approximations of high-resolution fuzzy components using nine points give a block tridiagonal Jacobi matrix. Apart from the numerical solution, it is easy to construct closed-form approximate solutions using a 2D spline interpolation polynomial from the available data with fuzzy components. The upper bounds of the approximation errors are estimated, as well as the convergence of the approximating solutions. Simulations with linear and nonlinear elliptical PDEs arising from quantum mechanics and convection-dominated diffusion phenomena are presented to confirm the usefulness of the new scheme and fourth-order convergence. To summarize:•The paper presents a high-resolution numerical method for the two-dimensions elliptic PDEs with nonlinear terms.•The combined effect of fuzzy transform and compact discretizations yields almost fourth-order accuracies to Schro¨dinger equation, convection-diffusion equation, and Burgers equation.•The high-order numerical scheme is computationally efficient and employs minimal data storage.

Sharp estimates for conditionally centered moments and for compact operators on Lp$L^p$ spaces

AbstractLet be a probability space, ξ be a random variable on , be a sub‐σ‐algebra of , and let be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of in terms of the moments of ξ. This allows us to find the optimal constant in the bounded compact approximation property of , .

Optimality conditions in set-valued optimization using approximations as generalized derivatives

Abstract The set criterion is an appropriate defining approach regarding the solutions for the set-valued optimization problems. By using approximations as generalized derivatives of set-valued mappings, we establish necessary optimality conditions for a constrained set-valued optimization problem in the sense of set optimization in terms of asymptotical pointwise compact approximations. Sufficient optimality conditions are then obtained through first-order strong approximations of data set-valued mappings.

Numerical analysis of fourth-order compact difference scheme for inhomogeneous time-fractional Burgers-Huxley equation

The time-fractional Burgers-Huxley (TFBH) equation is a typical fractional parabolic equation, it is an evolutionary model describing the propagation of neural pulses. The high-accuracy numerical method for studying TFBH equation has important scientific significance and engineering application value. In this paper, a high-order compact difference scheme is constructed for inhomogeneous TFBH equation. The time-fractional derivative is discretized by L1 formula, and the spatial derivative is approximated by fourth-order precision compact approximation. We analyzed the existence and uniqueness of difference scheme solution, and proved the stability and convergence of the fourth-order compact scheme using the energy method. Numerical experiments show that the scheme converges to O(τ2−α+h4) under the strong regularity assumption. Under the condition of weak regularity, the scheme converges to O(τα+h4). It shows that the scheme has good robustness and high accuracy for solving inhomogeneous TFBH equation.

Entanglement of electrons and nuclei: A most compact representation of the molecular wave function

Ultrafast pumping displaces both electrons and nuclei from equilibrium so that the wave function is a double sum of separable terms for the dynamics of the electrons and nuclei. We convert the double sum into a single one by a matricization of the wave function and then generate an exact separable expression for the entangled molecular wave function. A most compact approximation with a minimum number of terms is obtained via Singular Value Decomposition. LiH and N2 are used as an illustration: the two differ in their adiabatic electronic dynamics in the energy range accessible by a UV pulse.

Characterizations of bounded compact approximation property by Calkin representations

In this note we characterize the bounded compact approximation property via Calkin representations for Banach spaces.

High order compact split step finite difference method for two-dimensional coupled nonlinear Schrödinger system

In this paper, a decoupled and efficient finite difference method is developed for two-dimensional coupled nonlinear Schrödinger (CNLS) system. The proposed method uses split step technique for the temporal discretization and high order compact (HOC) difference approximation for the spatial discretization. The original problem is decomposed into a two-dimensional linear subproblem and a two-dimensional nonlinear subproblem. For the two-dimensional linear subproblem, the Lie-Trotter splitting formula is adopted in time to reduce computational cost. While for the nonlinear subproblem, it can be integrated directly and exactly. By the von Neumann approach, it is showed that the proposed method is unconditionally stable. Numerical examples are conducted to compare it with other scheme and numerical results verified the superiority of the proposed method in terms of accuracy and efficiency. The new method also exhibits good numerical performance in long-time simulation.

Coverage Axis: Inner Point Selection for 3D Shape Skeletonization

AbstractIn this paper, we present a simple yet effective formulation called Coverage Axis for 3D shape skeletonization. Inspired by the set cover problem, our key idea is to cover all the surface points using as few inside medial balls as possible. This formulation inherently induces a compact and expressive approximation of the Medial Axis Transform (MAT) of a given shape. Different from previous methods that rely on local approximation error, our method allows a global consideration of the overall shape structure, leading to an efficient high‐level abstraction and superior robustness to noise. Another appealing aspect of our method is its capability to handle more generalized input such as point clouds and poor‐quality meshes. Extensive comparisons and evaluations demonstrate the remarkable effectiveness of our method for generating compact and expressive skeletal representation to approximate the MAT.

Compact difference schemes for multidimensional Klein–Gordon equations

Abstract. In this article, we consider a compact difference approximation of the schemes of order O(| h|4 + τ2), h = (h1, h2, ..., hp) for the Klein–Gordon equations in the multidimensional case. In studying the stability of these difference schemes, the theory of operator-difference schemes by A. A. Samarskii is used, and the strong stability of difference schemes is proved with respect to a small perturbation of the initial conditions, the right-hand side and the coefficients of the equations. The theoretical results are confirmed by test numerical calculations.

High precision compact numerical approximation in exponential form for the system of 2D quasilinear elliptic BVPs on a discrete irrational region

This article presents a new approximation of order four in exponential form for two-dimensional (2D) quasilinear partial differential equation (PDE) of elliptic form with solution domain being irrational. It is further extended for application to a system of quasilinear elliptic PDEs with Dirichlet boundary conditions (DBCs). The main highlights of the method framed in this article are as under:•It uses a 9-point stencil with unequal mesh to approach the solution. The error analysis is discussed to authenticate the order of convergence of the proposed numerical approximation.•Various validating problems, for instance the Burgers’ equation, Poisson equation in cylindrical coordinates, Navier-Stokes (NS) equations in rectangular and cylindrical coordinates are solved using the proposed techniques to depict their stability. The proposed approximation produces solution free of oscillations for large values of Reynolds Number in the vicinity of a singularity.•The results of the proposed method are superior in comparison to the existing methods of [49] and [56].

Resistance scaling on 4N-carpets

Abstract The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F, let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.

A Crank-Nicolson-type compact difference method with the uniform time step for a class of weakly singular parabolic integro-differential equations

This paper is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t=0. A generalized Crank-Nicolson-type scheme for the time discretization is proposed by designing a product integration rule for the integral term, and a compact difference approximation is used for the space discretization. The proposed method is constructed on the uniform time mesh, but it can still achieve the second-order convergence in time for weakly singular solutions. The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L2-norm is obtained. The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space. The extension of the method to two-dimensional problems is also discussed. A simple comparison is made with several existing methods. Numerical results confirm the theoretical analysis result and show the effectiveness of the proposed method.

Effect of magnetic field, size and donor position on the absorption coefficients related a donor within the core/shell/shell quantum dot

In this study, linear, nonlinear and total optical absorption coefficients related a single shallow donor atom confined in semiconductor core/shell/shell quantum dot heterostructure are researched in detail within the compact density matrix formalism approximation. For this purpose, firstly, the energies and the wavefunctions are computed by the diagonalization method in the effective mass approach. Moreover, the effects of size modulation, donor position and magnetic field are analyzed. The numerical results indicate that the linear and nonlinear parts of the absorption coefficients related with intersubband \(1s\rightarrow 1p\) and \(1p\rightarrow 1d\) donor transitions undergo significant changes.

High resolution operator compact implicit half-step approximation for 3D quasi-linear hyperbolic equations and ADI method for 3D telegraphic equation on an irrational domain

We propose a novel three-level implicit half-step method of order two in time and four in space using unequal mesh for the solution of three-space dimensional quasi-linear hyperbolic equations on an irrational domain. The proposed method is directly applicable to 3D hyperbolic equation with singular coefficients, which is the key charisma of our scheme and we do not need any fictitious points for computation. The stability analysis of the model problem for unequal mesh has been discussed and it has been shown that the proposed method for Telegraphic equation is unconditionally stable. Alternating direction implicit (ADI) method by the help of a tri-diagonal solver is used to solve 3D linear schemes in three sweeps. The proposed method is scrutinized on several physical problems on dissimilar domain to exhibit the accuracy and effectiveness of the proposed method.