Numerical Algorithms最新文献

Let (A_alpha ) be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, ((A_alpha )_{11}=alpha ), where (alpha in mathbb C), and zero elsewhere. A basis ({P_0,P_1,P_2,ldots }) of the linear space (mathcal {P}_alpha ) spanned by the powers of (A_alpha ) is determined, where (P_0=I), (P_n=T_n+H_n), (T_n) is the symmetric Toeplitz matrix having ones in the nth super- and sub-diagonal, zeros elsewhere, and (H_n) is the Hankel matrix with first row ([theta alpha ^{n-2}, theta alpha ^{n-3}, ldots , theta , alpha , 0, ldots ]), where (theta =alpha ^2-1). The set (mathcal {P}_alpha ) is an algebra, and for (alpha in {-1,0,1}), (H_n) has only one nonzero anti-diagonal. This fact is exploited to provide a better representation of symmetric quasi-Toeplitz matrices (mathcal{Q}mathcal{T}_S), where, instead of representing a generic matrix (Ain mathcal{Q}mathcal{T}_S) as (A=T+K), where T is Toeplitz and K is compact, it is represented as (A=P+H), where (Pin mathcal {P}_alpha ) and H is compact. It is shown experimentally that the matrix arithmetic obtained this way is much more effective than that implemented in the toolbox CQT-Toolbox of Numer. Algo. 81(2):741–769, 2019.

A fourth-order improvised cubic B-spline collocation method (ICSCM) is proposed to numerically solve the equal width (EW) equation and the modified equal width (MEW) equation. The discretization of the spatial domain is done using the ICSCM and the Crank-Nicolson scheme is used for the discretization of the temporal domain. The nonlinear terms are processed using quasi-linearization techniques and the stability analysis of this method is performed using Fourier series analysis. The validity and accuracy of this method are verified through several numerical experiments using a single solitary wave, two solitary waves, Maxwellian initial condition, and an undular bore. Since there is an exact solution for the single wave, the error norms (varvec{L_2}) and (varvec{L_{infty }}) are first calculated and compared with some previous studies published in journal articles. In addition, the three conserved quantities (varvec{Q}), (varvec{M}), and (varvec{E}) of the problems raised during the simulation are also calculated and recorded in the table. Lastly, the comparisons of these error norms and conserved quantities show that the numerical results obtained with the proposed method are more accurate and agree well with the values of the conserved quantities obtained in some literatures using the same parameters. The main advantage of ICSCM is its ability to effectively capture solitary wave propagation and describe solitary wave collisions. It can perform solution calculations at any point in the domain, easily use larger time steps to calculate solutions at higher time levels, and produce more accurate calculation results.

In this paper, we introduce a modified projection method and give a strong convergence theorem for solving variational inequality problems in real Hilbert spaces. Under mild assumptions, there exists a novel line-search rule that makes the proposed algorithm suitable for non-Lipschitz continuous and pseudo-monotone operators. Compared with other known algorithms in numerical experiments, it is shown that our algorithm has better numerical performance.

In this paper, we first propose a version of FISTA, called C-FISTA type gradient projection algorithm, for quasi-variational inequalities in Hilbert spaces and obtain linear convergence rate. Our results extend the results of Nesterov for C-FISTA algorithm for strongly convex optimization problem and other recent results in the literature where linear convergence results of C-FISTA are obtained for strongly convex composite optimization problems. For a comprehensive study, we also introduce a new version of gradient projection algorithm with momentum terms and give linear rate of convergence. We show the adaptability and effectiveness of our proposed algorithms through numerical comparisons with other related gradient projection algorithms that are in the literature for quasi-variational inequalities.

In this paper, we conduct an in-depth investigation of the structural intricacies inherent to the Invariant Energy Quadratization (IEQ) method as applied to gradient flows, and we dissect the mechanisms that enable this method to keep linearity and the conservation of energy simultaneously. Building upon this foundation, we propose two methods: Invariant Energy Convexification and Invariant Energy Functionalization. These approaches can be perceived as natural extensions of the IEQ method. Employing our novel approaches, we reformulate the system connected to gradient flow, construct a semi-discretized numerical scheme, and obtain a commensurate modified energy dissipation law for both proposed methods. Finally, to underscore their practical utility, we provide numerical evidence demonstrating these methods’ accuracy, stability, and effectiveness when applied to both Allen-Cahn and Cahn-Hilliard equations.

An operator-splitting finite element scheme for the time-dependent radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite element method are used for the spatial-angular discretization of the radiative transfer equation, whereas the backward Euler scheme is used for temporal discretization. Error analysis of the proposed numerical scheme for the fully discrete radiative transfer equation is presented. The stability and convergence estimates for the fully discrete problem are derived. Moreover, an operator-splitting algorithm for the numerical simulation of high-dimensional equations is also presented. The validity of the derived estimates and implementation is illustrated with suitable numerical experiments.

We study the reconstruction of an unknown/inaccessible smooth inner boundary from the knowledge of the Dirichlet condition (temperature) on the entire boundary of a doubly connected domain occupied by a two-dimensional homogeneous anisotropic solid and an additional Neumann condition (normal heat flux) on the known, accessible, and smooth outer boundary in the framework of steady-state heat conduction with heat sources. This inverse geometric problem is approached through an operator that maps an admissible inner boundary belonging to the space of (2pi -)periodic and twice continuously differentiable functions into the Neumann data on the outer boundary which is assumed to be continuous. We prove that this operator is differentiable, and hence, a gradient-based method that employs the anisotropic single layer representation of the solution to an appropriate Dirichlet problem for the two-dimensional anisotropic heat conduction is developed for approximating the unknown inner boundary. Numerical results are presented for both exact and perturbed Neumann data on the outer boundary and show the convergence, stability, and robustness of the proposed method.

In this paper, a class of Runge-Kutta methods for solving neutral delay differential equations (NDDEs) is proposed, which was first introduced by Bassenne et al. (J. Comput. Phys. 424, 109847, 2021) for ODEs. In the study, the explicit Runge-Kutta method is multiplied by an operator, which is a Time-Accurate and highly-Stable Explicit operator (TASE-RK), resulting in higher stability than explicit RK. Recently, the multi-parameter TASE-W method was extended by González-Pinto et al. (Appl. Numer. Math. 188, 129–145, 2023). We generalized TASE-RK and TASE-W to NDDEs for the first time. Then, by applying the argument principle, sufficient conditions for delay-dependent stability of TASE-RK and TASE-W combined with Lagrange interpolation for NDDEs are investigated. Finally, numerical examples are carried out to verify the theoretical results.

The randomized Kaczmarz method, along with its recently developed variants, has become a popular tool for dealing with large-scale linear systems. However, these methods usually fail to converge when the linear systems are affected by heavy corruption, which is common in many practical applications. In this study, we develop a new variant of the randomized sparse Kaczmarz method with linear convergence guarantees, by making use of the quantile technique to detect corruptions. Moreover, we incorporate the averaged block technique into the proposed method to achieve parallel computation and acceleration. Finally, the proposed algorithms are illustrated to be very efficient through extensive numerical experiments.