Search

In this study, we extended the applicability of a derivative-free algorithm to encompass the solution of operators that may be either differentiable or non-differentiable. Conditions weaker than the ones in earlier studies are employed for the convergence analysis. The earlier results considered assumptions up to the existence of the ninth order derivative of the main operator, even though there are no derivatives in the algorithm, and the Taylor series on the finite Euclidian space restricts the applicability of the algorithm. Moreover, the previous results could not be used for non-differentiable equations, although the algorithm could converge. The new local result used only conditions on the divided difference in the algorithm to show the convergence. Moreover, the more challenging semi-local convergence that had not previously been studied was considered using majorizing sequences. The paper included results on the upper bounds of the error estimates and domains where there was only one solution for the equation. The methodology of this paper is applicable to other algorithms using inverses and in the setting of a Banach space. Numerical examples further validate our approach. [ABSTRACT FROM AUTHOR]

In this paper, based on the Lagrange multiplier method, we construct a maximum principle preserving scheme for the time-fractional Allen-Cahn equation of 2- α (0 < α < 1) order. The correction energy of this scheme is increased by a term compared to the original energy, which is O (τ α). We prove that our scheme is unconditionally stable related to the corrected energy and verify the convergence, maximum principle, and energy conservation properties of the algorithm through numerical examples. We also find that the larger the α , the faster the evolution of the time-fractional Allen-Cahn equation. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to consider a discrete data assimilation algorithm for the three dimensional planetary geostrophic equations of large-scale ocean circulation in the case that the observable measurements, obtained discretely in time, may be contaminated by systematic errors, which works for a general class of observable measurements, such as low Fourier modes and local spatial averages over finite volume elements. We will provide some suitable conditions to establish asymptotic in time estimates of the difference between the approximating solution and the unknown exact (reference) solution in some appropriate norms for these two different kinds of interpolation operators, which also shows that the approximation solution of the proposed discrete data assimilation algorithm will convergent to the unique unknown exact (reference) solution of the original system at an exponential rate, asymptotically in time if the observational measurements are free of error. [ABSTRACT FROM AUTHOR]

In this paper we provide a tetrahedra-free algorithm to compute low-cardinality quadrature rules with a given degree of polynomial exactness, positive weights and interior nodes on a polyhedral element with arbitrary shape. The key tools are the notion of Tchakaloff discretization set and the solution of moment-matching equations by Lawson-Hanson iterations for NonNegative Least-Squares. Several numerical tests are presented. The method is implemented in Matlab as open-source software. [ABSTRACT FROM AUTHOR]

In this paper, we propose a sparse least squares (SLS) optimization model for solving multilinear equations, in which the sparsity constraint on the solutions can effectively reduce storage and computation costs. By employing variational properties of the sparsity set, along with differentiation properties of the objective function in the SLS model, the first-order optimality conditions are analysed in terms of the stationary points. Based on the equivalent characterization of the stationary points, we propose the Newton Hard-Threshold Pursuit (NHTP) algorithm and establish its locally quadratic convergence under some regularity conditions. Numerical experiments conducted on simulated datasets including cases of Completely Positive(CP)-tensors and symmetric strong M-tensors illustrate the efficiency of our proposed NHTP method. [ABSTRACT FROM AUTHOR]

This paper investigates the convergence behavior of the Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for time-fractional sub-diffusion and diffusion-wave equations. The algorithms are applied to regular domains in 1D and 2D for multiple subdomains, and the impact of different constant values of the generalized diffusion coefficient on the algorithms' convergence is analyzed. The convergence rate of the algorithms is analyzed as the fractional order of the time derivative changes. The paper demonstrates that the algorithms exhibit slow superlinear convergence when the fractional order is close to zero, almost finite step convergence (exact finite step convergence for wave case) when the order approaches two, and faster superlinear convergence as the fractional order increases in between. The transitional nature of the algorithms' behavior is effectively captured through estimates with changes in the fractional order, and the results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we considered the existence of ground state solutions to the following Choquard equation { − Δ u = λ u + (I α ∗ F (u)) f (u) + μ | u | q − 2 u in R N , ∫ R N | u | 2 d x = a > 0 , where N ≥ 3 , I α is the Riesz potential of order α ∈ (0 , N) , 2 < q ≤ 2 + 4 N , μ > 0 and λ ∈ R is a Lagrange multiplier. Under general assumptions on F ∈ C 1 (R , R) , for a L 2 -subcritical and L 2 -critical of perturbation μ | u | q − 2 u , we established several existence or nonexistence results about the normalized ground state solutions. [ABSTRACT FROM AUTHOR]

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the nonlinear fractional pseudo‐parabolic equations (FPPEs). We propose an accurate numerical algorithm for solving the aforementioned well‐known equation. The problem is discretized in the temporal direction by utilizing a graded mesh linearized scheme and in the spatial direction by the Galerkin‐spectral scheme. We investigate the stability conditions of the proposed scheme. We also provide an H1$$ {H}^1 $$ error estimate of the proposed approach to demonstrate that it is convergent with temporal second‐order accuracy for fitted grading parameters. The proposed scheme is also extended to tackle coupled FPPEs. Numerical experiments are provided to validate the accuracy and reliability of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, a modified two-grid algorithm based on block-centred finite difference method is developed for the fourth-order nonlinear extended Fisher–Kolmogorov equation. To further improve the computational efficiency, an effective second-order accurate backward difference formula is considered. The modified two-grid method based on Newton iteration is constructed to linearize the nonlinear system. The method solves a miniature nonlinear system on a coarse grid accompanying a larger time step to get the numerical solution, then computes a linear system constructed by the previous result with the Taylor expansion on a fine grid accompanying a smaller time step to get the correct numerical solution. Theoretical analysis shows that the modified two-grid algorithm can achieve second-order convergence accuracy both in time and space domain. Several numerical experiments are provided to verify the theoretical result and the high efficiency of this approach. The practical problem illustrates the actual applicable value of the algorithm. [ABSTRACT FROM AUTHOR]

Structure-preserving doubling algorithms (SDAs) are efficient algorithms for solving Riccati-type matrix equations. However, breakdowns may occur in SDAs. To remedy this drawback, in this paper, we first introduce Ω -symplectic forms (Ω -SFs), consisting of symplectic matrix pairs with a Hermitian parametric matrix Ω. Based on Ω -SFs, we develop modified SDAs (MSDAs) for solving the associated Riccati-type equations. MSDAs generate sequences of symplectic matrix pairs in Ω -SFs and prevent breakdowns by employing a reasonably selected Hermitian matrix Ω. In practical implementations, we show that the Hermitian matrix Ω in MSDAs can be chosen as a real diagonal matrix that can reduce the computational complexity. The numerical results demonstrate a significant improvement in the accuracy of the solutions by MSDAs. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-grid lumped mass finite element algorithm for 2D Allen-Cahn equations, where Crank-Nicolson scheme and piecewise linear element are utilized for temporal and spatial discretization, respectively. Both the maximum-norm boundedness and H 1 -norm error estimates of the proposed two-grid scheme are discussed. Finally, a numerical example is given to verify the theoretical results. By comparing with the finite element scheme and the two-grid finite element scheme, we found that our scheme has a great advantage in calculation time. [ABSTRACT FROM AUTHOR]

This paper considers a high-order numerical method for a computed solution of multidimensional convection-diffusion-reaction equation with time-fractional derivative subjected to appropriate initial and boundary conditions. The stability and error estimates of the proposed numerical approach are analyzed using the L ∞ (0 , T ; L 2) -norm. The theoretical study suggests that the new technique is unconditionally stable and temporal accurate with order O(τ2+α), where τ denotes the time step and 0 < α < 1. This result shows that the developed algorithm is faster and more efficient than a broad range of numerical techniques widely studied in the literature for the considered problem. Numerical experiments confirm the theory and they indicate that the proposed numerical scheme converges with accuracy O(τ2+α + h4), where h represents the space step. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a second-order partitioned method with multiple-time-step technique for the nonstationary Stokes-Darcy model. This method allows different time steps in different subdomains and improves the accuracy by the time filters. Besides, by designing new error estimate and time step adjustment strategy, we extend this method to variable timestep and develop single and double adaptive algorithms. Constant and variable time step tests are given to confirm the theoretical analysis and illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, septic B-spline approach with linearization and without linearization technique with the help of Butcher's fifth-order Runge–Kutta scheme is implemented to obtain solitary wave solutions of the modified regularized long wave equation. The error norms L 2 and L ∞ and the three invariants I 1 , I 2 and I 3 are computed on three tests to confirm about the efficiency and accuracy of the proposed methods. The stability analysis is performed using Von-Neumann technique on the linearized septic B-spline approach. The results of the present work are compared with the analytical and existing solutions. [ABSTRACT FROM AUTHOR]

Node elimination is a numerical approach for obtaining cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by iteratively solving the moment equations. This paper introduces a new criterion for selecting which nodes to eliminate that is based on a linearization of the moment equation. In addition, a penalized iterative solver is introduced, that ensures that weights are positive and nodes are inside the integration domain. A strategy for constructing an initial cubature rule for various polytopes in several space dimensions is described. High efficiency rules are presented for two, three and four dimensional polytopes. The new rules are compared with rules that are obtained by combining tensor products of one dimensional quadrature rules and domain transformations, as well as with known analytically constructed cubature rules. [ABSTRACT FROM AUTHOR]

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston's gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a boundary-parabolic PSL (2 , ℂ) -representation to boundary-unipotent SL (2 , ℂ) -representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast Alikhanov algorithm with nonuniform time steps for a two dimensional distributed‐order time–space fractional advection–dispersion equation. First, an efficient fast Alikhanov algorithm on the general nonuniform time steps for the evaluation of Caputo fractional derivative is presented to sharply reduce the computational work and storage, and are applied to the distributed‐order time fractional derivative or multi‐term time fractional derivative under the nonsmooth regularity assumptions. And a generalized discrete fractional Grönwall inequality is extended to multi‐term fractional derivative or distributed‐order fractional derivative for analyzing theoretically our algorithm. Then the stability and convergence of time semi‐discrete scheme are investigated. Furthermore, we derive the corresponding fully discrete scheme by finite element method and discuss its convergence. At last, the given numerical examples adequately confirm the correctness of theoretical analysis and compare the computing effectiveness between the fast algorithm and the direct method. [ABSTRACT FROM AUTHOR]

Three-dimensional (3D) target localization by using hybrid time difference of arrival (TDOA) and angle of arrival (AOA) measurements from multiple sensors has been an active research area for several decades due to its extensive applications in various fields. For this nonlinear estimation problem, the pseudolinear system of equations constructed by using the measurements generally acts as the basis of numerous localization algorithms. In this paper, we aim to improve the performance of 3D TDOA–AOA localization by introducing the constrained total least squares (CTLS) framework wherein the inherent characteristics of the pseudolinear equations can be properly taken into consideration. On the basis of the total least squares model, the CTLS model for 3D TDOA–AOA localization is established by imposing the inherent characteristics of the pseudolinear equations as additional constraints. Then, the multi-constraint optimization problem in CTLS model is solved by using an iterative algorithm based on successive projections. Extensive numerical simulations are accomplished for evaluating the performance of the proposed CTLS algorithm. The results show that the proposed algorithm gives moderate accuracy enhancement with acceptable computational cost, and more importantly, it is more robust to large measurement noise than the compared algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a high order time discretization method by combining the time splitting scheme with semi-implicit spectral deferred correction method for the space-fractional Swift-Hohenberg equation (SFSH). Based on the operator splitting method, the original problem is split into linear and nonlinear subproblems, respectively. The Fourier spectral method is adopted for the linear part, and a first-order accurate finite difference scheme in time together with the Fourier spectral method in space is used for the nonlinear part. The stability and convergence of the obtained numerical scheme are analyzed theoretically. Moreover, the spectral deferred correction (SDC) method is then employed to improve the temporal accuracy. Various two and three dimensional numerical experiments are performed to validate the theoretical results and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, in order to penalize for lack of divergence-free solution, we propose a sparse grad-div stabilized algorithm for the incompressible magnetohydrodynamics equations, which just adds a minimally intrusive module that implements grad-div stabilization with a sparse block structure matrix. Unconditional stability and error estimates of the proposed algorithm are provided and numerical tests are carried out. Compared to other grad-div stabilizations, the sparse grad-div stabilized algorithm is more efficient with some large values of grad-div parameters. [ABSTRACT FROM AUTHOR]

A modified-upwind with block-centred finite difference scheme on the basis of the two-grid algorithm is presented for the convection-diffusion-reaction equations. This scheme can keep second-order accuracy in spatial mesh sizes for both state variables and fluxes in the convection–diffusion–reaction problem. Moreover, the two-grid algorithm is constructed in order to solve semilinear convection-dominated problems efficiently, in which the main idea is to settle an original semilinear equation on the coarse space, and next to settle a linearized equation on the fine space. The error estimate of the method proposed in this paper is given through theoretical analysis. It is indicated that the two-grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O (h 1 / 2). Thus, solving such a large-scale nonlinear problem is as easy as linearized problems. Besides, there are some numerical experiments to corroborate in practice that the algorithm is effective and robust to solve convection–diffusion–reaction problems. [ABSTRACT FROM AUTHOR]

Time-variant Yang-Baxter-like matrix equation (YBLME) with the disturbances of noises is the hotspot in various scientific disciplines. The existing research, either can not handle the noise, or rely on the build-in numerical algorithm provided by MATLAB. Therefore, it is necessary to further study the digital-computer discrete solution of this problem. In this paper, the authors propose a noise-handling ten-instant discrete solution for the time-variant YBLME. By defining the matrix-form error function, Zhang neural network (ZNN) is exploited to design the continuous-time noise-handling ZNN model. For potential digital hardware (i.e., digital computer) realization, a discrete-time ZNN-based noise-handling ten-instant (DTZNH10i) algorithm, which is capable of handling three types of noises, is proposed on the basis of a Zhang time discretization (ZTD) formula. The convergence and precision of the proposed DTZNH10i algorithm are investigated and discussed. Theoretical analyses show the correctness and effectiveness of the proposed algorithm. The paper offers three examples of time-variant YBLME with disturbances of constant, linear, and bounded random noises to illustrate the efficacy and convergence of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study three different types of algebraic symbolic 2-plithogenic equations. The symbolic 2-plithogenic linear Diophantine equations, symbolic 2-plithogenic quadratic equations, and linear system of symbolic 2-plithgenic equations will be discussed and handled, where algorithms to solve the previous types will be presented and proved by transforming them to classical algebraic systems of equations. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimally accurate second-order time-stepping algorithm for nonstationary magneto-hydrodynamics equations. The algorithm deals with a linear treatment for the nonlinear terms in both momentum equations and magnetic equations, and adds some stabilization terms of the discrete solutions for the velocity, the pressure and the magnetic. We derive that this algorithm satisfies unconditionally stable and optimally accurate error estimate. Numerical tests are given to validate the predicted convergence rate. [ABSTRACT FROM AUTHOR]

In this paper, a new real representation of reduced biquaternion matrix is proposed, and the solutions of the reduced biquaternion matrix equations XF−AX=BY$$ XF- AX= BY $$ and XF−AX˜=BY$$ XF-A\tilde{X}= BY $$ are solved by means of this method. The corresponding numerical algorithm is provided, and the effectiveness of this method is verified by numerical examples. [ABSTRACT FROM AUTHOR]

This paper presents a new numerical algorithm dealing with multi‐term time‐fractional Burgers‐type equation involving the Caputo derivative. The proposed method consists of temporal discretization of L2$$ L2 $$ formula and spatial discretization using the exponential B‐splines. The semi implicit approach is applied to discretize the nonlinear term u∂xu$$ u{\partial}_{\mathtt{x}}u $$. We adopt the Von–Neumann method to study stability. We also establish the convergence analysis. The proposed method is employed to solve a few numerical examples in order to test its efficiency and accuracy. Comparisons with the recent works confirm the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with a numerical spectral solution to a one-dimensional linear telegraph type equation with constant coefficients. An efficient Galerkin algorithm is implemented and analyzed for treating this type of equations. The philosophy of utilization of the Galerkin method is built on picking basis functions that are consistent with the corresponding boundary conditions of the telegraph type equation. A suitable combination of the orthogonal shifted Gegenbauer polynomials is utilized. The proposed method produces systems of especially inverted matrices. Furthermore, the convergence and error analysis of the proposed expansion are investigated. This study was built on assuming that the solution to the problem is separable. The paper ends by checking the applicability and effectiveness of the proposed algorithm by solving some numerical examples. [ABSTRACT FROM AUTHOR]

This paper revisits the study of optical solitons that is governed by one of the three forms of derivative nonlinear Schrödinger's equation that is also known as Chen–Lee–Liu model. This model is investigated by the aid of fully shifted Jacobi's collocation method with two independent approaches. The first is discretization of the spatial variable, while the other is discretization of the temporal variable. It is concluded that the method of the current paper is far more efficient and reliable for the considered model. Numerical results illustrate the performance efficiency of the algorithm. The results also point out that the scheme can lead to spectral accuracy of the studied model. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a quantum decomposition algorithm (QDA) that decomposes the problem ∂ ρ ∂ t = ℒ ρ = λ ρ into a summation of eigenvalues times phase–space variables. One interesting feature of QDA stems from its ability to simulate damped spin systems by means of pure quantum harmonic oscillators adjusted with the eigenvalues of the original eigenvalue problem. We test the proposed algorithm in the case of undriven qubit with spontaneous emission and dephasing. [ABSTRACT FROM AUTHOR]

This paper studies the problem of target control and how a virtual ellipsoid can avoid the static obstacle. During the motion to the target set, the virtual ellipsoid can achieve a motion under collision avoidance by keeping the distance between the ellipsoid and obstacle. We present solutions to this problem in the class of closed-loop (feedback) controls based on Hamilton–Jacobi–Bellman (HJB) equation. Simulation results verify the validity and effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, applying some properties of matrix inequality and Schur complement, we give new upper and lower bounds of the solution for the unified algebraic Lyapunov equation that generalize the forms of discrete and continuous Lyapunov matrix equations. We show that its positive definite solution exists and is unique under certain conditions. Meanwhile, we present three numerical algorithms, including fixed point iterative method, the acceleration fixed point method and the alternating direction implicit method, to solve the unified algebraic Lyapunov equation. The convergence analysis of these algorithms is discussed. Finally, some numerical examples are presented to verify the feasibility of the derived upper and lower bounds, and numerical algorithms. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to solve the viscous Cahn-Hilliard equation via a fast algorithm based on the two time-mesh (TT-M) finite element (FE) method to ease the problem caused by strong nonlinearities. The TT-M FE algorithm includes the following main computing steps. First, a nonlinear FE method is applied on a coarse time-mesh τc. Here, the FE method is used for spatial discretization and the implicit second-order θ scheme (containing both implicit Crank-Nicolson and second-order backward difference) is used for temporal discretization. Second, based on the chosen initial iterative value, a linearized FE system on time fine mesh is solved, where some useful coarse numerical solutions are found by Lagrange's interpolation formula. The analysis for both stability and a priori error estimates is made in detail. Numerical examples are given to demonstrate the validity of the proposed algorithm. Our algorithm is compared with the traditional Galerkin FE method and it is evident that our fast algorithm can save computational time. [ABSTRACT FROM AUTHOR]

The use of mathematical structures as a design tool by Luigi Moretti, the first theorist of Architettura Parametrica, is an unexplored theme even though he repeatedly stressed their importance in his writings. Moretti proposed that bringing design thinking closer to mathematical formalisations could profoundly renew the discipline of architecture. However, he also warns that these formalisations may overwhelm architecture’s values to which a numerical dimension cannot be associated. The reconstruction and discussion of the mathematics behind his swimming stadiums featured in the 1960 Architettura Parametrica exhibition permits us to understand how Moretti intended a design process informed by formalised methods. This paper proposes a novel reading, based on unpublished archival sources and presented with both theoretical and practical approaches, of one the first applications that interweaves formal methods, mathematics, digital tools and scientific thought in architecture. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the least-squares solution with the least norm to the following system of tensor equations over quaternions A 1 ∗ N X = D 1 , Y ∗ N B 2 = D 2 , A 3 ∗ N X ∗ N B 3 = D 3 , A 4 ∗ N Y ∗ N B 4 = D 4 , A 5 ∗ N X + Y ∗ N B 5 = D 5 , where X , Y are unknown quaternion tensors and the others are given quaternion tensors. Using the expressions of the Moore-Penrose inverses of partitioned tensors, we give a representation of the solution and an algorithm to compute this solution. [ABSTRACT FROM AUTHOR]

Many of the applied algorithms that have been developed for solving a system of logical equations or the Boolean satisfiability problem have solved the problem in the Boolean domain. However, other approaches have recently been developed and improved. One of these developments is the transformation of a system of logical equations to a real continuous domain. The essence of this development is that a system of logical equations is transformed into a system in a real domain and the solution is sought in a real continuous domain. A real continuous domain is a richer domain, as it involves many well-developed algorithms. In this paper, we have constructively transformed the solution of any system of logical equations with a unique solution into an optimization problem for a polylinear target function in a unit n -dimensional cube K n . The resulting polylinear target function in K n does not have a local minimum. We proved that only once by calculating the gradient of the polylinear target function at any interior point of the K n cube, we can determine the solution to the system of logical equations. [ABSTRACT FROM AUTHOR]

Generated Jacobian Equations have been introduced by Trudinger (Discrete Contin Dyn Syst A 34(4):1663–1681, 2014) as a generalization of Monge–Ampère equations arising in optimal transport. In this paper, we introduce and study a damped Newton algorithm for solving these equations in the semi-discrete setting, meaning that one of the two measures involved in the problem is finitely supported and the other one is absolutely continuous. We also present a numerical application of this algorithm to the near-field parallel reflector problem arising in non-imaging problems. [ABSTRACT FROM AUTHOR]

This paper focuses on efficient projection onto the intersection of a half-space and a box-like set and its generalized Jacobian. Based on the Lagrangian duality theory, we deal with the projection problem via a semismooth Newton algorithm with line search safeguard, which admits global and locally quadratic convergence, to solve a univariate semismooth equation. Numerical experiments show that our proposed algorithm outperforms favourably the existing state-of-the-art standard solvers and is able to reliably solve very large-scale projection problems. Besides, we derive an explicit expression of a generalized Jacobian of the studied projection, which is an essential component of second-order nonsmooth methods. [ABSTRACT FROM AUTHOR]

In connection with applications, the solution of a system of logical equations plays an important role in computational mathematics and in many other areas. As a result, many new directions and algorithms for solving systems of logical equations are being developed. One of these directions is transformation into the real continuous domain. The real continuous domain is a richer domain to work with because it features many algorithms, which are well designed. In this study, firstly, we transformed any system of logical equations in the unit n -dimensional cube K n into a system of polylinear–polynomial equations in a mathematically constructive way. Secondly, we proved that if we slightly modify the system of logical equations, namely, add no more than one special equation to the system, then the resulting system of logical equations and the corresponding system of polylinear–polynomial equations in K n + 1 is equivalent. The paper proposes an algorithm and proves its correctness. Based on these results, further research plans are developed to adapt the proposed method. [ABSTRACT FROM AUTHOR]

We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical solutions of one-dimensional (1D) and two-dimensional (2D) nonlinear coupled Schrödinger-Boussinesq equations (CSBEs) by a type of linearly energy- and mass- preserving finite difference methods (EMP-FDMs) because the existing EMP-FDMs for CSBEs are nonlinear and time-consuming, and corresponding theoretical analyses are not easy to generalize high-dimensional problems. Firstly, a linearized EMP-FDM is created for solving 1D CSBEs. By using the discrete energy analysis method, it is shown that this scheme is uniquely solvable and convergent with an order of O (Δ t 2 + h x 2) in L ∞ -, H 1 - and L 2 -norms, and corresponding numerical energy and mass are conservative. Then, by generalizing this type of EMP-FDM, a linearized EMP-FDM is developed for solving 2D CSBEs. Theoretical findings including the convergence, the discrete conservative laws, and the solvability of this numerical algorithm are strictly derived in detail by using the discrete energy analysis method as well. Finally, numerical results confirm the efficiency of our algorithms and the exactness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the kinetic description of the first-order Cucker-Smale (CS) flocking model on the real line. We reveal the equivalent relation between the measure-valued solution to the first- and second-order kinetic CS equations. The emergent behavior of the first-order kinetic CS equation and the characterization of the asymptotic solution are studied. We also provide the equivalent relation between classical/measure-valued solutions to the first-order kinetic CS equation and a classical solution to the second-order hydrodynamic CS equations, and present the corresponding analysis on the large-time behavior. The numerical experiments support our analysis and provide an efficient algorithm to obtain the asymptotic solution without simulating the model for a long time. [ABSTRACT FROM AUTHOR]

In this paper, a second-order hyperbolic equation is solved by a two-grid algorithm combined with the expanded mixed finite element method. The error estimate of the expanded mixed finite element method with discrete-time scheme is demonstrated. Moreover, we present a two-grid method and analyze its convergence. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O (h 1 2) . Finally, some numerical experiments are provided to illustrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with a fast finite element method for the three-dimensional Poisson equation in infinite domains. Both the exterior problem and the strip-tail problem are considered. Exact Dirichlet-to-Neumann (DtN)-type artificial boundary conditions (ABCs) are derived to reduce the original infinite-domain problems to suitable truncated-domain problems. Based on the best relative Chebyshev approximation for the square-root function, a fast algorithm is developed to approximate exact ABCs. One remarkable advantage is that one need not compute the full eigensystem associated with the surface Laplacian operator on artificial boundaries. In addition, compared with the modal expansion method and the method based on Pad |$\acute{\textrm{e}}$| approximation for the square-root function, the computational cost of the DtN mapping is further reduced. An error analysis is performed and numerical examples are presented to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Number theory is concerned with properties of integers and Diophantine equations. The objective of this paper is dedicated to introduce the basic concepts in refined neutrosophic number theory such as division, divisors, congruencies, and Pell's equation in the refined neutrosophic ring of integers Z(I1, I2). Also, algorithms to solve refined neutrosophic linear congruencies and refined neutrosophic Pell's equation will be presented and discussed. [ABSTRACT FROM AUTHOR]