Search

This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples. [ABSTRACT FROM AUTHOR]

As we know that, the parameterized Uzawa (PU) method can be very efficient when used to solve the standard saddle point problem, especially, when we have good and accurate estimation of preconditioned Schur complement matrix. In this paper, by taking full use of the special structure of coefficient matrix arising from complex Helmholtz equations, two types of optimized PU (OPU) methods are discussed theoretically and experimentally. Specifically, the convergence factors of these two OPU methods are less than 0.172, and the optimal result of first OPU method can reach 0.0396 with σ 1 ≥ σ 2 > 0 , which is currently the best theoretical result in the literatures. Moreover, the second OPU method has better computational advantages compared with the first OPU method as it avoids the inverse computation of W at each step, indicating the less CPU time will be costed by the second OPU method. In addition, the optimal parameters involved in our algorithms consist of constants, reducing the computational complexity associated with parameter selection. Finally, numerical results are given, not only show the effectiveness of OPU methods, but also confirm the rationality of theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider and analyze the stability of three implicit/explicit (IMEX) schemes for the time-dependent natural convection problem, these considered numerical schemes contain the first order backward Euler scheme, second order Crank-Nicolson IMEX scheme and BDF2-AB2 combination. All numerical schemes deal with the linear terms implicitly and the nonlinear terms explicitly. Then the original nonlinear problem is split into two linearized subproblems with constant coefficient matrixes, which reduces the computational complexity and saves a lot of storage. The biggest feature of this paper is to establish the unconditional stability of three IMEX schemes for incompressible flows, which improves and supplements the published theoretical findings. Finally, some numerical examples are provided to show the performances of the considered numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, several local and parallel finite element algorithms are proposed and analyzed for the 2D/3D stationary inductionless incompressible magnetohydrodynamic (MHD) equations. The core concept is to guarantee the charge-conservation property by choosing mixed finite element spaces in H 0 (div , Ω) × L 0 2 (Ω) to approximate (J , ϕ) , meantime combining the idea of domain decomposition method to realize parallel operation. The characteristic of the proposed algorithms is that the computational complexity is greatly reduced while ensuring the accuracy of the numerical simulation. With the local a prior estimate as the technical means of theoretical analysis, we give the error estimates of the algorithms. Finally, several numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an efficient dimension splitting method is applied to the two-dimensional (2D) and three-dimensional (3D) Cahn-Hilliard type equations with significant applications in physical, biological and computer science. The proposed method can greatly reduce the storage requirements and computational complexity in two and three dimensions, and preserve the high precision and mass conservation. The dimension splitting method is based on auxiliary variables for spatial derivative terms and the operator splitting approach. It converts the 2D or 3D problem into a series of one-dimensional (1D) problems which can be solved by using multi-thread. The stability and accuracy of the proposed method are improved by a local stabilizing approach and extrapolation. The analyses of discrete energy and discrete mass conservation property are shown. Numerical examples confirm the advantages of proposed method. And a variety of simulations, such as the phase separations, curvature driven flows, diblock copolymers, tumor growth and volume restoration, are performed with respect to practical applications. • Dimension splitting method for Cahn-Hilliard type equations. • The global stability is achieved by the local stability of solving each sub-problem. • Fourth-order mass conservative scheme based on the compact differencing and extrapolation. • A large number of numerical examples for practical applications. [ABSTRACT FROM AUTHOR]

In this paper, the unconditionally optimal H 1 -norm error estimate for a two-step fast L1 scheme is considered for a nonlinear subdiffusion equation, which involves the considerations of both the initial singularity of the solution and the computational complexity to evaluate the Caputo derivative. For the fully discrete scheme, we use the fast L1 scheme developed in [28] to speed up the evaluation of the Caputo derivative on nonuniform time step, employ the Galerkin finite element (FE) method to discretize the spatial direction, and take the Newton linearized method to approximate the nonlinear term. While the theory of the L 2 -norm error estimate is well studied, there are few works on the unconditional convergence of H 1 -norm error estimate of the linearized Galerkin FE scheme for nonlinear subdiffusion equations. The main reason is that it requires to find a suitable test function in the FE space. To this end, we here respectively take the time-discrete operator and Laplace operator as the suitable test function in the FE space, and combine an improved discrete fractional Grönwall inequality to achieve the unconditional error estimate in H 1 -norm. Numerical results are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

The framework of this paper is the improvement of direct-forcing immersed boundary methods in presence of moving obstacles. In particular, motivations for the use of the Direct Forcing (DF) method can be found in the advantage of a fixed computational mesh for fluid–structure interaction problems. Unfortunately, the direct forcing approach suffers a serious drawback in case of moving obstacles: the well known spurious force oscillations (SFOs). In this paper, we strengthen previous analyses of the origin of the SFO through a rigorous numerical evaluation based on Taylor expansions. We propose a remedy through an easy-to-implement regularization process (regularized DF). Formally, this regularization is related to the blending of the Navier–Stokes solver with the interpolation, but no modification of the numerical scheme is needed. This approach significantly cuts off the SFOs without increasing the computational cost. The accuracy and the space convergence order of the standard DF method are conserved. This is illustrated on numerical and physical validation test cases ranging from the Taylor–Couette problem to a cylinder with an imposed sinusoidal motion subjected to a cross-flow. [ABSTRACT FROM AUTHOR]

A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths which evolve through the indices of the matrix according to a suitable probability law. The computational complexity is proved in this paper to be significantly better than the classical Monte Carlo method, which allows the computation of much more accurate solutions. Furthermore, the positive features of the algorithm in terms of parallelism were exploited in practice to develop a highly scalable implementation capable of solving some test problems very efficiently using high performance supercomputers equipped with a large number of cores. For the specific case of shared memory architectures the performance of the algorithm was compared with the results obtained using an available Krylov-based algorithm, outperforming the latter in all benchmarks analyzed so far. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a fast singular boundary method (SBM) for three-dimensional (3D) Helmholtz equation. The SBM is a boundary-type meshless method which incorporates the advantages of the boundary element method (BEM) and the method of fundamental solutions (MFS). It is easy-to-program, and attractive to the problems with complex geometries. However, the SBM is usually limited to small-scale problems, because of the operation count of O (N 3) with direct solvers or O (N 2) with iterative solvers, as well as the memory requirement of O (N 2). To overcome this drawback, this study makes the first attempt to employ the precorrected-FFT (PFFT) to accelerate the SBM matrix–vector multiplication at each iteration step of the GMRES for 3D Helmholtz equation. Consequently, the computational complexity can be reduced from O (N 2) to O (N l o g N) or O (N). Three numerical examples are successfully tested on a desktop computer. The results clearly demonstrate the accuracy and efficiency of the developed fast PFFT-SBM strategy. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we have designed a simple method for the inverse scattering problem of stratified ocean waveguide. The simple approach applies the bisection technique in parallel to estimate the boundaries of inhomogeneous media from the received partial data. In addition, the algorithm only involves the matrix–vector operations and possesses the optimal computation complexity in both two and three dimensions. In practice, it is easy to carry out, robust against noise and capable of reconstructing the penetrable obstacles of different locations and shapes. We can consider the simple method as a direct and straightforward process to supply satisfactory initial positions for the implementation of any existing more refined but computationally more demanding techniques to achieve accurate reconstructions of physical features of scatterers. [ABSTRACT FROM AUTHOR]

For the dynamic load identification for stochastic structures, ill-posedness and randomness are main causes that lead to instability and low accuracy. Monte-Carlo simulation (MCS) method is a robust and effective random simulation technique for the dynamic load identification problems of stochastic structures. However, it needs large computational cost and is also inefficient for practical engineering applications because of the requirement of a large quantity of samples. In order to improve its computational efficiency, this paper proposes a novel computational algorithm for the dynamic load identification of stochastic structures. First, the newly developed algorithm transforms dynamic load identification problems for stochastic structures into equivalent deterministic dynamic load identification problems. Second, a new regularization method is proposed to realize the deterministic dynamic load identification. Third, the assessments of the statistics of identified loads are obtained based on statistical theory. Finally, the stability and robustness of the proposed algorithm are well validated by two engineering examples. It is demonstrated that the newly developed regularization method outperforms the traditional Tikhonov regularization method in computational accuracy. Moreover, the newly proposed algorithm can significantly improve the computational efficiency of MCS and is very stable and effective in solving the dynamic load identification for stochastic structures. [ABSTRACT FROM AUTHOR]

In this paper, linearly implicit predictor–corrector methods are proposed for solving space-fractional reaction–diffusion equations with non-smooth initial data. The methods are based on Matrix Transfer Technique for spatial discretization and are shown to be unconditionally stable. It is observed that the linearly implicit predictor–corrector method derived by using (1,1)-Padé approximation to matrix exponential function incurs oscillatory behavior for some time steps. These oscillations are due to high frequency components present in the solution and are diminished as the order of the space-fractional derivative decreases (slow diffusion). We present a priori reliability constraint to avoid these unwanted oscillations and generalize the constraints for all ( m , m ) -Padé approximants, m ∈ Z + , to the matrix exponential functions. These time stepping constraints are seen to be dependent on the order of the space-fractional derivative. The linearly implicit predictor–corrector method based on the (0,2)-Padé approximations to the matrix exponential function is shown to be oscillation-free for any time step. Error estimates are obtained for the methods and are theoretically shown to be second-order convergent. Computational complexity of the algorithms is discussed for solving multidimensional space-fractional reaction–diffusion systems. Several numerical experiments are performed to support our theoretical observations and to show the effectiveness, reliability, and efficiency of the methods. [ABSTRACT FROM AUTHOR]

We study the approximation of determinant for large scale matrices with low computational complexity. This paper develops a generalized stochastic polynomial approximation frame as well as a stochastic Legendre approximation algorithm to calculate log-determinants of large-scale positive definite matrices based on the prior eigenvalue distributions. The generalized frame is implemented by weighted L 2 orthogonal polynomial expansions with an efficient recursion formula and matrix–vector multiplications. So the proposed scheme is efficient both in computational complexity and data storage. Respective error bounds are given in theory which guarantee the convergence of the proposed algorithms. We illustrate the effectiveness of our method by numerical experiments on both synthetic matrices and counting spanning trees. [ABSTRACT FROM AUTHOR]

This paper presents a deep analysis of a time-dependent Schrödinger equation with fractional time derivative. After the discretization of the spatial operator the equation is reformulated in terms of a special system of fractional differential equations; it occurs that the eigenvalues of the coefficient matrix lay on the boundary of the stability region of fractional differential equations. The main difficulties in solving this system are hence related to the simultaneous presence of persisting oscillations (possibly with high frequency as it is typical with Schrödinger equations) and a persisting memory (as a consequence of the fractional order); moreover, an accurate spatial discretization gives rise to systems of large to very large size, involving a noteworthy computational complexity. By means of a theoretical analysis the exact solution is split into two or three terms (depending on the order of the fractional derivative), thus to face the numerical computation by different and suitably selected methods: direct evaluation of matrix functions for the terms characterized by smooth behaviour but with persistent memory and a step-by-step strategy, in conjunction with matrix function, for the oscillating term. In both cases, Krylov subspace methods are employed for the computation of matrix functions and convergence results are presented. [ABSTRACT FROM AUTHOR]

This paper focuses on providing the computation methods for the backward time tempered fractional Feynman–Kac equation, being one of the models recently proposed in Wu et al. (2016). The discretization for the tempered fractional substantial derivative is derived, and the corresponding finite difference and finite element schemes are designed with well established stability and convergence. The performed numerical experiments show the effectiveness of the presented schemes. [ABSTRACT FROM AUTHOR]

An enhanced version of a stochastic SParse Approximate Inverse (SPAI) preconditioner for general matrices is presented in this paper. This method is used in contrast to the standard deterministic preconditioners computed by the Modified SParse Approximate Inverse Preconditioner (MSPAI). Thus we present a Monte Carlo preconditioner that relies on the use of Markov Chain Monte Carlo (MCMC) methods to compute a rough approximate matrix inverse first, which can further be optimized by an iterative filter process and a parallel refinement, to enhance the accuracy of the inverse and the preconditioner respectively. The advantage of the proposed approach is that finding the sparse Monte Carlo matrix inversion has a computational complexity linear of the size of the matrix, it is inherently parallel and thus can be obtained very efficiently for large matrices and can be used also as an efficient preconditioner while solving systems of linear algebraic equations. The behaviour of the proposed algorithm is studied and its performance measured, evaluated and compared with MSPAI. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel variational model for salt and pepper noise removal, and an efficient numerical algorithm for solving it. The proposed model features the use of an approximating function of l 0 norm to measure the closeness of the reconstructed and observed images at the pixels which are not the candidates of the noisy pixels. In addition, the total variation (TV) of the image on the entire image domain is minimized for edge-preserving smoothing. When solving the proposed minimization problem, to reduce the computational complexity from the expression of the approximating function, we use the dual forms of both TV and data terms, and find the solution of the corresponding primal–dual problem. Numerous experiments on real images, and comparisons with TV-L2, TV-L1, and adaptive median filter (AMF) indicate the effectiveness and robustness of the proposed method in salt and pepper noise removal. [ABSTRACT FROM AUTHOR]

In this paper, we present two novel symbolic computational algorithms to solve periodic pentadiagonal (PP) linear systems. These two algorithms are based on a special matrix decomposition and the use of any fast pentadiagonal linear solver, respectively. Some numerical examples are given in order to demonstrate the performance of the proposed algorithms and their competitiveness with existing algorithms. All of the experiments are performed on a computer workstation with the aid of programs written in MATLAB. [ABSTRACT FROM AUTHOR]

In this paper, we focus on efficient second-order in time approximations of the Allen–Cahn and Cahn–Hilliard equations. First of all, we present the equations, generic second-order schemes (based on a mid-point approximation of the diffusion term) and some schemes already introduced in the literature. Then, we propose new ways of deriving second-order in time approximations of the potential term (starting from the main schemes introduced in Guillén-González and Tierra (2013)), yielding to new second-order schemes. For these schemes and other second-order schemes previously introduced in the literature, we study the constraints on the physical and discrete parameters that can appear to assure the energy-stability, unique solvability and, in the case of nonlinear schemes, the convergence of Newton’s method to the nonlinear schemes. Moreover, in order to save computational cost we have developed a new adaptive time-stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step. Finally, we compare the behaviour of the schemes and the effectiveness of the adaptive time-stepping algorithm through several computational experiments. [ABSTRACT FROM AUTHOR]