Showing papers in "Journal of Computational Mathematics in 2010"
TL;DR: Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.
Abstract: We present a Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. The unconditional convergence of the HSS iteration method is proved and an upper bound on the convergence rate is derived. Moreover, to reduce the computing cost, we establish an inexact variant of the HSS iteration method and analyze its convergence property in detail. Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.
137 citations
TL;DR: The original fixed-point continuation algorithm is extended to obtain better practical results, derive appropriate choices forM and under a given measurement model, and present numerical results for a variety of compressed sensing problems.
Abstract: Fixed-point continuation (FPC) is an approach, based on operator-splitting and continuation, for solving minimization problems with ‘1-regularization: minkxk1 + f (x): We investigate the application of this algorithm to compressed sensing signal recovery, in which f(x) = 1 kAx bk 2 , A2 R m n and m n. In particular, we extend the original algorithm to obtain better practical results, derive appropriate choices forM and under a given measurement model, and present numerical results for a variety of compressed sensing problems. The numerical results show that the performance of our algorithm compares favorably with that of several recently proposed algorithms.
62 citations
TL;DR: The first algorithm is a combination of the previous frame-based deconvolution algorithm and the iterative thresholding algorithm given by [14, 16] and the strong convergence of the algorithms in infinite dimensional settings is given by employing proximal forward-backward splitting (PFBS) method.
Abstract: In this paper, two framelet based deconvolution algorithms are proposed. The basic idea of framelet based approach is to convert the deconvolution problem to the problem of inpainting in a frame domain by constructing a framelet system with one of the masks being the given (discrete) convolution kernel via the unitary extension principle of [26], as introduced in [6, 7, 8, 9] . The first algorithm unifies our previous works in high resolution image reconstruction and infra-red chopped and nodded image restoration, and the second one is a combination of our previous frame-based deconvolution algorithm and the iterative thresholding algorithm given by [14, 16]. The strong convergence of the algorithms in infinite dimensional settings is given by employing proximal forward-backward splitting (PFBS) method. Consequently, it unifies iterative algorithms of infinite and finite dimensional setting and simplifies the proof of the convergence of the algorithms of [6].
60 citations
TL;DR: The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points as mentioned in this paper.
Abstract: The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implementations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step. Mathematics subject classification: 65F10, 65W05.
59 citations
TL;DR: In this paper, the Spectral-Lagrangian scheme developed by the authors in [30] for a wide range of homogeneous nonlinear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark.
Abstract: The numerical approximation of the Spectral-Lagrangian scheme developed by the authors in [30] for a wide range of homogeneous non-linear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark. Recognizing that the Boltzmann equation is an important tool in the analysis of formation of shock and boundary layer structures, we present the computational algorithm in Section 3.3 and perform a numerical study case in shock tube geometries well modeled in for 1D in x times 3D in v in Section 4. The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Aoki et al [2], where the wall temperature is suddenly increased or decreased, is also studied. We consider the problem of heat transfer between two parallel plates with difiusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefled state. Finally, the classical inflnite shock tube problem that generates a non-moving shock wave is studied. The point worth noting in this example is that the ∞ow in the flnal case turns from a supersonic ∞ow to a subsonic ∞ow across the shock.
59 citations
TL;DR: In this paper, Chen et al. analyzed the convergence of a general recursive linearization algorithm for solving inverse medium problems with multi-frequency measurements and showed that the algorithm is convergent with error estimates.
Abstract: This paper is devoted to the mathematical analysis of a general recursive linearization algorithm for solving inverse medium problems with multi-frequency measurements. Under some reasonable assumptions, it is shown that the algorithm is convergent with error estimates. The work is motivated by our effort to analyze recent significant numerical results for solving inverse medium problems. Based on the uncertainty principle, the recursive linearization allows the nonlinear inverse problems to be reduced to a set of linear problems and be solved recursively in a proper order according to the measurements. As an application, the convergence of the recursive linearization algorithm [Chen, Inverse Problems13(1997), pp.253-282] is established for solving the acoustic inverse scattering problem.
51 citations
TL;DR: In this article, the authors proposed a new and efficient numerical method for mul- ticriterion optimal control and single criterion optimal control under in-tegral constraints, based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically.
Abstract: We introduce a new and efficient numerical method for mul- ticriterion optimal control and single criterion optimal control under in- tegral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numeri- cally. The efficiency of our approach hinges on the causality i n that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We com- pare this to a recently introduced "weighted sum" based algorithm for the same problem (25). We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.
45 citations
TL;DR: An efficient and easy-to-implement coarsening algorithm is proposed for adaptive grids obtained using the newest vertex bisection method in two dimemsions that is efficient when applied for multilevel preconditioners and mesh adaptivity for time-dependent problems.
Abstract: In this paper, an ecient and easy-to-implement coarsening algorithm is proposed for adaptive grids obtained using the newest vertex bisection method in two dimemsions. The new coarsening algorithm does not require storing the binary renement tree explicitly. Instead, the structure is implicitly contained in a special ordering of triangular elements. Numerical experiments demonstrate that the proposed coarsening algorithm is ecient when applied for multilevel preconditioners and mesh adaptivity for time-dependent problems.
42 citations
TL;DR: In this article, the authors compare 13 different a posteriori error estimators for the Poisson problem with lowest-order finite element discretization and compare them with a wide range of averaging estimators and estimators based on local problems.
Abstract: We compare 13 different a posteriori error estimators for the Poisson problem with lowest-order finite element discretization. Residual-based error estimators compete with a wide range of averaging estimators and estimators based on local problems. Among our five benchmark problems we also look on two examples with discontinuous isotropic diffusion and their impact on the performance of the estimators.
38 citations
TL;DR: This work considers a parabolic optimal control problem with pointwise state constraints that is approximated by a discrete control problem based on a discretization of the state equation by linear finite elements in space and a discontinuous Galerkin scheme in time.
Abstract: We consider a parabolic optimal control problem with pointwise state constraints. The optimization problem is approximated by a discrete control problem based on a discretization of the state equation by linear finite elements in space and a discontinuous Galerkin scheme in time. Error bounds for control and state are obtained both in two and three space dimensions. These bounds follow from uniform estimates for the discretization error of the state under natural regularity requirements.
37 citations
TL;DR: In this paper, the existence and uniqueness of eigenvalues were studied and three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM), were investigated.
Abstract: Nonlinear rank-one modiflcation of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical flber. In this paper, we flrst study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM). The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method. Mathematics subject classiflcation: 65F15, 65H17, 15A18, 35P30, 65Y20.
TL;DR: In this article, the authors proposed an efiective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations, and analyzed the convergence of the flrst-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration.
Abstract: We propose an efiective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze the convergence of the flrst-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration. In addition, we propose a flfth-order Weighted PowerENO sweeping scheme for static Hamilton-Jacobi equations with convex Hamiltonians and present numerical examples that validate the efiectiveness of the new stopping criterion.
TL;DR: In this paper, an optimization approach was proposed to solve the problem by simply minimizing a discrepancy functional, where the information contained in the wavefield should be decomposed into two parts, a "near-field" part in the region around the anomaly and a "far field" part away from the anomaly.
Abstract: In magnetic resonance elastography, one seeks to reconstruct the shear modulus from measurements of the displacement field in the whole body. In this paper, we present an optimization approach which solves the problem by simply minimizing a discrepancy functional. In order to recover a complex anomaly in a homogenous medium, we first observe that the information contained in the wavefield should be decomposed into two parts, a “near-field” part in the region around the anomaly and a “far-field” part in the region away from the anomaly. As will be justified both theoretically and numerically, separating these scales provides a local and precise reconstruction.
TL;DR: In this paper, the authors present a second order upwind scheme for computing the viscosity solution of the Eikonal equation, based on the numerical observation that classical first order monotone upwind schemes yield numerical upwind gradient which is also first order accurate up to singularities.
Abstract: We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on: 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities. 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities. Numerical examples are presented. keyword: Eikonal equation, Upwind scheme, Hamilton-Jacobi, Viscosity Solution. Sweeping method.
TL;DR: A compact fourth order finite difference scheme is proposed to discrete the cavity scattering model in the rectangular domain, and a special treatment is enforced to approximate the boundary condition, which makes truncation errors reach O(h 4 ) in the whole computational domain.
Abstract: In this paper, the electromagnetic scattering from a rectangular large open cavity embedded in an infinite ground plane is studied. By introducing a nonlocal artificial boundary condition, the scattering problem from the open cavity is reduced to a bounded domain problem. A compact fourth order finite difference scheme is then proposed to discrete the cavity scattering model in the rectangular domain, and a special treatment is enforced to approximate the boundary condition, which makes truncation errors reach O(h 4 ) in the whole computational domain. A fast algorithm, exploiting the discrete Fourier transformation in the horizontal and a Gaussian elimination in the vertical direction, is employed, which reduces the discrete system to a much smaller interface system. An effective preconditioner is presented for the BICGstab iterative solver to solve this interface system. Numerical results demonstrate the remarkable accuracy and efficiency of the proposed method. In particular, it can be used to solve the cavity model for the large wave number up to 600�.
TL;DR: Computational experiments show that the simplex algorithm using a combination of these rules turned out to be even more e‐cient.
Abstract: Recently, computational results demonstrated remarkable superiority of a so-called \largest-distance" rule and
ested pricing" rule to other major rules commonly used in practice, such as Dantzig’s original rule, the steepest-edge rule and Devex rule. Our computational experiments show that the simplex algorithm using a combination of these rules turned out to be even more e‐cient.
TL;DR: In this article, the Crank-Nicolson mixed finite element methods are developed for three most popular dispersive medium models: the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium.
Abstract: In this paper, we consider the time dependent Maxwell’s equations when dispersive media are involved. The Crank-Nicolson mixed finite element methods are developed for three most popular dispersive medium models: the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Optimal error estimates are proved for all three models solved by the Raviart-Thomas-Nedelec spaces. Extensions to multiple pole dispersive media are presented also. Mathematics subject classification: 65N30, 35L15, 78-08.
TL;DR: For shape optimization of pressure driven capillary barriers between microchannels and reservoirs, a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation steplength along the barrier path is presented.
Abstract: We will be concerned with the mathematical modeling, numerical simulation, and shape optimization of micro∞uidic biochips that are used for various biomedical applications. A particular feature is that the ∞uid ∞ow in the ∞uidic network on top of the biochips is induced by surface acoustic waves generated by interdigital transducers. We are thus faced with a multiphysics problem that will be modeled by coupling the equations of piezoelectricity with the compressible Navier-Stokes equations. Moreover, the ∞uid ∞ow exhibits a multiscale character that will be taken care of by a homogenization approach. We will discuss and analyze the mathematical models and deal with their numerical solution by space-time discretizations featuring appropriate flnite element approximations with respect to hierarchies of simplicial triangulations of the underlying computational domains. Simulation results will be given for the propagation of the surface acoustic waves on top of the piezoelectric substrate and for the induced ∞uid ∞ow in the microchannels of the ∞uidic network. The performance of the operational behavior of the biochips can be signiflcantly improved by shape optimization. In particular, for such purposes we present a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation steplength along the barrier path. As a speciflc example, we will consider the shape optimization of pressure driven capillary barriers between microchannels and reservoirs.
TL;DR: In this paper, the high-order local absorbing boundary conditions (ABCs) for heat equation were designed for a class of nonlinear PDEs and proved that the coupled system yields a stable problem between the obtained highorder local ABCs and the computational domain.
Abstract: With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions (ABCs) for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been used widely in wave propagation models only recently. We extend the spirit of the methodology to parabolic ones, which will become a basis to design the local ABCs for a class of nonlinear PDEs. Some numerical tests show that the new treatment is very efficient and tractable.
TL;DR: Stability and convergence of the schemes are proved theoretically, and numerical simulation results are provided to compare with the scheme in [21].
Abstract: In this paper, we present further development of the local discontinuous Galerkin (LDG) method designed in [21] and a new dissipative discontinuous Galerkin (DG) method for the Hunter-Saxton equation. The numerical ∞uxes for the LDG and DG methods in this paper are based on the upwinding principle. The resulting schemes provide additional energy dissipation and better control of numerical oscillations near derivative singularities. Stability and convergence of the schemes are proved theoretically, and numerical simulation results are provided to compare with the scheme in [21].
TL;DR: It is proposed to use adaptive multi-meshes in developing efficient algorithms for the estimation problem with equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive finite element discretisation scheme.
Abstract: In this paper, we study adaptive finite element discretisation schemes for a class of parameter estimation problem. We propose to use adaptive multi-meshes in developing efficient algorithms for the estimation problem. We derive equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results. Mathematics subject classification: 49J20, 65N30.
TL;DR: O†ine-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small).
Abstract: In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial difierential equation. The flne solver is based on the flnite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. O†ine-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids flnite element method and a multi-degrees flnite element method are also presented. Some numerical results are reported.
TL;DR: In this paper, the smooth LU decomposition of a given analytic functional and its block-analogue is studied, and sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided.
Abstract: We study the smooth LU decomposition of a given analytic functional �-matrix A(�) and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A(�), and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.
TL;DR: In this paper, a class of rectangular finite elements for 2m-th-oder elliptic boundary value problems in n-dimension (m,n � 1) is proposed in a canonical fashion, which includes the (2m 1)-th Hermite interpolation element (n = 1), the n-linear finite element (m = 1) and the Adini element(m = 2).
Abstract: In this paper, a class of rectangular finite elements for 2m-th-oder elliptic boundary value problems in n-dimension (m,n � 1) is proposed in a canonical fashion, which includes the (2m 1)-th Hermite interpolation element (n = 1), the n-linear finite element (m = 1) and the Adini element (m = 2). A nonconforming triangular finite element for the plate bending problem, with convergent order O(h 2 ), is also proposed.
TL;DR: In this article, a modified block-SSOR preconditioned conjugate gradient method was proposed to solve the signal and image restoration problems with the half-quadratic regularization technique by making use of the Newton method.
Abstract: Signal and image restoration problems are often solved by minimizing a cost function consisting of an l(2) data-fidelity term and a regularization term. We consider a class of convex and edge-preserving regularization functions. In specific, half-quadratic regularization as a fixed-point iteration method is usually employed to solve this problem. The main aim of this paper is to solve the above-described signal and image restoration problems with the half-quadratic regularization technique by making use of the Newton method. At each iteration of the Newton method, the Newton equation is a structured system of linear equations of a symmetric positive definite coefficient matrix, and may be efficiently solved by the preconditioned conjugate gradient method accelerated with the modified block SSOR preconditioner. Our experimental results show that the modified block-SSOR preconditioned conjugate gradient method is feasible and effective for further improving the numerical performance of the half-quadratic regularization approach.
TL;DR: In this article, a numerical method based on flnite difierence method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems.
Abstract: A numerical method based on flnite difierence method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameteruniform convergence, a variable mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The uniform convergence analysis of the method is discussed. The original problem is reduced to its normal form and the reduced problem is solved by flnite difierence method taking variable mesh. To support the e‐ciency of the method, several numerical examples have been considered.
TL;DR: This paper studies the convergence of adaptive finite element methods for the general non-affine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes based on several basic ingredients, such as quasi-orthogonality, estimator.
Abstract: In this paper we study the convergence of adaptive finite element methods for the general non-affine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes. Based on several basic ingredients, such as quasi-orthogonality, estimator
TL;DR: Two versions of the tau-ROCK methods are discussed and their stability behavior is analyzed on a test, problem and a significant speed-up can be achieved for some stiff kinetic systems.
Abstract: Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit, methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345(10), 2007 and Commun. Math. Sci, 6(4), 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the tau-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but, our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the tau-ROCK methods are discussed and their stability behavior is analyzed on a test, problem. Compared to the tau-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.