Showing papers in "Computing and Visualization in Science in 2004"
TL;DR: This paper explains how the evaluation of integrals and the transfer between arbitrary finite element spaces can be implemented easily and computed efficiently.
Abstract: The basis of mapped finite element methods are reference elements where the components of a local finite element are defined. The local finite element on an arbitrary mesh cell will be given by a map from the reference mesh cell. This paper describes some concepts of the implementation of mapped finite element methods. From the definition of mapped finite elements, only local degrees of freedom are available. These local degrees of freedom have to be assigned to the global degrees of freedom which define the finite element space. We will present an algorithm which computes this assignment. The second part of the paper shows examples of algorithms which are implemented with the help of mapped finite elements. In particular, we explain how the evaluation of integrals and the transfer between arbitrary finite element spaces can be implemented easily and computed efficiently.
171 citations
TL;DR: A Galerkin Finite Element approximation of the Stokes–Darcy problem which models the coupling between surface and groundwater flows and an iterative subdomain method for its solution is proposed, inspired to the domain decomposition theory.
Abstract: We consider a Galerkin Finite Element approximation of the Stokes–Darcy problem which models the coupling between surface and groundwater flows. Then we propose an iterative subdomain method for its solution, inspired to the domain decomposition theory. The convergence analysis that we develop is based on the properties of the discrete Steklov–Poincare operators associated to the given coupled problem. An optimal preconditioner for Krylov methods is proposed and analyzed.
138 citations
TL;DR: It is illustrated that an energy minimizing basis first introduced in Wan, Chan and Smith (2000) for algebraic multigrid methods can be numerically obtained in an optimal fashion.
Abstract: This paper is devoted to the study of an energy minimizing basis first introduced in Wan, Chan and Smith (2000) for algebraic multigrid methods. The basis will be first obtained in an explicit and compact form in terms of certain local and global operators. The basis functions are then proved to be locally harmonic functions on each coarse grid "element". Using these new results, it is illustrated that this basis can be numerically obtained in an optimal fashion. In addition to the intended application for algebraic multigrid method, the energy minimizing basis may also be applied for numerical homogenization.
85 citations
TL;DR: Numerical simulation of flow through oil filters, describing coupled flows in the pure liquid subregions and in the porous filter media, as well as interface conditions between them, shows numerical results and validates the model and the algorithm.
Abstract: This paper concerns numerical simulation of flow through oil filters. Oil filters consist of filter housing (filter box), and a porous filtering medium, which completely separates the inlet from the outlet. We discuss mathematical models, describing coupled flows in the pure liquid subregions and in the porous filter media, as well as interface conditions between them. Further, we reformulate the problem in fictitious regions method manner, and discuss peculiarities of the numerical algorithm in solving the coupled system. Next, we show numerical results, validating the model and the algorithm. Finally, we present results from simulation of 3-D oil flow through a real car filter.
74 citations
TL;DR: In this paper, a semi-Lagrangian algorithm for first order Hamilton-Jacobi equations is proposed for solving the optimal control finite horizon problem in Rm with m ≥ 3.
Abstract: In this paper we develop a new version of the semi-Lagrangian algorithm for first order Hamilton–Jacobi equations. This implementation is well suited to deal with problems in high dimension, i.e. in Rm with m≥3, which typically arise in the study of control problems and differential games. Our model problem is the evolutive Hamilton–Jacobi equation related to the optimal control finite horizon problem. We will give a step-by-step description of the algorithm focusing our attention on two critical routines: the interpolation in high dimension and the search for the global minimum. We present some numerical results on test problems which range from m=3 to m=5 and deal with applications to front propagation, aerospace engineering, ecomomy and biology.
68 citations
TL;DR: In this paper, a direct method for solving the evolution of plane curves satisfying the geometric equation v=β(x,k,ν) where v is the normal velocity, k and ν are the curvature and tangential angle of a plane curve Γ⊂R2 at a point x∈Γ.
Abstract: We propose a direct method for solving the evolution of plane curves satisfying the geometric equation v=β(x,k,ν) where v is the normal velocity, k and ν are the curvature and tangential angle of a plane curve Γ⊂R2 at a point x∈Γ. We derive and analyze the governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves. The governing equations include a nontrivial tangential velocity functional yielding uniform redistribution of grid points along the evolving family of curves preventing thus numerically computed solutions from forming various instabilities. We also propose a full space-time discretization of the governing system of equations and study its experimental order of convergence. Several computational examples of evolution of plane curves driven by curvature and external force as well as the geodesic curvatures driven evolution of curves on various complex surfaces are presented in this paper.
64 citations
TL;DR: In this article, various block smoothers are tested in numerical experiments for equations of Black-Scholes-type (European options) in several dimensions and compared with other strategies including cascadic approaches and full approximation schemes.
Abstract: Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black---Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes.
56 citations
TL;DR: In this paper, an inner product of the residual error in the original p.d. is used to improve the accuracy of the output integrals of the PDEs, which is a correction term which is defined as the inner product this paper.
Abstract: When approximating the solutions of partial differential equations, it is a few key output integrals which are often of most concern. This paper shows how the accuracy of these values can be improved through a correction term which is an inner product of the residual error in the original p.d.e. and the solution of an appropriately defined adjoint p.d.e. A number of applications are presented and the challenges of smooth reconstruction on unstructured grids and error correction for shocks are discussed.
53 citations
TL;DR: In this article, a computational PDE method based on a nonlinear multigrid scheme for restoring noisy images is suggested, which starts with an isotropic ("smooth") problem and leads to smooth edges in the image.
Abstract: The classical image denoising technique introduced by Rudin, Osher, and Fatemi [17] a decade ago, leads to solve a constrained minimization problem for the total variation (TV) of the image. The formal first variation of the minimization problem is a nonlinear and highly anisotropic boundary value problem. In this paper, a computational PDE method based on a nonlinear multigrid scheme for restoring noisy images is suggested. Here, we examine different discretizations for the Euler---Lagrange equation as well as different smoothers within the multigrid scheme. Then we describe the iterative total variation regularization scheme, which starts with an isotropic ("smooth") problem and leads to smooth edges in the image. Within the iteration the problem becomes more and more anisotropic and converges to an image with sharp edges. Finally, we present some experimental results for synthetic and real images.
53 citations
TL;DR: In this article, a morphological multi-scale method for image sequence processing is presented, which results in a truly coupled spatio-temporal anisotropic diffusion, which denoise the whole sequence while retaining geometric features such as spatial edges and highly accelerated motions.
Abstract: We present a morphological multi-scale method for image sequence processing, which results in a truly coupled spatio-temporal anisotropic diffusion. The aim of the method is not to smooth the level-sets of single frames but to denoise the whole sequence while retaining geometric features such as spatial edges and highly accelerated motions. This is obtained by an anisotropic spatio-temporal level-set evolution, where the additional artificial time variable serves as the multi-scale parameter. The diffusion tensor of the evolution depends on the morphology of the sequence, given by spatial curvatures of the level-sets and the curvature of trajectories (=acceleration) in sequence-time. We discuss different regularization techniques and describe an operator splitting technique for solving the problem. Finally we compare the new method with existing multi-scale image sequence processing methodologies.
44 citations
TL;DR: ℋ2-matrices combine a multigrid-like structure with ideas from panel clustering algorithms in order to provide a very efficient method for discretizing and evaluating the integral operators found, e.g., in boundary element applications.
Abstract: Multigrid methods are typically used to solve partial differential equations, i.e., they approximate the inverse of the corresponding partial differential operators. At least for elliptic PDEs, this inverse can be expressed in the form of an integral operator by Green's theorem.
This implies that multigrid methods approximate certain integral operators, so it is straightforward to look for variants of multigrid methods that can be used to approximate more general integral operators.
?2-matrices combine a multigrid-like structure with ideas from panel clustering algorithms in order to provide a very efficient method for discretizing and evaluating the integral operators found, e.g., in boundary element applications.
TL;DR: A reliable and robust approximation scheme for biochemically reacting transport in the subsurface following Monod type kinetics is presented and it is shown that the higher order approximation scheme significantly reduces the amount of inherent numerical diffusion compared to lower order ones.
Abstract: In this work we present and analyze a reliable and robust approximation scheme for biochemically reacting transport in the subsurface following Monod type kinetics. Water flow is modeled by the Richards equation. The proposed scheme is based on higher order finite element methods for the spatial discretization and the two step backward differentiation formula for the temporal one. The resulting nonlinear algebraic systems of equations are solved by a damped version of Newton's method. For the linear problems of the Newton iteration Krylov space methods are used. In computational experiments conducted for realistic subsurface (groundwater) contamination scenarios we show that the higher order approximation scheme significantly reduces the amount of inherent numerical diffusion compared to lower order ones. Thereby an artificial transverse mixing of the species leading to a strong overestimation of the biodegradation process is avoided. Finally, we present a robust adaptive time stepping technique for the coupled flow and transport problem which allows efficient long-term predictions of biodegradation processes.
TL;DR: An algorithm to automatically detect regular subcurves in a set of digital curves based on Helmholtz Principle, which shows that contrast invariance is a very sound hypothesis for low level vision and that objects can be detected independently of contrast.
Abstract: In this paper, we propose an algorithm to automatically detect regular subcurves in a set of digital curves. The decision criterion is based on Helmholtz Principle introduced by Desolneux, Moisan and Morel and formulated in terms of number of false alarms. We apply our algorithm to low-level computer vision. Following Gestalt Theory, good continuation is indeed, one of the most important grouping laws entering into the early perception of objects. We check this on the level lines of images, which give a contrast invariant representation of images. The result is that most objects are good continuations. This experimentally shows that contrast invariance is a very sound hypothesis for low level vision and that objects can be detected independently of contrast. The parameters of the method may be reduced to a single one – the number of false alarms – and we can show that the detection has a very weak dependency on this number.
TL;DR: This work provides a concept combining techniques known from geometric multigrid methods for saddle point problems and AMG methods for scalar problems to a coupled algebraic multigrids solver for finite element discretizations of “real life” industrial problems.
Abstract: We provide a concept combining techniques known from geometric multigrid methods for saddle point problems (such as smoothing iterations of Braess- or Vanka-type) and from algebraic multigrid (AMG) methods for scalar problems (such as the construction of coarse levels) to a coupled algebraic multigrid solver. `Coupled' here is meant in contrast to methods, where pressure and velocity equations are iteratively decoupled (pressure correction methods) and standard AMG is used for the solution of the resulting scalar problems. To prove the efficiency of our solver experimentally, it is applied to finite element discretizations of "real life" industrial problems.
TL;DR: Numerical results presented here show an almost optimal behavior (with respect to the space discretization) and suggest that the new preconditioner is well suited also for “flexible” or “inexact” strategies, in which the systems for the preconditionser are solved inaccurately.
Abstract: The pressure matrix method is a well known scheme for the solution of the incompressible Navier-Stokes equations by splitting the computation of the velocity and the pressure fields (see, e.g., [17]). However, the set-up of effective preconditioners for the pressure matrix is mandatory in order to have an acceptable computational cost. Different strategies can be pursued (see, e.g., [4], [6], [7], [9], [22]). Inexact blockLU factorizations of the matrix obtained after the discretization and linearization of the problem, originally proposed as fractional step solvers, provide also a strategy for building effective preconditioners of the pressure matrix (see [23]). In this paper, we present numerical results about a new preconditioner, based on an inexact factorization. The new preconditioner applies to the case of the generalized Stokes problem and to the Navier-Stokes one, as well. In the former case, it improves the performances of the well known Cahouet-Chabard preconditioner (see [2]). In the latter one, numerical results presented here show an almost optimal behavior (with respect to the space discretization) and suggest that the new preconditioner is well suited also for "flexible" or "inexact" strategies, in which the systems for the preconditioner are solved inaccurately.
TL;DR: An efficient parallel code for the approximate solution of initial boundary value problems for hyperbolic balance laws is introduced, applied to the equations of compressible magnetohydrodynamics (MHD).
Abstract: An efficient parallel code for the approximate solution of initial boundary value problems for hyperbolic balance laws is introduced. The method combines three modern numerical techniques: locally-adaptive upwind finite-volume methods on unstructured grids, parallelization based on non-overlapping domain decomposition, and dynamic load balancing. Key ingredient is a hierarchical mesh in three space dimensions.
The proposed method is applied to the equations of compressible magnetohydrodynamics (MHD). Results for several testproblems with computable exact solution and for a realistic astrophysical simulation are shown.
TL;DR: In this article, a mathematical model for the fluidization of polydisperse suspensions is developed and the stationary concentration configurations for a given fluidizing velocity are analyzed and a mixing condition for bed inversion is derived.
Abstract: A mathematical model for the fluidization of polydisperse suspensions is developed. The stationary concentration configurations for a given fluidizing velocity are analyzed and a mixing condition for bed inversion is derived. A central finite difference scheme is applied to the simulation of the fluidization of a bidisperse suspension. The numerical simulations agree with the experimentally reported qualitative behaviour of fluidized suspensions such as bed expansion and bed inversion.
TL;DR: In this article, the Engquist-Osher scheme is applied to the clarifier-thickener setup for a nonconvex scalar conservation law whose flux depends on a vector of discontinuous parameters.
Abstract: Clarifier-thickener units treating ideal suspensions can be modeled as an initial-value problem for a nonconvex scalar conservation law whose flux depends on a vector of discontinuous parameters. This problem can be treated by the well-known Engquist–Osher scheme if the discontinuous paremeters are discretized on a grid staggered against that of the conserved variable. We prove convergence of this scheme to a weak solution of the problem and illustrate its application to the clarifier-thickener setup by a numerical example.
TL;DR: In this article, a well-posed vorticity-velocity-pressure formulation for the Stokes problem is introduced and its finite element discretization, which needs some stabilization, is then studied.
Abstract: We analyze here the bidimensional boundary value problems, for both Stokes and Navier-Stokes equations, in the case where non standard boundary conditions are imposed. A well-posed vorticity-velocity-pressure formulation for the Stokes problem is introduced and its finite element discretization, which needs some stabilization, is then studied. We consider next the approximation of the Navier-Stokes equations, based on the previous approximation of the Stokes equations. For both problems, the convergence of the numerical approximation and optimal error estimates are obtained. Some numerical tests are also presented.
TL;DR: A rigorous a posteriori error bound framework for the case in which the parametrization of the partial differential equation is exact is developed; the situation in which this mathematical model is not “complete” is addressed.
Abstract: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with (approximately) affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N, typically very small, and the (approximate) parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.In our earlier work, we develop a rigorous a posteriori error bound framework for the case in which the parametrization of the partial differential equation is exact; in this paper, we address the situation in which our mathematical model is not “complete.” In particular, we permit error in the data that define our partial differential operator: this error may be introduced, for example, by imperfect specification, measurement, calculation, or parametric expansion of a coefficient function. We develop both accurate predictions for the outputs of interest and associated rigorous a posteriori error bounds; and the latter incorporate both numerical discretization and “model truncation” effects. Numerical results are presented for a particular instantiation in which the model error originates in the (approximately) prescribed velocity field associated with a three-dimensional convection-diffusion problem.
TL;DR: It is demonstrated, that the tool of Local Fourier Analysis can also be profitably applied in this cell-centred multigrid approach and the use of constant, problem-independent transfer operators even in complicated situations.
Abstract: In this paper we treat the cell-centred multigrid approach, which distinguishes itself from the classical vertex-centred multigrid by a non-nested hierarchy of grid nodes and the use of constant, problem-independent transfer operators even in complicated situations. We demonstrate, that the tool of Local Fourier Analysis can also be profitably applied in this setting. We consider in detail the standard transfer operators from literature and their respective polynomial and Fourier orders, paying special attention to the combination of piecewise constant interpolation and its adjoint. Furthermore, we give several numerical examples for model problems and an application from biomedical engineering.
TL;DR: The efficiency of the new algorithm is elucidated by test calculations for an obstacle problem and for a Signorini problem, and a cg-method is proposed as smoother and as solver on coarse meshes.
Abstract: When classical multigrid methods are applied to discretizations of variational inequalities, several complications are frequently encountered mainly due to the lack of simple feasible restriction operators. These difficulties vanish in the application of the cascadic version of the multigrid method which in this sense yields greater advantages than in the linear case. Furthermore, a cg-method is proposed as smoother and as solver on coarse meshes. The efficiency of the new algorithm is elucidated by test calculations for an obstacle problem and for a Signorini problem.
TL;DR: In this paper, a finite element algorithm for the two-dimensional Stefan problem is described, where the free boundary is represented as a level set. The accuracy of the method is verified and several numerical simulations, including topological changes of the free boundaries, are presented.
Abstract: Dendritic growth is a nonlinear process, which falls into the category of self-organizing pattern formation phenomena. It is of great practical importance, since it appears frequently and, in the case of alloys, affects the engineering properties of the resulting solid. We describe a new finite element algorithm for the two---dimensional Stefan problem, where the free boundary is represented as a level set. This allows to handle topological changes of the free boundary. The accuracy of the method is verified and several numerical simulations, including topological changes of the free boundary, are presented.
TL;DR: In this paper, the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, is studied.
Abstract: In this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann---Oden and for the symmetric DG method, we give a detailed analysis of the convergence for cell- and point-wise block-relaxation strategies.
We show that, for a suitably constructed two-dimensional polynomial basis, point-wise block partitioning gives much better results than the classical cell-wise partitioning. Independent of the mesh size, for Poisson's equation, simple MG cycles with block-Gauss---Seidel or symmetric block-Gauss---Seidel smoothing, yield a convergence rate of 0.4---0.6 per iteration sweep for both DG-methods studied.
TL;DR: The purpose of this paper is to provide a necessary and sufficient condition for the B-stability of additive Runge-Kutta methods and to present a family of B-stable fractional step Runge–KUTta methods.
Abstract: An important requirement of numerical methods for the integration of nonlinear stiff initial value problems is B-stability. In many applications it is also convenient to use splitting methods to take advantage of the special structure of the differential operator that defines the model. The purpose of this paper is to provide a necessary and sufficient condition for the B-stability of additive Runge–Kutta methods. We also present a family of B-stable fractional step Runge–Kutta methods.
TL;DR: In this article, the authors discuss topics for a fast and accurate solution of continuous American-style Asian option problems from computational finance, which lead to 2D time-dependent convection-dominated partial differential equations with a free boundary.
Abstract: In this paper, we discuss topics for a fast and accurate solution of continuous American-style Asian option problems from computational finance. These problems lead to 2D time-dependent convection-dominated partial differential equations with a free boundary. As a pre-study for accurate discretization schemes in “asset price space” and in time, we solve numerically reference problems based on the Black–Scholes equation with small volatility and with discontinuous final conditions.
TL;DR: In this paper, the authors describe flux-based methods of characteristics (FLOM) that have no restriction on time steps and that produce numerical solutions with valid discrete minimum and maximum principle.
Abstract: Mathematical models of transport of radioactive contaminants in flowing groundwater involve large systems of coupled advection dominated transport equations. High-resolution explicit finite volume methods, if applied to advective part of model and combined with appropriate numerical methods for diffusion-dispersion-reaction part, can offer precise and monotone numerical solutions, but they require small time steps. This paper describes Flux-Based Methods Of Characteristics that are extension of explicit finite volume methods, that have no restriction on time steps and that produce numerical solutions with valid discrete minimum and maximum principle. Such particular method was implemented in software package R3T (Retardation, Reaction, Radionuclides and Transport) and it was used successfully to solve large systems of coupled transport equations with different retardation factors for transport.
TL;DR: In this paper, the authors consider the Rational Large Eddy Simulation model for turbulent flows (RLES in the sequel), introduced by Galdi and Layton [11] and recall some analytical results regarding the RLES model and the main result they will prove is the convergence of the strong solutions to the Rles model to those of the Navier-Stokes (in some Sobolev spaces), as the averaging radius goes to zero.
Abstract: In this paper we consider the Rational Large Eddy Simulation model for turbulent flows (RLES in the sequel), introduced by Galdi and Layton [11]. We recall some analytical results regarding the RLES model and the main result we will prove is the convergence of the strong solutions to the RLES model to those of the Navier–Stokes (in some Sobolev spaces), as the averaging radius goes to zero. Estimates on the rates of convergence are also obtained. These results give more weight to the validity of the method in either computational or physical experiments.We also consider the error arising from the derivation of the model in presence of boundaries. In particular, the equations present an extra-term involving the boundary value of the stress tensor. By using some recent estimates on this “commutation error” we show that, with a Smagorinsky sub-grid scale term, the kinetic energy remains bounded.
TL;DR: Results of numerical experiments demonstrate robustness of the AMG scheme with respect to changes of the weight of the cost of the control and show that the computational performance of the proposedAMG scheme is comparable to that of AMG applied to single scalar equations.
Abstract: An algebraic multigrid method (AMG) for solving convection-diffusion optimality systems is presented. Results of numerical experiments demonstrate robustness of the AMG scheme with respect to changes of the weight of the cost of the control and show that the computational performance of the proposed AMG scheme is comparable to that of AMG applied to single scalar equations.
TL;DR: Estimates for the physical splitting errors and the numerical splitting errors are established and lead to the selection of optimal sequences and coupling of physical phenomena and adequate use of implicitness and explicitness.
Abstract: The aim of this paper is to study discretizations of convection-diffusion-reaction equations using splitting methods. Estimates for the physical splitting errors and the numerical splitting errors are established. These estimates lead to the selection of optimal sequences and coupling of physical phenomena and adequate use of implicitness and explicitness. Numerical simulations of two chemical industry problems are included.