Showing papers on "Discretization published in 2011"

Markov models of molecular kinetics: Generation and validation

Abstract: Markov state models of molecular kinetics (MSMs), in which the long-time statistical dynamics of a molecule is approximated by a Markov chain on a discrete partition of configuration space, have seen widespread use in recent years. This approach has many appealing characteristics compared to straightforward molecular dynamics simulation and analysis, including the potential to mitigate the sampling problem by extracting long-time kinetic information from short trajectories and the ability to straightforwardly calculate expectation values and statistical uncertainties of various stationary and dynamical molecular observables. In this paper, we summarize the current state of the art in generation and validation of MSMs and give some important new results. We describe an upper bound for the approximation error made by modeling molecular dynamics with a MSM and we show that this error can be made arbitrarily small with surprisingly little effort. In contrast to previous practice, it becomes clear that the best MSM is not obtained by the most metastable discretization, but the MSM can be much improved if non-metastable states are introduced near the transition states. Moreover, we show that it is not necessary to resolve all slow processes by the state space partitioning, but individual dynamical processes of interest can be resolved separately. We also present an efficient estimator for reversible transition matrices and a robust test to validate that a MSM reproduces the kinetics of the molecular dynamics data.

Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems

Abstract: 1. Introduction.- 1.1 Problem classification.- 1.2 Problem formulation and exact definition of the subject.- 1.2.1 Application of the different models.- 1.2.2 Remarks on the term model.- 1.2.3 Objective and structure of this book.- 2. Fundamental principles of conceptual modeling.- 2.1 Preliminary remarks.- 2.1.1 General remarks.- 2.1.2 Definitions and fundamental terms.- 2.2 System properties.- 2.2.1 Mass and mole fractions.- 2.2.2 Density.- 2.2.3 Viscosity.- 2.2.4 Specific enthalpy, specific internal energy.- 2.2.5 Surface tension.- 2.2.6 Specific heat capacity.- 2.3 Phase state, phase transition, phase change.- 2.3.1 Phase state.- 2.3.2 Phase transition, phase change.- 2.4 Capillarity.- 2.4.1 Microscopic capillarity.- 2.4.2 Macroscopic capillarity.- 2.4.3 Capillarity in fractures.- 2.5 Hysteresis.- 2.6 Definition of different saturations.- 2.7 Relative permeability.- 2.7.1 Permeability.- 2.7.2 Relative permeability at the micro scale.- 2.7.3 Relative permeability at the macro scale.- 2.7.4 Relative permeability-saturation relation in fractures.- 2.7.5 Fracture-matrix interaction.- 2.8 Pressure and temperature dependence of porosity.- Mathematical modeling.- 3.1 General balance equation.- 3.1.1 Preconditions and assumptions.- 3.1.2 The Reynolds transport theorem in integral form.- 3.1.3 Derivation of the general balance equation.- 3.1.4 Initial and boundary conditions.- 3.1.5 Choice of the primary variables.- 3.2 Continuity equation per phase.- 3.2.1 Time derivative.- 3.3 Momentum equation and Darcy's law.- 3.3.1 General remarks.- 3.3.2 Darcy's law of single-phase flow.- 3.3.3 Generalization of Darcy's law for multiphase flow.- 3.4 General form of the multiphase flow equation.- 3.4.1 Pressure formulation.- 3.4.2 Pressure-saturation formulation.- 3.4.3 Saturation formulation.- 3.4.4 Mathematical modeling for three-phase infiltration and remobilization processes.- 3.5 Transport equation.- 3.5.1 Basic transport equation.- 3.5.2 Transport in a multiphase system.- 3.5.3 Description of the mass transfer between phases.- 3.5.4 Multicomponent transport processes in the gas phase.- 3.6 Energy equation.- 3.7 Multiphase/multicomponent system.- 4. Numerical modeling.- 4.1 Classification.- 4.1.1 Problem and special solution methods.- 4.1.2 Fundamentals of discretization.- 4.1.3 Conservative discretization.- 4.1.4 Weighted residual method.- 4.2 Finite element and finite volume methods.- 4.2.1 Spatial discretization.- 4.2.2 Choice of element types.- 4.2.3' Galerkin finite element method.- 4.2.4 Sub domain collocation - finite volume method.- 4.2.5 Time discretization.- 4.3 Linearization of the multiphase problem.- 4.3.1 Weak nonlinearities.- 4.3.2 Strong nonlinearities.- 4.3.3 Handling of the nonlinearities.- 4.3.4 Example: Linearized two-phase equation.- 4.4 Discussion of the instationary hyperbolic (convective) transport equation.- 4.4.1 Classification of hyperbolic differential equations.- 4.4.2 A linear hyperbolic transport equation.- 4.4.3 A quasilinear hyperbolic transport equation - Buckley-Levereit equation.- 4.4.4 Analytical solutions for the Buckley-Lev ereit problem.- 4.5 Special discretization methods.- 4.5.1 Motivation.- 4.5.2 Upwind method - finite difference method.- 4.5.3 Explicit upwind method of first order - Fully Upwind.- 4.5.4 Multidimensional upwind method of first order.- 4.5.5 Explicit upwind method of higher order - TVD techniques.- 4.5.6 Implicit upwind method of first order - Fully Upwind.- 4.5.7 Petrov-Galerkin finite element method.- 4.5.8 Additional remarks on conservative discretization.- 4.5.9 Flux-corrected method.- 4.5.10 Mixed-hybrid finite element methods.- 5. Comparison of the different discretization methods.- 5.1 Discretization.- 5.1.1 Finite element Galerkin method.- 5.1.2 Sub domain collocation finite volume method (box method).- 5.2 Boundedness principle - discussion of a monotonic solution.- 5.3 Comparative study of the different methods in homogeneous porous media.- 5.3.1 Multiphase flow without capillary pressure effects - Buckley-Lev ereit problem.- 5.3.2 Multiphase flow with capillary pressure effects - McWhorter problem.- 5.4 Heterogeneity effects.- 5.5 Comparative study of the methods for flow in heterogeneous porous media.- 5.6 Five-spot waterflood problem.- 6. Test problems - applications.- 6.1 DNAPL-Infiltration.- 6.2 LNAPL-Infiltration.- 6.3 Non-isothermal multiphase/multicomponent flow.- 6.3.1 Heat pipe.- 6.3.2 Study of bench-scale experiments.- 7. Final remarks.

Iterative Solution of Large Sparse Systems of Equations

Abstract: In the second edition of this classic monograph, complete with four new chapters and updated references, readers will now have access to content describing and analysing classical and modern methods with emphasis on the algebraic structure of linear iteration, which is usually ignored in other literature. The necessary amount of work increases dramatically with the size of systems, so one has to search for algorithms that most efficiently and accurately solve systems of, e.g., several million equations. The choice of algorithms depends on the special properties the matrices in practice have. An important class of large systems arises from the discretization of partial differential equations. In this case, the matrices are sparse (i.e., they contain mostly zeroes) and well-suited to iterative algorithms. The first edition of this book grew out of a series of lectures given by the author at the Christian-Albrecht University of Kiel to students of mathematics. The second edition includes quite novel approaches

Filters in topology optimization based on Helmholtz‐type differential equations

Abstract: The aim of this paper is to apply a Helmholtz-type partial differential equation as an alternative to standard density filtering in topology optimization problems. Previously, this approach has been successfully applied as a sensitivity filter. The usual filtering techniques in topology optimization require information about the neighbor cells, which is difficult to obtain for fine meshes or complex domains and geometries. The complexity of the problem increases further in parallel computing, when the design domain is decomposed into multiple non-overlapping partitions. Obtaining information from the neighbor subdomains is an expensive operation. The proposed filter technique requires only mesh information necessary for the finite element discretization of the problem. The main idea is to define the filtered variable implicitly as a solution of a Helmholtz-type differential equation with homogeneous Neumann boundary conditions. The properties of the filter are demonstrated for various 2D and 3D topology optimization problems in linear elasticity, solved on serial and parallel computers. Copyright © 2010 John Wiley & Sons, Ltd.

Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications

Abstract: Introducing delay.- Basic delay differential equations.- Newtonian examples.- Engineering applications.- Summary.- References.

Discretization of Processes

Abstract: Part I Introduction and Preliminary Material.- 1.Introduction .- 2.Some Prerequisites.- Part II The Basic Results.- 3.Laws of Large Numbers: the Basic Results.- 4.Central Limit Theorems: Technical Tools.- 5.Central Limit Theorems: the Basic Results.- 6.Integrated Discretization Error.- Part III More Laws of Large Numbers.- 7.First Extension: Random Weights.- 8.Second Extension: Functions of Several Increments.- 9.Third Extension: Truncated Functionals.- Part IV Extensions of the Central Limit Theorems.- 10.The Central Limit Theorem for Random Weights.- 11.The Central Limit Theorem for Functions of a Finite Number of Increments.- 12.The Central Limit Theorem for Functions of an Increasing Number of Increments.- 13.The Central Limit Theorem for Truncated Functionals.- Part V Various Extensions.- 14.Irregular Discretization Schemes. 15.Higher Order Limit Theorems.- 16.Semimartingales Contaminated by Noise.- Appendix.- References.- Assumptions.- Index of Functionals.- Index.

Adaptive Moving Mesh Methods

Abstract: Preface.- Introduction.- Adaptive Mesh Movement in 1D.- Discretization of PDEs on Time-Varying Meshes.- Basic Principles of Multidimensional Mesh Adaption.- Monitor Functions.- Variational Mesh Adaptive Methods.- Velocity-Based Adaptive Methods.- Appendix: Sobolev Spaces.- Appendix: Arithmetic Mean Geometric Mean Inequality and Jensen's Inequality.- Bibliography.

Analytic regularity and polynomial approximation of parametric and stochastic elliptic pde's

Abstract: Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDE's in a bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space of weak solutions of the elliptic problem with a controlled number of terms N. The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality. The discretization of the coefficients from a family of continuous, piecewise linear finite element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number Ndof of degrees of freedom is the minimum of the convergence rates afforded by the best N-term sequence approximations in the parameter space and the rate of finite element approximations in D for a single instance of the parametric problem.

The Finite Cell Method

Abstract: Thanks to its versatility, the Finite Element Method (FEM) has become the most frequently applied numerical method in Computational Mechanics. Although the FEM is in a mature state there are still problems where its application is difficult. These problems arise, for example, when considering heterogeneous materials or more generally when discretizing structures which have a very complex geometry. In such cases mesh generation can become very involved. To overcome these problems we propose to apply the Finite Cell Method (FCM) [1] which can be considered as a combination of a fictitious domain method with high-order finite elements. The main idea is to embed the physical domain into an extended domain which can be easily discretized with structured hexahedral meshes. In this way, the meshing generation is dramatically simplified and the burden is shifted towards the integration of the stiffness matrices. However, the integration can be performed adaptively in a fully automatic way. During the integration the geometry and varying material properties are taken into account while the discretization error is controlled by adjusting the polynomial degree of the hexahedrals. The proposed method will be illustrated by considering the discretization of thin-walled structures, the numerical homogenization of heterogeneous materials as well as problems of topology optimization.

Review of Zero-D and 1-D Models of Blood Flow in the Cardiovascular System

Abstract: Zero-dimensional (lumped parameter) and one dimensional models, based on simplified representations of the components of the cardiovascular system, can contribute strongly to our understanding of circulatory physiology. Zero-D models provide a concise way to evaluate the haemodynamic interactions among the cardiovascular organs, whilst one-D (distributed parameter) models add the facility to represent efficiently the effects of pulse wave transmission in the arterial network at greatly reduced computational expense compared to higher dimensional computational fluid dynamics studies. There is extensive literature on both types of models. The purpose of this review article is to summarise published 0D and 1D models of the cardiovascular system, to explore their limitations and range of application, and to provide an indication of the physiological phenomena that can be included in these representations. The review on 0D models collects together in one place a description of the range of models that have been used to describe the various characteristics of cardiovascular response, together with the factors that influence it. Such models generally feature the major components of the system, such as the heart, the heart valves and the vasculature. The models are categorised in terms of the features of the system that they are able to represent, their complexity and range of application: representations of effects including pressure-dependent vessel properties, interaction between the heart chambers, neuro-regulation and auto-regulation are explored. The examination on 1D models covers various methods for the assembly, discretisation and solution of the governing equations, in conjunction with a report of the definition and treatment of boundary conditions. Increasingly, 0D and 1D models are used in multi-scale models, in which their primary role is to provide boundary conditions for sophisticate, and often patient-specific, 2D and 3D models, and this application is also addressed. As an example of 0D cardiovascular modelling, a small selection of simple models have been represented in the CellML mark-up language and uploaded to the CellML model repository http://models.cellml.org/ . They are freely available to the research and education communities. Each published cardiovascular model has merit for particular applications. This review categorises 0D and 1D models, highlights their advantages and disadvantages, and thus provides guidance on the selection of models to assist various cardiovascular modelling studies. It also identifies directions for further development, as well as current challenges in the wider use of these models including service to represent boundary conditions for local 3D models and translation to clinical application.

Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method

Abstract: The deformation of an initially spherical capsule, freely suspended in simple shear flow, can be computed analytically in the limit of small deformations [D. Barthes-Biesel, J.M. Rallison, The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech. 113 (1981) 251-267]. Those analytic approximations are used to study the influence of the mesh tessellation method, the spatial resolution, and the discrete delta function of the immersed boundary method on the numerical results obtained by a coupled immersed boundary lattice Boltzmann finite element method. For the description of the capsule membrane, a finite element method and the Skalak constitutive model [R. Skalak, A. Tozeren, R.P. Zarda, S. Chien, Strain energy function of red blood cell membranes, Biophys. J. 13 (1973) 245-264] have been employed. Our primary goal is the investigation of the presented model for small resolutions to provide a sound basis for efficient but accurate simulations of multiple deformable particles immersed in a fluid. We come to the conclusion that details of the membrane mesh, as tessellation method and resolution, play only a minor role. The hydrodynamic resolution, i.e., the width of the discrete delta function, can significantly influence the accuracy of the simulations. The discretization of the delta function introduces an artificial length scale, which effectively changes the radius and the deformability of the capsule. We discuss possibilities of reducing the computing time of simulations of deformable objects immersed in a fluid while maintaining high accuracy.

The PLUTO Code for Adaptive Mesh Computations in Astrophysical Fluid Dynamics

Abstract: We present a description of the adaptive mesh refinement (AMR) implementation of the PLUTO code for solving the equations of classical and special relativistic magnetohydrodynamics (MHD and RMHD). The current release exploits, in addition to the static grid version of the code, the distributed infrastructure of the CHOMBO library for multidimensional parallel computations over block-structured, adaptively refined grids. We employ a conservative finite-volume approach where primary flow quantities are discretized at the cell-center in a dimensionally unsplit fashion using the Corner Transport Upwind (CTU) method. Time stepping relies on a characteristic tracing step where piecewise parabolic method (PPM), weighted essentially non-oscillatory (WENO) or slope-limited linear interpolation schemes can be handily adopted. A characteristic decomposition-free version of the scheme is also illustrated. The solenoidal condition of the magnetic field is enforced by augmenting the equations with a generalized Lagrange multiplier (GLM) providing propagation and damping of divergence errors through a mixed hyperbolic/parabolic explicit cleaning step. Among the novel features, we describe an extension of the scheme to include non-ideal dissipative processes such as viscosity, resistivity and anisotropic thermal conduction without operator splitting. Finally, we illustrate an efficient treatment of point-local, potentially stiff source terms over hierarchical nested grids by taking advantage of the adaptivity in time. Several multidimensional benchmarks and applications to problems of astrophysical relevance assess the potentiality of the AMR version of PLUTO in resolving flow features separated by large spatial and temporal disparities.

A class of discontinuous Petrov–Galerkin methods. II. Optimal test functions

Abstract: We lay out a program for constructing discontinuous Petrov-Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well-posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated through several theoretical and numerical examples.

Numerical simulation of laminar reacting flows with complex chemistry

Abstract: We present an adaptive algorithm for low Mach number reacting flows with complex chemistry. Our approach uses a form of the low Mach number equations that discretely conserves both mass and energy. The discretization methodology is based on a robust projection formulation that accommodates large density contrasts. The algorithm uses an operator-split treatment of stiff reaction terms and includes effects of differential diffusion. The basic computational approach is embedded in an adaptive projection framework that uses structured hierarchical grids with subcycling in time that preserves the discrete conservation properties of the underlying single-grid algorithm. We present numerical examples illustrating the performance of the method on both premixed and non-premixed flames.

Multiscale space---time fluid---structure interaction techniques

Abstract: We present the multiscale space---time techniques we have developed for fluid---structure interaction (FSI) computations. Some of these techniques are multiscale in the way the time integration is performed (i.e. temporally multiscale), some are multiscale in the way the spatial discretization is done (i.e. spatially multiscale), and some are in the context of the sequentially-coupled FSI (SCFSI) techniques developed by the Team for Advanced Flow Simulation and Modeling $${({\rm T} \bigstar {\rm AFSM})}$$ . In the multiscale SCFSI technique, the FSI computational effort is reduced at the stage we do not need it and the accuracy of the fluid mechanics (or structural mechanics) computation is increased at the stage we need accurate, detailed flow (or structure) computation. As ways of increasing the computational accuracy when or where needed, and beyond just increasing the mesh refinement or decreasing the time-step size, we propose switching to more accurate versions of the Deforming-Spatial-Domain/Stabilized Space---Time (DSD/SST) formulation, using more polynomial power for the basis functions of the spatial discretization or time integration, and using an advanced turbulence model. Specifically, for more polynomial power in time integration, we propose to use NURBS, and as an advanced turbulence model to be used with the DSD/SST formulation, we introduce a space---time version of the residual-based variational multiscale method. We present a number of test computations showing the performance of the multiscale space---time techniques we are proposing. We also present a stability and accuracy analysis for the higher-accuracy versions of the DSD/SST formulation.

A large deformation frictional contact formulation using NURBS‐based isogeometric analysis

Abstract: This paper focuses on the application of NURBS-based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar-based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C0-continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface. Copyright © 2011 John Wiley & Sons, Ltd.

Variationally consistent discretization schemes and numerical algorithms for contact problems

Abstract: We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-order a priori convergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-based a posteriori error estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

Numerical Methods for Two-phase Incompressible Flows

Abstract: Introduction.- Part I One-phase incompressible flows.- Mathematical models.- Finite element discretization.- Time integration.-

Discrete mechanics and optimal control: an analysis ∗

Abstract: The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.

Efficient elasticity for character skinning with contact and collisions

Abstract: We present a new algorithm for near-interactive simulation of skeleton driven, high resolution elasticity models. Our methodology is used for soft tissue deformation in character animation. The algorithm is based on a novel discretization of corotational elasticity over a hexahedral lattice. Within this framework we enforce positive definiteness of the stiffness matrix to allow efficient quasistatics and dynamics. In addition, we present a multigrid method that converges with very high efficiency. Our design targets performance through parallelism using a fully vectorized and branch-free SVD algorithm as well as a stable one-point quadrature scheme. Since body collisions, self collisions and soft-constraints are necessary for real-world examples, we present a simple framework for enforcing them. The whole approach is demonstrated in an end-to-end production-level character skinning system.

Plane Wave Discontinuous Galerkin Methods for the 2D Helmholtz Equation: Analysis of the $p$-Version

Abstract: Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator $-\Delta-\omega^2$, $\omega>0$. They include the so-called ultra weak variational formulation from [O. Cessenat and B. Despres, SIAM J. Numer. Anal., 35 (1998), pp. 255-299]. This paper is concerned with the a priori convergence analysis of PWDG in the case of $p$-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [P. Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121-136], and plane wave approximation theory.

A probabilistic numerical method for fully nonlinear parabolic PDEs

Abstract: We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in \cite{cstv}, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations a! rising in the theory of portfolio optimization in financial mathematics.

Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation

Abstract: The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable-coefficient Helmholtz equation including very-high-frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The generalized minimal residual method (GMRES) solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three-dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach.