PATCH DYNAMICS: MACROSCOPIC SIMULATION OF MULTISCALE SYSTEMS

[1]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[2]  A J Chorin,et al.  Optimal prediction and the Mori-Zwanzig representation of irreversible processes. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Radek Erban,et al.  From Signal Transduction to Spatial Pattern Formation in E. coli: A Paradigm for Multiscale Modeling in Biology , 2005 .

[4]  Hans Christian Öttinger,et al.  Variance reduced simulations of stochastic differential equations , 1995 .

[5]  Sabine Attinger,et al.  Generalized Coarse Graining Procedures for Flow in Porous Media , 2003 .

[6]  G. Allaire Homogenization and two-scale convergence , 1992 .

[7]  Pierre Lallemand,et al.  Lattice Gas Hydrodynamics in Two and Three Dimensions , 1987, Complex Syst..

[8]  Andrew M. Stuart,et al.  Fitting SDE models to nonlinear Kac–Zwanzig heat bath models , 2004 .

[9]  S. Wolfram Cellular automaton fluids 1: Basic theory , 1986 .

[10]  Giovanni Samaey,et al.  Coarse-Grained Simulation and Bifurcation Analysis Using Microscopic Time-Steppers , 2006 .

[11]  C. Kelley,et al.  Newton-Krylov solvers for time-steppers , 2004, math/0404374.

[12]  Graeme W. Milton,et al.  Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics , 2003 .

[13]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[14]  H. S. Wijesinghe,et al.  Discussion of Hybrid Atomistic-Continuum Methods for Multiscale Hydrodynamics , 2004 .

[15]  Ioannis G. Kevrekidis,et al.  Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes , 2005, SIAM J. Appl. Dyn. Syst..

[16]  Ioannis G. Kevrekidis,et al.  Dynamics on Microcomposite Catalytic Surfaces: The Effect of Active Boundaries , 1999 .

[17]  Ioannis G. Kevrekidis,et al.  Coarse projective kMC integration: forward/reverse initial and boundary value problems , 2003, nlin/0307016.

[18]  W. Saarloos,et al.  PROPAGATION AND STRUCTURE OF PLANAR STREAMER FRONTS , 1997, patt-sol/9702006.

[19]  C. W. Gear,et al.  'Coarse' integration/bifurcation analysis via microscopic simulators: Micro-Galerkin methods , 2002 .

[20]  I. Kevrekidis,et al.  An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal , 2005, physics/0505179.

[21]  Christophe Vandekerckhove,et al.  Numerical stability analysis of an acceleration scheme for step size constrained time integrators , 2005 .

[22]  Dirk Roose,et al.  An Adaptive Newton-Picard Algorithm with Subspace Iteration for Computing Periodic Solutions , 1998, SIAM J. Sci. Comput..

[23]  George Papanicolaou,et al.  Convection of microstructure and related problems , 1985 .

[24]  E Weinan,et al.  The Heterogeneous Multiscale Method Based on the Discontinuous Galerkin Method for Hyperbolic and Parabolic Problems , 2005, Multiscale Model. Simul..

[25]  Paul Andries Zegeling,et al.  Moving Grid Techniques , 1998 .

[26]  R. Zwanzig Nonlinear generalized Langevin equations , 1973 .

[27]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[28]  Hans Christian Öttinger,et al.  Variance reduced simulations of polymer dynamics , 1996 .

[29]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[30]  Ludwig Boltzmann,et al.  Lectures on Gas Theory , 1964 .

[31]  E. Vanden-Eijnden,et al.  Analysis of multiscale methods for stochastic differential equations , 2005 .

[32]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[33]  Kun Xu,et al.  Numerical Navier-Stokes solutions from gas kinetic theory , 1994 .

[34]  Björn Engquist,et al.  Heterogeneous multiscale methods for stiff ordinary differential equations , 2005, Math. Comput..

[35]  Bharat K. Soni,et al.  Handbook of Grid Generation , 1998 .

[36]  H. Mori Transport, Collective Motion, and Brownian Motion , 1965 .

[37]  Ioannis G. Kevrekidis,et al.  Coarse bifurcation analysis of kinetic Monte Carlo simulations: A lattice-gas model with lateral interactions , 2002 .

[38]  Ivo Babuska,et al.  Generalized p-FEM in homogenization , 2000, Numerische Mathematik.

[39]  Yacine Ait-Sahalia,et al.  Estimating Affine Multifactor Term Structure Models Using Closed-Form Likelihood Expansions , 2002 .

[40]  E Weinan,et al.  The Heterognous Multiscale Methods , 2003 .

[41]  Tom De Wolf,et al.  Decentralised Autonomic Computing: Analysing Self-Organising Emergent Behaviour using Advanced Numerical Methods , 2005, Second International Conference on Autonomic Computing (ICAC'05).

[42]  Ioannis G. Kevrekidis,et al.  “Coarse” stability and bifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples , 2001, nlin/0111038.

[43]  B. Nadler,et al.  Diffusion maps, spectral clustering and reaction coordinates of dynamical systems , 2005, math/0503445.

[44]  Desmond J. Higham,et al.  An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..

[45]  Andrew J. Majda,et al.  A mathematical framework for stochastic climate models , 2001 .

[46]  Ioannis G. Kevrekidis,et al.  Coarse-grained numerical bifurcation analysis of lattice Boltzmann models , 2005 .

[47]  Ioannis G. Kevrekidis,et al.  Constraint-Defined Manifolds: a Legacy Code Approach to Low-Dimensional Computation , 2005, J. Sci. Comput..

[48]  Thierry Goudon,et al.  Homogenization of Transport Equations: Weak Mean Field Approximation , 2005, SIAM J. Math. Anal..

[49]  James P. Keener,et al.  Homogenization and propagation in the bistable equation , 2000 .

[50]  Markus Bär,et al.  Composite Catalyst Surfaces: Effect of Inert and Active Heterogeneities on Pattern Formation , 1996 .

[51]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[52]  L. Reichl A modern course in statistical physics , 1980 .

[53]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[54]  I. G. Kevrekidis,et al.  Higher order accuracy in the gap-tooth scheme for large-scale solutions using microscopic simulators , 2004 .

[55]  H. Risken Fokker-Planck Equation , 1984 .

[56]  Constantinos Theodoropoulos,et al.  An Input/Output Model Reduction-Based Optimization Scheme for Large-Scale Systems , 2005, Multiscale Model. Simul..

[57]  W. T. Grandy,et al.  Kinetic theory : classical, quantum, and relativistic descriptions , 2003 .

[58]  Giovanni Samaey,et al.  Damping factors for the gap-tooth scheme , 2003 .

[59]  Chi-Wang Shu Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[60]  A. Gorban,et al.  Invariant Manifolds for Physical and Chemical Kinetics , 2005 .

[61]  Pingwen Zhang,et al.  Analysis of the heterogeneous multiscale method for parabolic homogenization problems , 2007, Math. Comput..

[62]  Y. Pomeau,et al.  Lattice-gas automata for the Navier-Stokes equation. , 1986, Physical review letters.

[63]  Eleuterio F. Toro,et al.  Derivative Riemann solvers for systems of conservation laws and ADER methods , 2006, J. Comput. Phys..

[64]  Andrew M. Stuart,et al.  The Moment Map: Nonlinear Dynamics of Density Evolution via a Few Moments , 2006, SIAM J. Appl. Dyn. Syst..

[65]  Bastien Chopard,et al.  Cellular Automata and Lattice Boltzmann Techniques: an Approach to Model and Simulate Complex Systems , 2002, Adv. Complex Syst..

[66]  Andrew J. Majda,et al.  A priori tests of a stochastic mode reduction strategy , 2002 .

[67]  Ioannis G. Kevrekidis,et al.  The gap-tooth method in particle simulations , 2003 .

[68]  B. Engquist,et al.  Wavelet-Based Numerical Homogenization , 1998 .

[69]  Giovanni Samaey,et al.  Finite Difference Patch Dynamics for Advection Homogenization Problems , 2006 .

[70]  Thomas G. Kurtz,et al.  A limit theorem for perturbed operator semigroups with applications to random evolutions , 1973 .

[71]  Giovanni Samaey,et al.  Patch dynamics with buffers for homogenization problems , 2006, J. Comput. Phys..

[72]  I. Kevrekidis,et al.  Low‐dimensional models for complex geometry flows: Application to grooved channels and circular cylinders , 1991 .

[73]  N. Hadjiconstantinou Regular Article: Hybrid Atomistic–Continuum Formulations and the Moving Contact-Line Problem , 1999 .

[74]  Lowell L. Baker,et al.  Variance reduction for Monte Carlo solutions of the Boltzmann equation , 2005 .

[75]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[76]  Ioannis G. Kevrekidis,et al.  Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum , 2002, SIAM J. Sci. Comput..

[77]  Marcel Lesieur,et al.  Large-Eddy Simulations of Turbulence , 2005 .

[78]  Jacques Ferber,et al.  Multi-agent systems - an introduction to distributed artificial intelligence , 1999 .

[79]  K Asakura,et al.  Effects of Boundaries on Pattern Formation: Catalytic Oxidation of CO on Platinum , 1994, Science.

[80]  D. Gillespie A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[81]  Sidney Yip,et al.  Statistical field estimators for multiscale simulations. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[82]  Ioannis G. Kevrekidis,et al.  Application of Coarse Integration to Bacterial Chemotaxis , 2003, Multiscale Model. Simul..

[83]  Eric Vanden-Eijnden,et al.  NUMERICAL TECHNIQUES FOR MULTI-SCALE DYNAMICAL SYSTEMS WITH STOCHASTIC EFFECTS ⁄ , 2003 .

[84]  O'Connell,et al.  Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[85]  Thomas Y. Hou,et al.  Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , 1999, Math. Comput..

[86]  Holmes,et al.  Large-scale statistics of the Kuramoto-Sivashinsky equation: A wavelet-based approach. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[87]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[88]  N. Goldenfeld Lectures On Phase Transitions And The Renormalization Group , 1972 .

[89]  W. E,et al.  Matching conditions in atomistic-continuum modeling of materials. , 2001, Physical review letters.

[90]  Gautam M. Shroff,et al.  You have printed the following article : Stabilization of Unstable Procedures : The Recursive Projection Method , 2007 .

[91]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[92]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[93]  G. Bird Molecular Gas Dynamics and the Direct Simulation of Gas Flows , 1994 .

[94]  I. Kevrekidis,et al.  Coarse molecular dynamics of a peptide fragment: Free energy, kinetics, and long-time dynamics computations , 2002, physics/0212108.

[95]  L. Luo,et al.  A priori derivation of the lattice Boltzmann equation , 1997 .

[96]  Li-Shi Luo,et al.  Unified Theory of Lattice Boltzmann Models for Nonideal Gases , 1998 .

[97]  Marco Picasso,et al.  Variance reduction methods for CONNFFESSIT-like simulations , 1999 .

[98]  Giovanni Samaey,et al.  The Gap-Tooth Scheme for Homogenization Problems , 2005, Multiscale Model. Simul..

[99]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases , 1939 .

[100]  G. Milton The Theory of Composites , 2002 .

[101]  A. Stuart,et al.  Extracting macroscopic dynamics: model problems and algorithms , 2004 .

[102]  Alexandre J. Chorin,et al.  Optimal prediction with memory , 2002 .

[103]  Björn Engquist,et al.  Computation of oscillatory solutions to partial differential equations , 1987 .

[104]  Wim Vanroose,et al.  Numerical and Analytical Spatial Coupling of a Lattice Boltzmann Model and a Partial Differential Equation , 2005 .

[105]  C. D. Boor,et al.  On Calculating B-splines , 1972 .

[106]  W. Cai,et al.  Minimizing boundary reflections in coupled-domain simulations. , 2000, Physical review letters.

[107]  Tom De Wolf,et al.  Development of Self-organising Emergent Applications with Simulation-Based Numerical Analysis , 2005, Engineering Self-Organising Systems.

[108]  Giovanni Samaey,et al.  Analysis of a lattice Boltzmann model for planar streamer fronts , 2006 .

[109]  E Weinan,et al.  A dynamic atomistic-continuum method for the simulation of crystalline materials , 2001 .

[110]  I. Kevrekidis,et al.  Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation , 1990 .

[111]  I. G. Kevrekidis,et al.  Coarse Brownian dynamics for nematic liquid crystals: Bifurcation, projective integration, and control via stochastic simulation , 2003 .

[112]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[113]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[114]  M. Ortiz,et al.  Quasicontinuum analysis of defects in solids , 1996 .

[115]  Roger G. Ghanem,et al.  A Multiscale Data Assimilation with the Ensemble Kalman Filter , 2005, Multiscale Model. Simul..

[116]  M. Ortiz,et al.  An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method , 1997, cond-mat/9710027.

[117]  D. Lemons,et al.  Paul Langevin’s 1908 paper “On the Theory of Brownian Motion” [“Sur la théorie du mouvement brownien,” C. R. Acad. Sci. (Paris) 146, 530–533 (1908)] , 1997 .

[118]  Constantinos Theodoropoulos,et al.  Effective bifurcation analysis: a time-stepper-based approach , 2002 .

[119]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[120]  Giovanni Samaey,et al.  Combining the Gap-Tooth Scheme with Projective Integration: Patch Dynamics , 2005 .

[121]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[122]  Radek Erban,et al.  From Individual to Collective Behavior in Bacterial Chemotaxis , 2004, SIAM J. Appl. Math..

[123]  Shivkumar Chandrasekaran,et al.  Fast and Stable Algorithms for Banded Plus Semiseparable Systems of Linear Equations , 2003, SIAM J. Matrix Anal. Appl..

[124]  Richard B. Lehoucq,et al.  Large‐scale eigenvalue calculations for stability analysis of steady flows on massively parallel computers , 2001 .

[125]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[126]  A J Chorin,et al.  Optimal prediction of underresolved dynamics. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[127]  T. Hou,et al.  Why nonconservative schemes converge to wrong solutions: error analysis , 1994 .

[128]  Francis Heylighen,et al.  Self-organization, Emergence and the Architecture of Complexity , 1989 .

[129]  Stanley Osher,et al.  Convergence of Generalized MUSCL Schemes , 1985 .

[130]  Eirik Grude Flekkøy,et al.  Hybrid model for combined particle and continuum dynamics , 2000 .

[131]  Panayotis G. Kevrekidis,et al.  Deciding the Nature of the Coarse Equation through Microscopic Simulations: The Baby-Bathwater Scheme , 2003, Multiscale Model. Simul..

[132]  R. Fisher THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[133]  P. Prescott,et al.  Monte Carlo Methods , 1964, Computational Statistical Physics.

[134]  H. Van Dyke Parunak,et al.  Agent-Based Modeling vs. Equation-Based Modeling: A Case Study and Users' Guide , 1998, MABS.

[135]  Constantinos Theodoropoulos,et al.  Coarse bifurcation studies of bubble flow lattice Boltzmann simulations , 2004 .

[136]  Nicolas G. Hadjicostantinou COMBINING ATOMISTIC AND CONTINUUM SIMULATIONS OF CONTACT-LINE MOTION , 1999 .

[137]  Ioannis G. Kevrekidis,et al.  Boundary processing for Monte Carlo Simulations in the Gap-Tooth Scheme , 2002 .

[138]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[139]  Geri Wagner,et al.  Coupling molecular dynamics and continuum dynamics , 2002 .

[140]  P. Donato,et al.  An introduction to homogenization , 2000 .

[141]  Berend Smit,et al.  Understanding molecular simulation: from algorithms to applications , 1996 .

[142]  M. Ortiz,et al.  An analysis of the quasicontinuum method , 2001, cond-mat/0103455.

[143]  Randall J. LeVeque Finite Volume Methods for Hyperbolic Problems: Conservation Laws and Differential Equations , 2002 .

[144]  Shiyi Chen,et al.  Lattice Boltzmann computations for reaction‐diffusion equations , 1993 .

[145]  C. W. Gear,et al.  Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis , 2003 .