The Heterognous Multiscale Methods
01 Mar 2003-Communications in Mathematical Sciences (International Press of Boston)-Vol. 1, Iss: 1, pp 87-132
TL;DR: The heterogenous multiscale method (HMM) as mentioned in this paper is a general methodology for the efficient numerical computation of problems with multiscales and multiphysics on multigrids.
Abstract: The heterogenous multiscale method (HMM) is presented as a general methodology for the efficient numerical computation of problems with multiscales and multiphysics on multigrids. Both variational and dynamic problems are considered. The method relies on an efficent coupling between the macroscopic and microscopic models. In cases when the macroscopic model is not explicity available or invalid, the microscopic solver is used to supply the necessary data for the microscopic solver. Besides unifying several existing multiscale methods such as the ab initio molecular dynamics [13], quasicontinuum methods [73,69,68] and projective methods for systems with multiscales [34,35], HMM also provides a methodology for designing new methods for a large variety of multiscale problems. A framework is presented for the analysis of the stability and accuracy of HMM. Applications to problems such as homogenization, molecular dynamics, kinetic models and interfacial dynamics are discussed.
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TL;DR: A number of simple model systems where the coarse-grained or macroscopic behaviour of a system can be explicitly determined from the full, or microscopic, description are described.
Abstract: In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of coarse-grained, or macroscopic, quantities. The purpose of this review is twofold: first, to describe a number of simple model systems where the coarse-grained or macroscopic behaviour of a system can be explicitly determined from the full, or microscopic, description; and second, to overview some of the emerging algorithmic approaches that have been introduced to extract effective, lower-dimensional, macroscopic dynamics. The model problems we describe may be either stochastic or deterministic in both their microscopic and macroscopic behaviour, leading to four possibilities in the transition from microscopic to macroscopic descriptions. Model problems are given which illustrate all four situations, and mathematical tools for their study are introduced. These model problems are useful in the evaluation of algorithms. We use specific instances of the model problems to illustrate these algorithms. As the subject of algorithm development and analysis is, in many cases, in its infancy, the primary purpose here is to attempt to unify some of the emerging ideas so that individuals new to the field have a structured access to the literature. Furthermore, by discussing the algorithms in the context of the model problems, a platform for understanding existing algorithms and developing new ones is built.
440 citations
TL;DR: Over the last few years with several collaborators, a mathematically inspired, computational enabling technology is developed and validated that allows the modeler to perform macroscopic tasks acting on the microscopic models directly, and can lead to experimental protocols for the equation-free exploration of complex system dynamics.
Abstract: The best available descriptions of systems often come at a fine level (atomistic, stochastic, microscopic, agent based), whereas the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization, and control) are at a much coarser, macroscopic level. Traditional modeling approaches start by deriving macroscopic evolution equations from microscopic models, and then bringing an arsenal of computational tools to bear on these macroscopic descriptions. Over the last few years with several collaborators, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the “equation-free” approach, since it circumvents the step of obtaining accurate macroscopic descriptions. The backbone of this approach is the design of computational “experiments”. In traditional numerical analysis, the main code “pings“ a subroutine containing the model, and uses the returned information (time derivatives, etc.) to perform computer-assisted analysis. In our approach the same main code “pings“ a subroutine that runs an ensemble of appropriately initialized computational experiments from which the same quantities are estimated. Traditional continuum numerical algorithms can, thus, be viewed as protocols for experimental design (where “experiment“ means a computational experiment set up, and performed with a model at a different level of description). Ultimately, what makes it all possible is the ability to initialize computational experiments at will. Short bursts of appropriately initialized computational experimentation -through matrix-free numerical analysis, and systems theory tools like estimationbridge microscopic simulation with macroscopic modeling. If enough control authority exists to initialize laboratory experiments “at will” this computational enabling technology can lead to experimental protocols for the equation-free exploration of complex system dynamics.
391 citations
TL;DR: This review paper summarizes surrogate modeling techniques in three categories: data‐driven, projection, and hierarchical‐based approaches, which approximate a groundwater model through an empirical model that captures the input‐output mapping of the original model.
Abstract: The spatially and temporally variable parameters and inputs to complex groundwater models typically result in long runtimes which hinder comprehensive calibration, sensitivity, and uncertainty analysis. Surrogate modeling aims to provide a simpler, and hence faster, model which emulates the specified output of a more complex model in function of its inputs and parameters. In this review paper, we summarize surrogate modeling techniques in three categories: data-driven, projection, and hierarchical-based approaches. Data-driven surrogates approximate a groundwater model through an empirical model that captures the input-output mapping of the original model. Projection-based models reduce the dimensionality of the parameter space by projecting the governing equations onto a basis of orthonormal vectors. In hierarchical or multifidelity methods the surrogate is created by simplifying the representation of the physical system, such as by ignoring certain processes, or reducing the numerical resolution. In discussing the application to groundwater modeling of these methods, we note several imbalances in the existing literature: a large body of work on data-driven approaches seemingly ignores major drawbacks to the methods; only a fraction of the literature focuses on creating surrogates to reproduce outputs of fully distributed groundwater models, despite these being ubiquitous in practice; and a number of the more advanced surrogate modeling methods are yet to be fully applied in a groundwater modeling context.
355 citations
Cites background from "The Heterognous Multiscale Methods"
...The gain in computational efficiency opens the door for exploration of structural model uncertainty by simultaneous simulation and calibration of alternative model structures [Matott and Rabideau, 2008] or inclusion of data and physical processes at multiple scales [Weinan and Engquist, 2003]....
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TL;DR: In this article, the authors review Lagrangian, multiresolution, particle methods such as vortex methods and smooth particle hydrodynamics for continuous flows and molecular dynamics for the simulation of flows at the atomistic scale.
Abstract: ▪ Abstract Flow simulations are one of the archetypal multiscale problems. Simulations of turbulent and unsteady separated flows have to resolve a multitude of interacting scales, whereas molecular phenomena determine the structure of shocks and the validity of the no-slip boundary condition. Particle simulations of continuum and molecular phenomena can be formulated by following the motion of interacting particles that carry the physical properties of the flow. In this article we review Lagrangian, multiresolution, particle methods such as vortex methods and smooth particle hydrodynamics for the simulation of continuous flows and molecular dynamics for the simulation of flows at the atomistic scale. We review hybrid molecular-continuum simulations with an emphasis on the computational aspects of the problem. We identify the common computational characteristics of particle methods and discuss their properties that enable the formulation of a systematic framework for multiscale flow simulations.
332 citations
TL;DR: A framework for computer-aided multiscale analysis is reviewed, which enables macroscopic computational tasks (over extended spatiotemporal scales) using only appropriately initialized microscopic simulation on short time and length scales.
Abstract: In traditional physicochemical modeling, one derives evolution equations at the (macroscopic, coarse) scale of interest; these are used to perform a variety of tasks (simulation, bifurcation analysis, optimization) using an arsenal of analytical and numerical techniques. For many complex systems, however, although one observes evolution at a macroscopic scale of interest, accurate models are only given at a more detailed (fine-scale, microscopic) level of description (e.g., lattice Boltzmann, kinetic Monte Carlo, molecular dynamics). Here, we review a framework for computer-aided multiscale analysis, which enables macroscopic computational tasks (over extended spatiotemporal scales) using only appropriately initialized microscopic simulation on short time and length scales. The methodology bypasses the derivation of macroscopic evolution equations when these equations conceptually exist but are not available in closed form—hence the term equation-free. We selectively discuss basic algorithms and underlyin...
305 citations