Multiscale Theory and Computation

Open House and Reception — 2:00pm to 7:00pm, Lind 400

Breakfast — 8:30am - 9:00am, Lind 400

Lectures — 9:00am - 12:00pm

9:00 - Opening Remarks
 

9:10 - David Kinderlehrer (CMU): "Towards a Gradient Flow for Microstructure"

A central problem of microstructure is to develop technologies capable of producing an arrangement, or ordering, of the material, in terms of mesoscopic parameters like geometry and crystallography, appropriate for a given application. Is there such an order in the first place? We describe the emergence of the grain boundary character distribution (GBCD), a statistic that details texture evolution, and illustrate why it should be considered a material property. Its identification as a gradient flow by our method is tantamount to exhibiting the harvested statistic as the iterates in a mass transport JKO implicit scheme, which we found astonishing. Consequently the GBCD is the solution of a Fokker-Planck Equation. The development exposes the question of how to understand the circumstances under which a harvested empirical statistic is a property of the underlying process.
(joint work with P. Bardsley, K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, X.-Y. Lu and S. Ta'asan)

9:50 - Richard James (Minnesota): "Atomistically Inspired Origami"

World population is growing approximately linearly at about 80 million per year. Thus there is necessarily less space per person, and this is particularly acute in cities. Perhaps this is why the scientific community seems to be obsessed with folding things. We develop a mathematical approach to “rigid folding” based on the use of piecewise rigid isometric mappings that have a group structure. The ideas are inspired by the way atomistic structures form naturally. Their characteristic features in molecular science imply highly desirable features for macroscopic structures, particularly 4D structures that deform. We illustrate these by constructing some “objective origami” structures.

10:30 - Coffee Break

11:00 - Irene Fonseca (CMU): "Epitaxially Strained Elastic Films: Quantum Dots and Dislocations"

The formation and assembly patterns of quantum dots play a central role on the optoelectronic properties of semiconductors. We will address quasistatic equilibria and regularity of islands (quantum dots), and using De Giorgi's minimizing movements we will obtain short time existence for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained films. A variational model for the nucleation of misfit dislocations will be analyzed.

11:40 - Dio Margetis (Maryland): "The Problem of Small Temperatures in a Quantum Gas"

In this talk, I will discuss recent progress in understanding the fundamentals of a system of interacting particles with Boson statistics at low temperature. A topic to be addressed is the description of this system beyond the usual mean-field-type approach. Mathematically, the underlying formalism involves the passage from the microscopic dynamics of mixed states to coupled PDEs at the macroscopic scale.

Lunch — 12:00pm - 2:00pm, Lind 400

Lectures — 2:00pm - 5:00pm

2:00 - Stanley Osher (UCLA): "Overcoming the Curse of Dimensionality"

It is well known that certain Hamilton-Jacobi partial differential equations (HJ PDE’s) play an important role in analyzing control theory and differential games. The cost of standard numerical algorithms for HJ PDE’s is exponential in the space dimension and time, with huge memory requirements. Here we propose and test methods for solving a large class of these problems without the use of grids or significant numerical approximation. We begin with the classical Hopf and Hopf-Lax formulas which enable us to solve state independent problems via variational methods originating in compressive sensing with remarkable results.
We can evaluate the solution in 10-4 to 10-8 seconds per evaluation on a laptop. The method is embarrassingly parallel and has low memory requirements. Recently, with a slightly more complicated, but still embarrassingly parallel method, we have extended this in great generality to state dependent HJ equations,, with the help of parallel computers, overcoming the curse of dimensionality for these problems, deriving new Lax and Hopf type formulas in very general situations The term, “curse of dimensionality” was coined by Richard Bellman in 1957 when he did his classic work on dynamic optimization.
This is joint work with Jerome Darbon, Yat-Tin Chow, Matthew Kirchner, Gary Hewer and Alex Lin.

2:40 - Thomas Hou (Caltech): "Sparse Operator Compression for Higher Elliptic PDEs with Rough Coefficients and Graph Laplacians"

We introduce the sparse operator compression to compress a self-adjoint higher order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy minimizing functions on local patches. On a regular mesh with mesh size h, the localized basis functions have supports of diameter O(h log(1/h)), and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter O(h log(1/h)), our method achieves the optimal compression rate of the solution operator. From the perspective of the multiscale finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate O(h^k) for a (2k)th-order elliptic problem in the energy norm. From the perspective of the Sparse PCA, our results show that a large set of Matern covariance functions can be approximated by a rank-n operator with a localized basis and with the optimal accuracy. Finally, we discuss some recent progress in extending this method to solve graph Laplacians with nearly linear complexity.

3:20 - Coffee Break
 

3:40 - Bjorn Engquist (Texas): "Multiscale Dynamical Systems"

Molecular and planetary systems and are example of multiscale ordinary differential equations with highly oscillatory solutions and they pose severe challenges for computer simulations. Distributed computing is a natural tool for large-scale simulations but causality makes it challenging to develop parallel algorithms for dynamical systems. The parareal framework has been successful for dissipative systems. We will discuss applications to the more difficult case of Hamiltonian systems by using phase plane maps as coarse solver.

4:20 - John Lowengrub (UC Irvine): "BioPFC: A Hybrid Continuum-discrete Mechanical Collective Cell Model"

We present a mechanistic hybrid continuum-discrete model to simulate the dynamics of epithelial cell colonies. Collective cell dynamics are modeled using continuum equations that capture plastic, viscoelastic and elastic deformations in the clusters while providing single-cell resolution. The continuum equations can be viewed as a coarse-grained version of previously developed discrete models that treat epithelial clusters as a two-dimensional network of vertices or stochastic interacting particles, following the framework of dynamic density functional theory appropriately modified to account for cell size and shape variability. The discrete component of the model implements cell division and thus influences cell sizes and shapes that couple to the continuum component. The model is validated against recent in vitro studies of epithelial cell colonies using Madin-Darby canine kidney cells. In good agreement with the experiments, we find that mechanical interactions and constraints on the local expansion of cell size cause inhibition of cell motion and reductive cell division. This leads to successively smaller cells and a transition from exponential to quadratic growth of the colony that is associated with the emergence of short-range ordering and a constant-thickness proliferating rim of cells at the cluster edge. A detailed analysis of the model reveals a scale-invariance of the growth and provides insight on the generation of stresses and their influence on the dynamics of the colonies. Compared to previous models, our approach has several advantages: (i) it is dimension independent, (ii) it can be parametrized using classical elastic properties (Poisson's ratio and Young's modulus), and (iii) it can easily be extended to incorporate multiple cell types and general substrate geometries.

4:40 - Todd. F Dupont (Chicago) "A Simple Step-Doubling Scheme for Stiff Differential Equations"

In his thesis, Mitchell Luskin analyzed a "box scheme" for flow in pipeline networks. This scheme for simple first order systems of one-space dimensional time-dependent partial differential equations had been popularized by Herb Keller, and was based on earlier work by Vidar Thomee. Luskin's analysis provided a solid scientific basis for a nonlinear problem that had received scant attention, and this method is still widely used in pipeline engineering.  In the box scheme, the time and space discretizations are closely related and the most useful versions of this method result in a first-order time integration method.

In this talk I will discuss a method that combines step-doubling with local extrapolation and a Taylor-method linearization that applies to systems of ordinary differential equations. The motivation for examining this method came from optimal control problems in pipeline simulation. The simplicity of the method is particularly useful in forming the adjoint systems that arise in gradient computations for these problems.

Banquet — 6:00pm - 9:30pm, Weisman Art Museum 

(5:30pm Museum open for banquet guests, 6:00pm Drinks and Appetizers, 7:00pm Dinner) 

Breakfast — 8:30am - 9:00am, Lind 400

Lectures — 9:00am - 12:00pm

9:00 - Efthimios Kaxiras (Harvard) "Wandering in Flatland: the Wonders of 2D Materials" 

Two-dimensional materials, having started as a curiosity with graphene, are becoming ripe for real applications. In this presentation we survey some theoretical predictions of how 2D layers of different materials behave when stacked in different arrangements, in terms of their response to external fields and to excitation by light.

9:40 - Eric Cances (ENPC) "Computational Non-commutative Geometry for Materials Science: The Example of Multilayer 2D Materials"

After recalling the standard mathematical formalism used to model disordered systems such as random composite materials (mesoscale disorder), doped semiconductors, alloys, or amorphous materials (atomic-scale disorder), I will present a tight-binding model for computing the electrical conductivity of incommensurate multilayer 2D materials. All these models fall into the scope of the mathematical framework, based on non-commutative geometry, introduced by Bellissard to study the physical properties of aperiodic systems. Surprisingly, this rather abstract theoretical framework leads to completely new numerical schemes allowing one to perform simulations out of the scope of usual methods.

10:20 - Coffee Break
 

10:40 - Claude Le Bris (ENPC and INRIA) "Nonperiodic Multiscale Problems: Some Recent Mathematical and Numerical Advances" 

We will review a set of works devoted to the modelling and numerical simulation of multiscale materials. The general purpose of the set of approaches developed is to be able to address realistic materials with nonperiodic microstructures, and possibly defects. Several aspects will be presented, both theoretical and numerical. In particular, one line of research described aims at making multiscale finite element methods more versatile and robust.

The works are joint works with Xavier Blanc (Paris 7), Pierre-Louis Lions (Collège de France), Frédéric Legoll (Ecole des Ponts & Inria), and many other collaborators.

11:20 - Yalchin Efendiev (Texas A&M) "Multiscale Model Reduction for Heterogeneous Problems" 

In this talk, I will discuss some general strategies for coarse-grid multiscale model reduction techniques for heterogeneous problems. The proposed methods are designed for problems with high contrast and multiple scales. I will also describe applications to inverse problems.

11:40 - Poster Blitz 

Lunch — 12:15pm - 1:00pm, Lind 400 

Poster Session — 1:00pm - 1:50pm, Lind 400

Lectures — 2:00pm - 5:00pm

2:00 - Art Voter (LANL) "Increasing the Power of Accelerated Molecular Dynamics Methods"

Many important materials processes take place on time scales that vastly exceed the few microseconds accessible to molecular dynamics simulation.  Typically, this long-time dynamical evolution is characterized by a succession of thermally activated infrequent events involving defects in the material.  In the accelerated molecular dynamics (AMD) methodology, known characteristics of infrequent-event systems are exploited to make reactive events take place more frequently, in a dynamically correct way.  The key feature is that the trajectory itself is allowed to find its own way out of each state, so that no prior assumptions need to be made about the transition states or available reaction paths.  These methods have proven powerful for a range of processes, including metallic surface diffusion and growth, radiation damage annealing processes, and nanotube and nanowire dynamics.  In this talk, I will discuss some recent advances that extend the power and range of applicability of the AMD methods.

2:40 - Tony Lelievre (ENPC) "Metastability: a Journey from Stochastic Processes to Semiclassical Analysis"

A stochastic process is metastable if it stays for a very long period of time in a region of the phase space (called a metastable region) before going to another metastable region, where it again remains trapped. Such processes naturally appear in many applications, metastability being related to a two time scale mechanism: the small time scale corresponds to the vibration period within the metastable regions and the large time scale is associated with the transitions between metastable states. For example, in molecular dynamics, the metastable regions are typically associated with the atomic conformations of a molecule (or an ensemble of molecules), and one is actually interested in simulating and studying the transitions between these conformations.

In this talk, I will explain how the exit events from a metastable state can be studied using an eigenvalue problem for a differential operator. This point of view is useful to build very efficient algorithms to simulate metastable stochastic processes (using in particular parallel architectures). It also gives a new way to prove the Eyring-Kramers laws and to justify the parametrization of an underlying Markov chain (Markov state model), using techniques form semiclassical analysis.

3:20 - Coffee Break
 

3:40 - Gideon Simpson (Drexel) "Spin-Diffusions and Diffusive Molecular Dynamics"

Diffusive Molecular Dynamics (DMD) is a novel approach that aims to reach diffusive time scales of milliseconds and beyond while retaining atomistic detail. To accomplish this, DMD “averages out” the vibrational time scale of femtoseconds and evolves probability densities at atomistic sites. This requires the approximation of a probability distribution in an extended state space by a synthetic distribution that can easily be sampled. The mean occupancies at the atomic sites are then evolved under a system of ODEs, using a so-called Master Equation, but no underlying stochastic process is given in the current formulation.  Here, we propose and examine a stochastic process, coupling a diffusion to a spin system, which gives rise to dynamics close to that of existing DMD models. This highlights the key assumptions and approximations of DMD, constrains the form of the system of ODEs, and offers a way to connect DMD to classical MD. 

4:00 - Vladimir Sverak (Minnesota) "On Regularity and Long-time Behavior of 1D Models"

In the 1980s, P. Constantin, P. Lax, and A. Majda introduced a 1d equation which models some of the features of the 3d incompressible Euler equation, showed that the model equation is explicitly solvable, and has solutions which can develop singularities from smooth data. There are modifications of the 1d equation for which the regularity question remains open. I will discuss recent results concerning these models. Some of the results were motivated by numerical simulations.

Reception — 5:00pm - 6:15pm , Lind 400

Dinner — 6:30pm - 9:00pm, University Hall, McNamara Alumni Center

Breakfast — 8:30am - 9:00am, Lind 400

Lectures — 9:00am - 12:00pm

9:00 - Arnd Scheel (Minnesota) - "Moduli Spaces of Growth Patterns"

Interfaces or boundaries affect the formation of crystalline phases in sometimes quite dramatic ways. Examples range from the alignment of convection roles in Benard convection perpendicular to the boundary, to the robust patterning through presomites in limb formation. Mathematically, the object of interest is a moduli space of solutions to elliptic equations in unbounded domains. This moduli space contains the relation between rate of growth and crystallographic parameters such as the width and orientation of convection rolls or presomites. I will explain the role of this moduli space and give results and conjectures on its shape in examples, starting with simple convection-diffusion and phase separation problems, and concluding with Turing patterns.

9:40 -  Petr Plechac (Delaware) "Parallel Replica Dynamics for Sampling and Sensitivity Analysis of Multi-scale Stochastic Reaction Networks" 

Stochastic reaction networks that exhibit metastable behavior are common in chemical reaction kinetics, systems biology as well as materials science. Sampling of the stationary distribution is crucial for understanding and characterizing the long term dynamics of stochastic dynamical systems. However, this task is normally hindered by the insufficient sampling of the rare transitions between metastable regions.

We present parallel replica dynamics for accelerating simulations of continuous time Markov chains in the presence of metastability. We demonstrate that the proposed method accelerates stationary distribution sampling and yields correct stationary averaging. We explain how it can be combined with path-space information bounds on path-dependent functionals and risk sensitive functionals. Such bounds provide error estimates on quantities of interest as well as bounds on parametric sensitivity in the complex reaction networks.

10:00 - Frederic Legoll (ENPC) "Approximating the Fluctuations in Random Heterogeneous Problems"

Computing the homogenized properties of random materials is often very expensive. A standard approach consists in considering a large domain, and solving the so-called corrector problem on that domain, submitted to e.g. periodic boundary conditions. Because the computational domain is finite, the approximate homogenized properties are random, and fluctuate from one realization of the microstructure to another. We have recently introduced several efficient numerical approaches to reduce the statistical noise. These approaches allow to compute the expectation of the homogenized coefficients in a more efficient manner than brute force Monte Carlo methods.

Beside the (averaged) behavior of the material response on large space scales (which is given by its homogenized limit), another question of interest is to understand how much this response fluctuates around its coarse approximation, before the homogenized regime is attained. More generally, we aim at understanding which parameters of the distribution of the material coefficients affect the distribution of the response, and whether it is possible to compute that latter distribution without resorting to a brute force Monte Carlo approach.

This talk, based on joint works with P.-L. Rothe, will review the recent progresses made on these questions.

10:20 - Coffee Break
 

10:40 - Christoph Ortner (Warwick) "The Numerical Analysis of Hybrid Models for Crystalline Defects"

For many questions of scientific interest, all-atom molecular dynamics simulations are still out of reach, for example when accurate and hence computationally expensive force-fields are used, or in materials modelling where large numbers of atoms are required. A variety of multi-scale / hybrid models has been developed to reduce computational cost.

In this talk, I will review the {\em numerical analysis} of these hybrid techniques which was spear-headed by Mitch Luskin and his research group. I will demonstrate how classical numerical analysis concepts can be applied to (1) identify the various errors committed in the coarse-graining process and (2) how to apply this analysis to optimise practical computational schemes.

11:20 - Alexander Shapeev (Skoltech) "Machine Learning-based Interatomic Potentials"

Molecular simulations (such as molecular dynamics) is the largest consumer of supercomputing time worldwide. Molecular simulations rely on one of the two models: very accurate and very computationally expensive quantum-mechanical models that resolve the electronic structure, and empirical interatomic models that postulate a simple functional form of interatomic interaction that is fast to compute. Application of ideas of machine learning has recently been put forward as a promising way to get the best out of these two models: accuracy of quantum mechanics and computational efficiency of the interatomic potentials.

11:40 - Bo Li (UCSD) "Multi-Scale Modeling and Simulation of the Growth of Bacterial Colony with Cell-Cell Mechanical Interactions"

The growth of bacterial colony exhibits striking patterns that are determined by the interactions among individual, growing and dividing bacterial cells, and that between cells and the surrounding nutrient and waste. Understanding the principles that underlie such growth has far-reaching consequences in biological and health sciences. In this work, we construct a multi-scale model of the growth of E. coli cells on agar surface. Our model consists of detailed, microscopic descriptions of the cell growth, cell division with fluctuations, and cell movement due to the cell-cell and cell-environment mechanical interactions, and macroscopic diffusion equations for the nutrient and waste. We use the velocity Verlet algorithm to simulate the motion of individual cells and iterative algorithm to update the nutrient. Our large-scale simulations reproduce experimentally observed growth scaling laws, strip patterns, and many other features of an E. coli colony. This work is the first step toward detailed multi-scale computational modeling of three-dimensional bacterial growth with mechanical and chemical interactions. This is joint work with Mya Warren, Hui Sun, Yue Yan, and Terry Hwa.

Lunch — 12:00pm - 2:00pm

Afternoon Informal Discussion — 2:00pm - 4:00pm