Petr Plechac

On Functional Separately Convex Hulls

Abstract. Let D be a set of vectors in Rd . A function f:Rd→R is called D-convex if its restriction to each line parallel to a nonzero vector of D is a convex function. For a set A⊆Rd , the functional D-convex hull of A, denoted by coD(A) , is the intersection of the zero sets of all nonnegative D -convex functions that are 0 on A . We prove some results concerning the structure of functional D -convex hulls, e.g., a Krein—Milman-type theorem and a result on separation of connected components. We give a polynomial-time algorithm for computing coD(A) for a finite point set A (in any fixed dimension) in the case of D being a basis of Rd (the case of separate convexity). This research is primarily motivated by questions concerning the so-called rank-one convexity, which is a particular case of D -convexity and is important in the theory of systems of nonlinear partial differential equations and in mathematical modeling of microstructures in solids. As a direct contribution to the study of rank-one convexity, we construct a configuration of 20 symmetric 2 x 2 matrices in a general (stable) position with a nontrivial functionally rank-one convex hull (answering a question of K. Zhang on the existence of higher-dimensional nontrivial configurations of points and matrices).

Numerical Analysis of Compatible Phase Transitions in Elastic Solids

The variational model of phase transitions for elastic materials based on linearized elasticity leads to a nonconvex minimization problem (P) in which a minimum need not be attained. In the design of advanced materials, the main interest is in reliable numerical predictions of certain macroscopic quantities such as the global deformation and the stress field determined in a relaxed problem (QP). An explicit formula of the quasi-convexified energy density in (QP) due to R. V. Kohn provides us with a well-posed numerical problem. First, a mathematical a priori and a posteriori error analysis is established for the finite element approximation of the stress variable; then the residual based error indicator is implemented within an adaptive mesh-refinement algorithm. Numerical examples illustrate that the macroscopic properties of the materials are computed efficiently with appropriate error control.

Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems.

In this paper, we focus on the development of new methods suitable for efficient and reliable coarse-graining of non-equilibrium molecular systems. In this context, we propose error estimation and controlled-fidelity model reduction methods based on Path-Space Information Theory, combined with statistical parametric estimation of rates for non-equilibrium stationary processes. The approach we propose extends the applicability of existing information-based methods for deriving parametrized coarse-grained models to Non-Equilibrium systems with Stationary States. In the context of coarse-graining it allows for constructing optimal parametrized Markovian coarse-grained dynamics within a parametric family, by minimizing information loss (due to coarse-graining) on the path space. Furthermore, we propose an asymptotically equivalent method-related to maximum likelihood estimators for stochastic processes-where the coarse-graining is obtained by optimizing the information content in path space of the coarse variables, with respect to the projected computational data from a fine-scale simulation. Finally, the associated path-space Fisher Information Matrix can provide confidence intervals for the corresponding parameter estimators. We demonstrate the proposed coarse-graining method in (a) non-equilibrium systems with diffusing interacting particles, driven by out-of-equilibrium boundary conditions, as well as (b) multi-scale diffusions and the corresponding stochastic averaging limits, comparing them to our proposed methodologies.

Singular and regular solutions of a nonlinear parabolic system

We study a dissipative nonlinear equation modelling certain features of the Navier–Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions n≤4. For dimensions n>4, we present strong numerical evidence supporting the existence of blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding the existence of self-similar singular solutions to a semi-linear heat equation.

Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms

We present a mathematical framework for constructing and analyzing parallel algorithms for lattice kinetic Monte Carlo (KMC) simulations. The resulting algorithms have the capacity to simulate a wide range of spatio-temporal scales in spatially distributed, non-equilibrium physiochemical processes with complex chemistry and transport micro-mechanisms. Rather than focusing on constructing exactly the stochastic trajectories, our approach relies on approximating the evolution of observables, such as density, coverage, correlations and so on. More specifically, we develop a spatial domain decomposition of the Markov operator (generator) that describes the evolution of all observables according to the kinetic Monte Carlo algorithm. This domain decomposition corresponds to a decomposition of the Markov generator into a hierarchy of operators and can be tailored to specific hierarchical parallel architectures such as multi-core processors or clusters of Graphical Processing Units (GPUs). Based on this operator decomposition, we formulate parallel Fractional step kinetic Monte Carlo algorithms by employing the Trotter Theorem and its randomized variants; these schemes, (a) are partially asynchronous on each fractional step time-window, and (b) are characterized by their communication schedule between processors. The proposed mathematical framework allows us to rigorously justify the numerical and statistical consistency of the proposed algorithms, showing the convergence of our approximating schemes to the original serial KMC. The approach also provides a systematic evaluation of different processor communicating schedules. We carry out a detailed benchmarking of the parallel KMC schemes using available exact solutions, for example, in Ising-type systems and we demonstrate the capabilities of the method to simulate complex spatially distributed reactions at very large scales on GPUs. Finally, we discuss work load balancing between processors and propose a re-balancing scheme based on probabilistic mass transport methods.

Path-Space Information Bounds for Uncertainty Quantification and Sensitivity Analysis of Stochastic Dynamics

Uncertainty quantification is a primary challenge for reliable modeling and simulation of complex stochastic dynamics. Such problems are typically plagued with incomplete information that may enter as uncertainty in the model parameters, or even in the model itself. Furthermore, due to their dynamic nature, we need to assess the impact of these uncertainties on the transient and long-time behavior of the stochastic models and derive corresponding uncertainty bounds for observables of interest. A special class of such challenges is parametric uncertainties in the model and in particular sensitivity analysis along with the corresponding sensitivity bounds for stochastic dynamics. Moreover, sensitivity analysis can be further complicated in models with a high number of parameters that render straightforward approaches, such as gradient methods, impractical. In this paper, we derive uncertainty and sensitivity bounds for path-space observables of stochastic dynamics in terms of new goal-oriented divergences; the latter incorporate both observables and information theory objects such as the relative entropy rate. These bounds are tight, depend on the variance of the particular observable and are computable through Monte Carlo simulation. In the case of sensitivity analysis, the derived sensitivity bounds rely on the path Fisher Information Matrix, hence they depend only on local dynamics and are gradient-free. These features allow for computationally efficient implementation in systems with a high number of parameters, e.g., complex reaction networks and molecular simulations.

Numerical and Statistical Methods for the Coarse-Graining of Many-Particle Stochastic Systems

In this article we discuss recent work on coarse-graining methods for microscopic stochastic lattice systems. We emphasize the numerical analysis of the schemes, focusing on error quantification as well as on the construction of improved algorithms capable of operating in wider parameter regimes. We also discuss adaptive coarse-graining schemes which have the capacity of automatically adjusting during the simulation if substantial deviations are detected in a suitable error indicator. The methods employed in the development and the analysis of the algorithms rely on a combination of statistical mechanics methods (renormalization and cluster expansions), statistical tools (reconstruction and importance sampling) and PDE-inspired analysis (a posteriori estimates). We also discuss the connections and extensions of our work on lattice systems to the coarse-graining of polymers.

PARALLELIZATION, PROCESSOR COMMUNICATION AND ERROR ANALYSIS IN LATTICE KINETIC MONTE CARLO ∗

In this paper we study from a numerical analysis perspective the Fractional Step Kinetic Monte Carlo (FS-KMC) algorithms proposed in (1) for the parallel simulation of spatially distributed particle systems on a lattice. FS-KMC are fractional step algorithms with a time-stepping windowt, and as such they are inherently partially asynchronous since there is no processor com- munication during the periodt. In this contribution we primarily focus on the error analysis of FS-KMC algorithms as approximations of conventional, serial kinetic Monte Carlo (KMC). A key as- pect of our analysis relies on emphasizing a goal-oriented approach for suitably defined macroscopic observables (e.g., density, energy, correlations, surface roughness), rather than focusing on strong topology estimates for individual trajectories. One of the key implications of our error analysis is that it allows us to address systematically the processor communication of different parallelization strategies for KMC by comparing their (partial) asynchrony, which in turn is measured by their respective fractional time stept for a prescribed error tolerance.

Multibody Interactions in Coarse-Graining Schemes for Extended Systems

In this paper we address the role of multibody interactions for the coarse-grained approximation of stochastic lattice systems. Such interaction potentials are often not included in coarse-graining schemes, as they can be computationally expensive. The multibody interactions are obtained from the error expansion of the reference measure which is, in many cases, chosen as a Gibbs measure corresponding to a local mean-field approximation. We identify the parameter

Path-space variational inference for non-equilibrium coarse-grained systems

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