Amit Varshney

Reduced order modeling and dynamic optimization of multiscale PDE/kMC process systems

Abstract The problem of dynamic optimization for multiscale systems comprising of coupled continuum and discrete descriptions is considered. The solution of such problems is challenging owing to large computational requirements of the multiscale process model. This problem is addressed by developing a reduced multiscale model. This is achieved by combining order reduction techniques for dissipative partial-differential equations with adaptive tabulation of microscopic simulation data. The multiscale process optimization problem is subsequently solved using standard search algorithms. The proposed method is applied to two representative catalytic oxidation processes where optimal inlet concentration profiles are computed to guide the microscopic system from one stable stationary state to another stable stationary state.

An Efficient Optimization Approach for Computationally Expensive Timesteppers Using Tabulation

A methodology is outlined for the efficient solution of dynamic optimization problems when the system evolution is described by computationally expensive timestepper-based models. The computational requirements issue is circumvented by extending the notion of in situ adaptive tabulation to stochastic systems. Conditions are outlined that allow unbiased estimation of the mapping gradient matrix and, subsequently, expressions to compute the ellipsoid of attraction are derived. The proposed approach is applied towards the solution of two representative dynamic optimization problems, (a) a bistable reacting system describing catalytic oxidation of CO and, (b) a homogeneous chemically reacting system describing dimerization of a monomer. In both cases, tabulation resulted in significant reduction in the solution time of the optimization problem.

Nonlinear control of dissipative PDE systems employing adaptive model reduction

The problem of feedback control of distributed processes is considered. Typically this problem is addressed through model reduction where finite dimensional approximations to the original infinite dimensional system are derived. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ODE models using the method of weighted residuals. The most common approach for this task is the Karhunen-Loeve expansion combined with the method of snapshots. However, this approach requires a priori availability of a sufficiently large ensemble of PDE solution data, a requirement which is difficult to satisfy. In this work we focus on the recursive computation of eigenfunctions using a relatively small number of snapshots. The empirical eigenfunctions are continuously modified as additional data from the process becomes available. We use ideas from the recursive projection method to keep track of the dominant invariant eigenspace of the covariance matrix which is subsequently utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is continuously modified with the addition of each snapshot with possible increase or decrease in its dimensionality, while simultaneously the computational burden is kept relatively small. The proposed approach is applied to control temperature in a jacketed tubular reactor where first order chemical reaction is taking place and the closed-loop system is successfully stabilized at an unstable steady-state.

Implication of dynamics in signal transduction and targeted disruption analyses of signaling networks

Modeling and analysis of the dynamics of signaling transduction networks can be powerful tools to understand and predict how cells will respond to native signals and artificial perturbations. This is of special interest for analyzing disease processes associated with signal transduction malfunctioning and to contribute to the development of efficient drug treatment strategies. In this work we examine the advantages of a kinetics-based framework as compared with purely topological approaches to identify input sets and disruption strategies that preserve desired cellular functions while blocking undesired disease states in signaling networks. These differences are highlighted through two examples where the mechanistic-based approach captures information that the topological-based analysis is unable to reveal.

Dynamic optimization of stochastic systems using in situ adaptive tabulation

The problem of efficient formulations for the optimization of stochastic dynamical systems modeled by timestep-per based descriptions is investigated. The issue of computational requirements for the system evolution is circumvented by extending the notion of in situ adaptive tabulation to stochastic systems. Conditions are outlined that allow unbiased estimation of the mapping gradient matrix and, subsequently, expressions to compute the ellipsoid of attraction are derived. The proposed approach is applied towards the solution of dynamic optimization problems for a bistable reacting system describing catalytic oxidation of CO and an illustrative homogeneous chemically reacting system describing dimerization of a monomer. The dynamic evolution of both systems is modeled using kinetic Monte Carlo simulations. In both cases, tabulation resulted in significant reduction in the solution time of the optimization problem

Optimization of thin film growth using multiscale process systems

In this work we consider optimization problems for transport-reaction processes, when the cost function and/or equality constraints necessitate the consideration of phenomena that occur over widely disparate length scales. Initially, we develop multiscale process models that link continuum conservation laws with microscopic scale simulators. Subsequently, we combine nonlinear order reduction techniques for dissipative partial-differential equation systems with adaptive tabulation methods for microscopic simulators to reduce the computational requirements of the process description. The optimization problem is subsequently solved using standard search algorithms. The proposed method is applied to a representative thin film deposition process, where we compute optimal surface temperature profiles that simultaneously maximize deposition-rate uniformity (macroscale objective) and minimize surface roughness (microscale objective) across the film surface.

Dynamic optimization of multiscale PDE/kMC process systems using nonlinear order reduction and tabulation techniques

The problem of dynamic optimization for multi- scale systems comprising of coupled continuum and discrete descriptions is considered. The solution of such problems is challenging owing to large computational requirements of the multiscale process model. This problem is addressed by developing a reduced multiscale model. This is achieved by combining order reduction techniques for dissipative partial-differential equations with adaptive tabulation of microscopic simulation data. The optimization problem is subsequently formulated and solved using standard search algorithms. The proposed method is applied to a representative catalytic oxidation process where optimal inlet concentration profiles are computed to guide the microscopic system from one stable stationary state to another stable stationary state.

OPTIMAL OPERATION OF THIN FILM GROWTH WITH MULTISCALE PROCESS OBJECTIVES

Abstract The issue of optimal time-varying operation for transport-reaction processes is considered, when the cost functional and/or equality constraints necessitate the consideration of phenomena that occur over disparate length scales. Multiscale process models are initially developed, linking continuum conservation laws with microscopic scale simulators. Subsequently, order reduction techniques for dissipative partial-differential equations are combined with adaptive tabulation methods for microscopic simulators to reduce the computational requirements of the optimization problem, which is then solved using standard search algorithms. The method is demonstrated on a thin film deposition process, where optimal surface temperature profiles and inlet switching times that simultaneously maximize thickness uniformity and minimize surface roughness across the film surface are computed.