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Computational Tools for LargeScale Linear Systems
Abstract
While the theoretical analysis of linear dynamical systems with finite statespaces is a mature topic, in situations where the underlying model has a large number of dimensions, modelers must turn to computational tools to better visualize and analyze the dynamic behavior of interest. In these situations, we are confronted with the Curse of Dimensionality: computational and storage complexity grows exponentially in the number of dimensions.
This doctoral project focuses on two main classes of largescale linear systems which arise in system biology. The Chemical Master Equation (CME) is a FokkerPlanck equation which describes the evolution of the probability mass function of a countable state space Markov process. Each state of the CME is labelled with an ordered Stuple corresponding to one configuration of a wellmixed chemical system, where S is the number of distinct chemical species of interest. Even in cases where one only considers a projection of the CME to a finite subset of the states, one still must contend with the Curse of Dimensionality: the computational complexity grows exponentially in the number of chemical species. This dissertation describes a computational methodology for efficient solution of the CME which, in the best cases, will scale linearly in the number of chemical species.
The second main class of highdimensional problems requiring computational tools are coupled linear reactiondiffusion equations. For this class of models, we focus primarily on the computation of certain highdimensional matrices which describe in a quantitative sense the inputtostate and statetooutput relationships. We describe algorithms for extracting useful information stored in these matrices and use this information to efficiently compute both reduced order models and openloop control laws for steering the full system. A key feature of this approach is that the method is completely simulation or experiment free, in fact, in our numerical experiments, the computation of a reduced model or openloop control law is an order of magnitude faster on a laptop than simulation of the full system on a 32 core node of a highperformance cluster.
In both projects, the enabling computational technology is the recently proposed Tensor Train (TT) structured lowparametric representation of highdimensional data. The TTformat effectively exploits lowrank structure of the "unfolding matrices" for compression and computational efficiency. Formally, the computational complexity of basic TT arithmetics scale linearly in the number of dimensions, potentially circumventing the curse of dimensionality. To demonstrate the effectiveness of this approach, we performed numerous numerical experiments whose results are reported here.
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