Model reduction methods usually focus on the error performance analysis; however, in presence of uncertainties, it is important to analyze the robustness properties of the error in model reduction as well. This problem is particularly relevant for engineered biological systems that need to function in a largely unknown and uncertain environment. We give robustness guarantees for structured model reduction of linear and nonlinear dynamical systems under parametric uncertainties. We consider a model reduction problem where the states in the reduced model are a strict subset of the states of the full model, and the dynamics for all of the other states are collapsed to zero (similar to quasi‐steady‐state approximation). We show two approaches to compute a robustness guarantee metric for any such model reduction—a direct linear analysis method for linear dynamics and a sensitivity analysis based approach that also works for nonlinear dynamics. Using the robustness guarantees with an error metric and an input‐output mapping metric, we propose an automated model reduction method to determine the best possible reduced model for a given detailed system model. We apply our method for the (1) design space exploration of a gene expression system that leads to a new mathematical model that accounts for the limited resources in the system and (2) model reduction of a population control circuit in bacterial cells.
Nonlinear methods for model reduction
Typical model reduction methods for parametric partial differential equations construct a linear space V n which approximates well the solution manifold M consisting of all solutions u ( y ) with y the vector of parameters. In many problems of numerical computation, nonlinear methods such as adaptive approximation, n term approximation, and certain treebased methods may provide improved numerical efficiency over linear methods. Nonlinear model reduction methods replace the linear space V n by a nonlinear space Σ n . Little is known in terms of their performance guarantees, and most existing numerical experiments use a parameter dimension of at most two. In this work, we make a step towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation, we give a first comparison of their performance with the performance of standard linear approximation for any compact set. We then study these methods for solution manifolds of parametrized elliptic PDEs. We study a specific example of library approximation where the parameter domain is split into a finite number N of rectangular cells, with affine spaces of dimension m assigned to each cell, and give performance guarantees with respect to accuracy of approximation versus m and N .
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 Award ID(s):
 1817691
 NSFPAR ID:
 10279446
 Date Published:
 Journal Name:
 ESAIM: Mathematical Modelling and Numerical Analysis
 Volume:
 55
 Issue:
 2
 ISSN:
 0764583X
 Page Range / eLocation ID:
 507 to 531
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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