(357b) Affine Lumping Formalism for Comparison of Projection-Based Model Reduction Techniques | AIChE

(357b) Affine Lumping Formalism for Comparison of Projection-Based Model Reduction Techniques

Authors 

Oxberry, G. M. - Presenter, Massachusetts Institute of Technology
Barton, P. I. - Presenter, Massachusetts Institute of Technology


Many practical problems involve combustion under inhomogeneous, transient conditions, and therefore require the use of numerical methods that solve large systems of coupled, nonlinear partial differential equations of the kind typically found in reacting flow solvers. These practical problems typically require large, detailed chemical models in order to simulate faithfully the physics involved. However, reacting flow simulations of large chemical models have prohibitively large computational costs. Consequently, smaller reduced models are used in place of large chemical models in order to obtain approximate numerical solutions to the large chemical models at decreased computational cost.

Several methods are available for generating reduced models from detailed chemical models (see [5], [9], [6] and [7] for examples). However, these different methods originate from different theoretical backgrounds, including methods based on singular perturbation, methods based on a graph-theoretic interpretation of chemistry, and others. Given this variation in the theoretical development of model reduction methods, it is difficult to compare two given model reduction techniques. Previous work has made progress in this area by showing that the intrinsic low-dimensional manifold (ILDM) technique is a special case of computational singular perturbation (CSP) [5, 12]. In addition, comparisons have been made between ILDM and the Fraser method (see [8] for the Fraser method and [4] for the comparison), between CSP and the Fraser method [3], between CSP and conventional asymptotics [10], and between other methods (such as [13, 2]). All of these comparisons have made progress in comparing model reduction techniques that have been derived from the theory of asymptotics. To make further progress in comparing model reduction techniques, we propose a formalism for comparing projection-based model reduction techniques called ?affine lumping.?

In this work, we provide a definition of affine lumping. Affine lumping defines two affine mappings. The first affine mapping transforms a detailed model to a reduced-dimension representation called the reduced model; this process is called lumping. The second affine mapping lifts the reduced state variables into the space of the original state variables, recovering a representation of the reduced model in the original state variables that approximates the detailed model; this process is called unlumping. It can be shown that applying these two affine mappings in sequence is equivalent to projecting the original model onto an affine subspace. This projection is the reduced model, lifted into the space of the original state variables. It can be shown that under certain conditions, lifting the solution of the reduced model into the space of the original state variables yields a solution of the detailed model. In other words, under certain conditions,

the solution of the detailed model lies in an affine subspace so that the solution of the reduced model can be used to construct the exact solution of the detailed model, and the reduced model can be described with fewer differential equations and state variables than the detailed model.

After presenting a definition of affine lumping, we present examples of existing model reduction techniques that are special cases of affine lumping. We show that the species lumping technique of Li et al. [6], CSP and the technique of reaction invariants reviewed by Waller and Mäkilä [11] are all special cases of affine lumping by translating the nomenclature and mathematics used in each of these techniques to the nomenclature and mathematics of affine lumping. We also show that reaction elimination methods like [1] are not projection-based methods. These results suggest that a model reduction technique can be expressed using the affine lumping formalism if and only if it is projection-based. This property can be used to show that apparently different techniques could have similar properties despite different theoretical backgrounds, and it can also be used to classify model reduction techniques as projection-based or non-projection based. The formalism could then be used in future work to assess the relative merits of different projection-based model reduction techniques.

References

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[8] M. R. Roussel and S. J. Fraser. Geometry of the steady-state approximation: Perturbation and accelerated convergence methods. Journal of Chemical Physics, 93(2):1072?1081, 1990.

[9] S. Singh, J. M. Powers, and S. Paolucci. On slow manifolds of chemical reactive systems. Journal of Chemical Physics, 117(4):1482?1496, 2002.

[10] M. Valorani, D. A. Goussis, F. Creta, and H. N. Najm. Higher order corrections in the approximation of low-dimensional manifolds and the construction of simplified problems with the CSP method. Journal of Computational Physics, 209:754?786, 2005.

[11] K. V. Waller and P. M. Mäkilä. Chemical reaction invariants and variants and their use in reactor modeling, simulation, and control. Industrial and Engineering Chemistry Process Design and Development, 20:1?13, 1981.

[12] A. Zagaris, H. G. Kaper, and T. J. Kaper. Analysis of computational singular perturbation reduction method for chemical kinetics. Journal of Nonlinear Science, 14:59?91, 2004.

[13] A. Zagaris, H. G. Kaper, and T. J. Kaper. Two perspectives on reduction of ordinary differential equations. Mathematische Nachtrichten, 278(12-13):1629?1642, 2005.

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