Index-aware Model Order Reduction for higher index DAEs

There exists many Model Order Reduction (MOR) methods for ODEs but little had been done to reduce DAEs especially higher index DAEs. In principle, if the matrix pencil of a DAE is regular, it is possible to use conventional MOR techniques to obtain reduced order models, which are generally ODEs. However, as far as their numerical treatment is concerned, the reduced models may be close to higher index models, that is, to DAEs. Thus the numerical solution of the reduced models might be computationally expensive, or even not feasible. In the worst cases, the reduced models may be unsolvable, i.e. their matrix pencil is singular. This problem is very pronounced for systems with index higher than 1, but it may occur even if the index of the problem does not exceed 1. Thus MOR methods for ODEs cannot generally be used for DAEs. This motivated us to introduce a new MOR method for DAEs which we call the index-aware MOR (IMOR) which can reduce DAEs while preserving the index of the system. This method involves first splitting the DAEs into differential and algebraic parts. Then, we use the existing MOR methods to reduce the differential part. We observed that the reduction of the differential part induces a reduction in the algebraic part. This enabled us to construct a method which reduces both the differential and the algebraic part. As a result a DAE is reduced. This method can also be used as a new method to solve DAEs. In this paper, we generalize the IMOR method to higher index DAEs and we shall call this method the GIMOR method. We use index-3 systems for testing and validating the accuracy of the GIMOR method.