Marta D Elia
Reduced basis method and model order reduction for initial value problems of differential algebraic equations
Thursday 20th December 2007
This Thesis deals with the numerical solution of systems of Diﬀerential Algebraic Equations (DAEs), which typically describe the dynamics of Integrated Circuits (ICs). Since over the last years the design of modern ICs has become more complex, there is an increasing demand for new, eﬀective and eﬃcient optimization tools, in order to avoid costly and repeated re-designs. In particular, there is the need of approximations of the original models (which describe the complete dynamics of the devices), which satisfy the following requirements: (i) they should provide feasible and accurate reproductions of the original models; (ii) the cost of the simulations with such models should be signiﬁcantly lower than the cost of analogous simulations with the original models (this prerequisite is actually obtained by reducing the dimension of the original problems in the construction of the approximated ones, the “reduced” models). The design of an IC is associated with speciﬁc “design parameters”, such as the values of resistors, capacitors, input signals, etc...; hence, the reduced model should be able to reproduce the dynamics of an IC according to the speciﬁc choices of such parameters. In this work we ﬁrst focus on non-parametrized problems; in this context several reduction methods are available. Among them, the most widespread are the Model Order Reduction (MOR) ones , , , which are eﬃcient and reliable for the solution of linear systems of DAEs. Firstly, we present the extension of the MOR techniques to nonlinear problems and the formulation of the Reduced Basis method, introduced by Porsching , for nonlinear systems of DAEs; then, we compare their performances in terms of accuracy and computational costs. The second part of this work deals with the solution of parametrized problems. In this context the reduction methods are less widespread and constitute an area of investigation of the Numerical Analysis: in this Thesis we propose a new method for the solution of parametrized systems of DAEs in analogy with the Reduces Basis method for parametrized Partial Diﬀerential Equations (PDEs) . Firstly, we present the formulation of the proposed method for linear and nonlinear problems with a linear and nonlinear dependence on the parameter; then, we propose an eﬃcient adaptive procedure for the generation of the reduced subspace (in which the solution of the reduced model lies) able to capture the parametric dependence of the problem and to provide accurate solutions with low computational costs. Moreover, we also propose an a priori estimate for the reduction and global approximation error. Finally, we compare the parametrized Reduced Basis method with a parametrized MOR technique  in order to highlight the eﬀectiveness of the proposed method in terms of accuracy and computational costs savings.