OT research project
OT-10-038 PMORA: Multi-parameter Model Order Reduction and Applications (1-10-2010 to 30-09-2014)
Description
Mathematical models are frequently used for studying physical phenomena or for tuning commercial products to customer and environmental demands. Computer models often give rise to extremely large dynamical systems whose manipulation is computationally expensive. In this project, a dynamical system is formulated in the frequency domain, so the behaviour of the system is described by the frequency response (or transfer function). Because of the size of such models, model order reduction (MOR) has become an important tool for numerical simulation. Moreover, the frequency response of a model can depend on one or more parameters. E.g., the acoustic behaviour of a vehicle may depend on physical parameters such as plate thickness or damping material. Simulation of such large scale parametrized models is not feasible without the use of model reduction that takes into account the dependency on these parameters. Model order reduction for parametrized systems is called parametrized or parametric MOR (PMOR).
The aim of the research is to design accurate and encient PMOR methods that can be ectively used in the context of design optimization. In min-max design optimization, the parameters are determined that minimize a cost function that depends on the maximum of the frequency response. Just evaluating the frequency response and its gradient towards the parameters is expensive. As the number of parameters grows, there is a risk that the computational cost becomes unacceptably high. This is known as the curse of dimensionality.
Standard MOR methods have been able to speed up the computation of the cost function by a factor five to twenty. This might not be enough for optimization with many parameters. Therefore, PMOR must be utilized. However, deriving a reduced model for the entire parameter range is not required, since optimization methods only focus on specific regions in the parameter range.
A distinction can be made based on the information that is available from the model. One can consider PMOR methods when a specific representation of the model is known, e.g., a state space representation or a polynomial matrix fraction description. On the other hand, when only "measurements" are available from the model, i.e., the model is considered as a black box, a representation for the model or a reduced model can be found by PMOR based only on these measurements. For the optimization method we will consider two different MOR approaches, which we now explain.
Given a reduced model for fixed parameters, the idea is to build another reduced model for slightly different values of the parameters. We will therefore "recycle" as much information as possible from the model corresponding to the previous values of the parameters and use a "continuation" approach to update the model. The motivation for a continuation approach is that a line search or trust region method converges in small steps towards the minimum.
Instead of recycling, a MOR method can be used to "compute" measurements for specific points in the parameter domain. To obtain complete coverage of the parameter domain considered, it is important to limit the number of interpolation points. Hence, the choice of these points will play a crucial role in the enciency of the method. The choice of points in our project will be based on "magic points". Also, the representation of the reduced model when many parameters are involved is important. Instead of a state-space representation, a coprime polynomial matrix fraction description looks much more viable. Hence, the existing MOR methods for model reduction without parameters based on measurements have to be generalized in several directions. On the one hand, the parameters have to be taken into account and on the other hand, the computed reduced model has to be represented into forms different than the state-space representation.
In addition, we will study systems with multiple inputs or outputs. In many simulations, parameters only have a low rank contribution to the system matrix. This is, e.g., the case when the parameters are related to boundary conditions of a partial differential equation. Such problems can be written as systems without parameters but with multiple right-hand sides. The number of right-hand sides can be of the order of hundreds. In other cases, the output is a non-linear function of the state variables, which can be reformulated as a linear problem with many right-hand sides.