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Parametric model order reduction

Model order reduction is an efficient method to directly extract accurate compact models from extended finite element models. For design, it is desirable to leave some parameters (material properties, dimensions, boundary conditions) undetermined in the compact model, so that optimization can be performed with the more efficient reduced order model.

Generic example of a device with a single heat source. The generated heat dissipates through the device to the surroundings. The exchange between surrounding and the device is modelled by convection boundary conditions with different film coefficients at the top, ht, bottom, hb, and the side, hs. The dimension of the system (5) is n=4257. Figure 1 
Comparison of the numer of matrix vector products. Figure 2 
Comparison of the numer of matrix vector products. Figure 3 
MKR-GMRES is compared with SimGCRODR(m,k) by CPU time spent on solving each system when they are run on the same computer. Figure 4 
2-norm relative error between the output of the reduced model and that of the original system on the whole range of the time interval. Figure 5 

Introduction

Many model order reduction methods for parameterized systems need to construct a projection matrix V which requires computing several moment matrices of the parameterized systems. For computing each moment matrix, the solution of a linear system with multiple right-hand sides is required. Furthermore, the number of linear systems increases with both the number of moment matrices used and the number of parameters in the system. Usually, a considerable number of linear systems have to be solved when the system includes more than two parameters. The standard way of solving these linear systems in case sparse direct solvers are not feasible is to use conventional iterative methods such as GMRES [1] or CG [2]. The fast recycling algorithm SimGCRO-DR which is developed based on the method in [3] is applied to solve the whole sequence of linear systems and is shown to be much more efficient than the standard iterative solver GMRES as well as the newly proposed recycling method MKR-GMRES in [4]. As a result, the computation of the reduced-order model can be significantly accelerated.
 

References

[1] Y. Saad and M.H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems", SIAM J. Sci. Stat. Comput. 7 (1986) 856-869
[2] Kendell A. Atkinson (1988), An introduction to numerical analysis (2nd ed.), Section 8.9, John Wiley and Sons.ISBN 0-471-50023-2
[3] M. L. Parks, E. de Sturler, G. Mackey, D. D. Johnson, and S. Maiti, “Recycling Krylov subspaces for sequences of linear systems”. SIAM J. Sci. Comput. 28 (5) (2006) 1651-1674
[4] Z. Ye, Z. Zhu, and J. R. Phillips, “Generalized Krylov Recycling Methods for Solution of Multiple Related Linear equation Systems in Electromagnetic Analysis”. In Proc. Design Automation Conference (DAC08) (2008), 682-687

 

Example

An example is shown in figure 1. The model is a generic example of a device with a single heat source when the generated heat dissipates through the device to the surroundings. The exchange between surrounding and the device is modeled by convection boundary conditions with different film coefficients at the top, ht, bottom, hb, and the side, hs. The dimension of the system (5) is n=4257. The reduced model is of dimension q = 325. All the simulations are run with MATLAB 2007b.

We employ the same comparison criterion as the one used in [3] to compare SimGCRO-DR(m,k) with GMRES, i.e. the matrix vector (MV) products used in the algorithm for solving each linear system. The comparison is shown in figures 2 and 3. Here, GMRES (∞) means GMRES without restarts and GMRES (m) means the one with restarts. We compare SimGCRO-DR(m,k) with both GMRES(∞) and GMRES(m). In comparing SimGCRO-DR and restarted GMRES, we let the solution produced by both methods be minimized over the subspace of the same dimension, i.e. we use the same m = 40 in SimGCRO-DR(m,k) and GMRES(m). k is the number of Harmonic Ritz vectors which are recycled in solving the systems. Two different incomplete LU factorizations are used as the preconditioners: in MATLAB notation, one is luinc(·,0.005), the other is luinc(·,0.001).

In figure 4, MKR-GMRES is compared with SimGCRODR(m,k) by CPU time spent on solving each system when they are run on the same computer. In order to show that the use of SimGCRO-DR in the context of PMOR does not cause accuracy problems, we depict the model reduction error in figure 5 as the 2-norm relative error between the output of the reduced model and that of the original system on the whole range of the time interval. Figure 5 indicates the error varying with the two parameters hs, hb. We take 529 groups of hs and hb which cover the whole range [1, 109]. The maximum error of the reduced model with all the groups of parameters is around 10-7, and it is much smaller than the requirement of the maximum error 0.05 for real applications.

 

Relevant publications

  • L. Feng, P. Benner, and J. G. Korvink, Parametric Model Order Reduction Accelerated by Recycling Algorithm, accepted by 48th IEEE conference on Decision and Control

Contact person

Lihong Feng
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