Dhaene, Tom

Ferranti, Francesco

Knockaert, Luc

Samuel, Elizabeth Rita

English

Ghent University Academic Bibliography

Multipoint Model Order Reduction using Reflective ExplorationElizabeth Rita Samuel1, Francesco Ferranti1, Luc Knockaert1, and Tom Dhaene1Ghent University - iMinds, Gaston Crommenlaan 8 Bus 201, B-9050 Gent, Belgium {elizabeth.ritasamuel,francesco.ferranti, luc.knockaert, tom.dhaene}@ugent.be.Summary. Reduced order models obtained by model or-der reduction methods must be accurate over the whole fre-quency range of interest. Multipoint reduction algorithmsallow to generate accurate reduced models. In this paper,we propose the use of reflective exploration technique forobtaining the expansion points adaptively for the reductionalgorithm. At each expansion point the corresponding pro-jection matrix is computed. Then, the projection matricesare merged and truncated based on their singular values toobtain a compact reduced order model.1 IntroductionElectromagnetic (EM) methods are used for the anal-ysis of complex high-speed systems and usually gen-erate very large systems of equations. Therefore, modelorder reduction (MOR) techniques are crucial to re-duce the complexity of large scale models and thecomputational cost of the simulations, while retain-ing the important physical features of the original sys-tem [1–3]. Multipoint MOR methods have been de-veloped over the years [1,4,5], which allows to gener-ate accurate reduced models over the whole frequencyrange of interest. In this paper, the expansion pointsare selected adaptively using a reflective explorationtechnique. It is a selective sampling algorithm, wherethe model is improved incrementally using the bestpossible data at each iteration, allowing it to proposecandidate exploration points [6]. An error-based ex-ploration is performed to find the expansion points.After obtaining the expansion points, the correspond-ing projection matrices are computed using any of theKrylov based MOR techniques. The projection matri-ces are then merged and truncated based on their sin-gular values to obtain a compact reduced order model.Then the reduced order models are obtained by con-gruence transformation using the truncated projectionmatrix. The technique is validated using a multicon-ductor transmission line example.2 Projection MatrixFor this paper, the PRIMA algorithm [3] has beenused for obtaining the projection matrices at the ex-pansion points.For n expansion points we obtain the correspond-ing projection matrices Vqi for i= 1,2, . . . ,n, then thecommon projection matrix is defined as:Vcomm = [Vq1 Vq2 . . . Vqn ]. (1)The common projection matrix is not truncatedusing its singular values during the iterative procedureof the adaptive reflective exploration. It is truncatedafter all the expansion points have been adaptivelychosen using reflective exploration.3 Reflective ExplorationThe reflective exploration requires a reflective func-tion to select the expansion points. The reflectivefunction used for the proposed algorithm is the rootmean square (RMS) (2) error between the obtainedbest models:Err(I)est =√√√√∑Ksk=1∑Pini=1∑Poutj=1 |HI,(i j)(sk)−HI−1,(i j)(sk)|2|HI,(i j)(sk)|2PinPoutKs(2)where, Ks, Pin and Pout are the number of frequencysamples considered on a dense grid, input and out-put ports of the system, respectively. The explorationconsists of an adaptive modeling loop and an adaptivesampling loop.1. Adaptive Modeling Loop: The algorithm startswith two expansion points selected at [ωmin,ωmax]of the frequency range of interest. The reducedorder q at these points is equal to the number ofports of the system. Then with a common projec-tion matrix as explained in Section 2, the reducedmodel is obtained. Then in the next iteration againthe projection matrix is computed for a reducedorder equal to two times the port of the system. Ifthe RMS error between the two best models (i.e.,the model obtained in the Ith and the (I − 1)thiteartion) exceeds a certain threshold, then the re-duced order q is again increased by the number ofports for the respective expansion points.2. Adaptive Sampling Loop: When the difference inRMS error between the Ith and (I− 1)th, is lessthan 10%, a new expansion point is selected. Forselecting the new expansion point the error perfrequency is computed by taking the norm L2, of2Fig. 1. Flowchart 1: Reflective Exploration.Fig. 2. Flowchart 2:Truncation of the projection matrix.the frequency response of the best model (HI) andthe original model (Hact ):Errsk = norm(Hact,(i j)(sk)−HI,(i j)(sk));k = 1, . . . ,Ks, (3)and the frequency at which Errsk is maximum is con-sidered as the new expansion point.This process is iteratively repeated until the RMSerror between the original frequency response and thereduced model is 0.001. Figure 1 shows the reflectiveexploration algorithm.4 Model compactingAfter obtaining the best reduced order model fromthe iterative procedure, it might be possible to fur-ther compact the model with the information obtainedfrom the singular values Σ of Vcomm (1). Figure 2shows the flowchart for the truncation of the singularvalues. The projection matrix Qcomm with congruencetransformation gives the reduced state-space matricesof order qcomm.5 Numerical ResultsA multiconductor transmission line described by anoriginal state-space of order 1202 and 4 ports is con-Fig. 3. Magnitude of Y11.sidered. As described in Section 3, 4 expansion pointsare chosen adaptively. Table 1 gives the dimension ofTable 1. Dimension of the Original and Reduced Model.Models DimensionOriginal 1202Model after reflective exploration 64Model after compacting 42the reduced models. The reduced model after trunca-tion has an RMS error of 4.128e− 4 when evalutedover a dense grid of Ks = 200 frequency samples asshown in Figure 3.Acknowledgement. This work was supported by the Re-search Foundation Flanders (FWO-Vlaanderen) and by theInteruniversity Attraction Poles Programme BESTCOM ini-tiated by the Belgian Science Policy Office. Francesco Fer-ranti is a post-doctoral research fellow of FWO-Vlaanderen.References1. K. Gallivan, E. Grimme, and P. V. Dooren. A rationalLanczos algorithm for model reduction. Numerical Al-gorithms vol. 12, no. 1, pp. 3363, Mar. 1996.2. L. Knockaert and D. De Zutter. Laguerre-SVD reduced-order modeling. IEEE Transactions on Microwave The-ory and Techniques vol. 48, no. 9, pp. 1469 1475, Sept.2000.3. A. Odabasioglu, M. Celik, and L. Pileggi. PRIMA: pas-sive reduced order interconnect macromodeling algo-rithm. EEE Transactions on Computer-Aided Design ofIntegrated Circuits and Systems vol. 17, no. 8, pp. 645654, Aug. 1998.4. F. Ferranti, M. Nakhla, G. Antonini, T. Dhaene, L.Knockaert, and A. E. Ruehli. Multipoint Full-WaveModel Order Reduction for Delayed PEEC Models withLarge Delays. IEEE Transaction on ElectromagneticCompatibility vol. 53, no. 4, pp. 959-967, Nov. 2011.5. G. J. Burke , E. K. Miller , S. Chakrabarti and K.Demarest. Using Model-Based Parameter Estimation toIncrease the Efficiency of Computing ElectromagneticTransfer Functions. IEEE Transactions on Magneticsvol. 25, no. 4, pp.2807 -2809, 1989.6. U. Beyer and U. B. Frank. Data Exploration with Reflec-tive Adaptive Models. Computational Statistics & DataAnalysis vol. 22, no. 2, pp. 193-211, 1996.

Multipoint model order reduction using reflective exploration

https://biblio.ugent.be/publication/5736910/file/5736911.pdf

Multipoint model order reduction using reflective exploration

Abstract

Similar works

Full text

Available Versions

Ghent University Academic Bibliography