1. A more reliable reduction algorithm for behavioral model extraction Dmitry Vasilyev, Jacob White Massachusetts Institute of Technology

2. Outline Background Projection framework for model reduction Balanced Truncation algorithm and approximations AISIAD algorithm Description of the proposed algorithm Modified AISIAD and a low-rank square root algorithm Efficiency and accuracy Conclusions

3. Model reduction problem Reduction should be automatic Must preserve input-output properties Many (> 10 4 ) internal states inputsoutputs few (<100) internal states inputsoutputs

4. Differential Equation Model Model can represent: Finite-difference spatial discretization of PDEs Circuits with linear elements A – stable, n x n (large) E – SPD, n x n - state - vector of inputs - vector of outputs

5. Model reduction problem n – large (thousands)! Need the reduction to be automatic and preserve input-output properties (transfer function) q – small (tens)

6. Approximation error Wide-band applications: model should have small worst-case error ω => maximal difference over all frequencies

7. Projection framework for model reduction Pick biorthogonal projection matrices W and V Projection basis are columns of V and W Vx r x x n x xrxr V q W T AVx r Ax Most reduction methods are based on projection

8. LTI SYSTEM X (state) t u t y input output P (controllability) Which modes are easier to reach? Q (observability) Which modes produce more output? Reduced model retains most controllable and most observable modes Mode must be both very controllable and very observable Projection should preserve important modes

9. Reduced system: ( W T AV, W T B, CV, D ) Compute controllability and observability gramians P and Q : (~n 3 ) AP + PA T + BB T =0 A T Q + QA + C T C = 0 Reduced model keeps the dominant eigenspaces of PQ : (~n 3 ) PQ v i = λ i v i w i PQ = λ i w i Balanced truncation reduction (TBR) Very expensive. P and Q are dense even for sparse models

10. Arnoldi [Grimme ‘97]: V = colsp{A -1 B, A -2 B, …}, W=V T, approx. P dom only Padé via Lanczos [Feldman and Freund ‘95] colsp(V) = {A -1 B, A -2 B, …}, - approx. P dom colsp(W) = {A -T C T, (A -T ) 2 C T, …}, - approx. Q dom Frequency domain POD [Willcox ‘02], Poor Man’s TBR [Phillips ‘04] Most reduction algorithms effectively separately approximate dominant eigenspaces of P and Q : However, what matters is the product PQ colsp(V) = {(jω 1 I-A) -1 B, (jω 2 I-A) -1 B, …}, - approx. P dom colsp(W) = {(jω 1 I-A) -T C T, (jω 2 I-A) -T C T, …}, - approx. Q dom

11. RC line (symmetric circuit) Symmetric, P=Q all controllable states are observable and vice versa V(t) – input i(t) - output

12. RLC line (nonsymmetric circuit) P and Q are no longer equal! By keeping only mostly controllable and/or only mostly observable states, we may not find dominant eigenvectors of PQ Vector of states:

13. Lightly damped RLC circuit Exact low-rank approximations of P and Q of order < 50 leads to PQ ≈ 0!! R = 0.008, L = 10 -5 C = 10 -6 N=100

14. Lightly damped RLC circuit Union of eigenspaces of P and Q does not necessarily approximate dominant eigenspace of PQ. Top 5 eigenvectors of P Top 5 eigenvectors of Q

15. AISIAD model reduction algorithm Idea of AISIAD approximation: Approximate eigenvectors using power iterations: V i converges to dominant eigenvectors of PQ Need to find the product (PQ)V i X i = (PQ)V i => V i+1 = qr(X i ) “iterate” How?

16. Approximation of the product V i+1 =qr(PQV i ), AISIAD algorithm W i ≈ qr(QV i ) V i+1 ≈ qr(PW i ) Approximate using solution of Sylvester equation

17. More detailed view of AISIAD approximation Right-multiply by W i X X H, q x q (original AISIAD) M, n x q

18. X X H, q x q Modified AISIAD approximation Right-multiply by V i Approximate! M, n x q ^

19. Modified AISIAD approximation Right-multiply by V i We can take advantage of numerous methods, which approximate P and Q ! X X H, q x q Approximate! M, n x q ^

20. n x qn x q n x nn x n Specialized Sylvester equation A X + X H = -M q x qq x q Need only column span of X

21. Solving Sylvester equation Schur decomposition of H : A X + X = -M ~ ~ Solve for columns of X ~ ~ X

22. Solving Sylvester equation Applicable to any stable A Requires solving q times Schur decomposition of H : Solution can be accelerated via fast MVP Another methods exists, based on IRA, needs A>0 [Zhou ‘02]

23. Solving Sylvester equation Applicable to any stable A Requires solving q times Schur decomposition of H : For SISO systems and P = 0 equivalent to matching at frequency points –Λ(W T AW) ^

24. Modified AISIAD algorithm 1.Obtain low-rank approximations of P and Q 2.Solve AX i +X i H + M = 0, => X i ≈ PW i where H=W i T A T W i, M = P(I - W i W i T )A T W i + BB T W i 3. Perform QR decomposition of X i =V i R 4. Solve A T Y i +Y i F + N = 0, => Y i ≈ QV i where F=V i T AV i, N = Q(I - V i V i T )AV + C T CV i 5.Perform QR decomposition of Y i =W i+1 R to get new iterate. 6.Go to step 2 and iterate. 7.Bi-orthogonalize W and V and construct reduced model: ( W T AV, W T B, CV, D ) LR-sqrt ^^ ^ ^

25. For systems in the descriptor form Generalized Lyapunov equations: Lead to similar approximate power iterations

26. mAISIAD and low-rank square root Low-rank gramians LR-square root mAISIAD (inexpensive step) (more expensive) For the majority of non-symmetric cases, mAISIAD works better than low-rank square root (cost varies)

27. RLC line example results H-infinity norm of reduction error (worst-case discrepancy over all frequencies) N = 1000, 1 input 2 outputs

28. Steel rail coolling profile benchmark Taken from Oberwolfach benchmark collection, N=1357 7 inputs, 6 outputs

29. mAISIAD is useless for symmetric models For symmetric systems ( A = A T, B = C T ) P=Q, therefore mAISIAD is equivalent to LRSQRT for P,Q of order q RC line example ^^

30. Cost of the algorithm Cost of the algorithm is directly proportional to the cost of solving a linear system: (where s jj is a complex number) Cost does not depend on the number of inputs and outputs (non-descriptor case) (descriptor case)

31. Conclusions The algorithm has a superior accuracy and extended applicability with respect to the original AISIAD method Very promising low-cost approximation to TBR Applicable to any dynamical system, will work (though, usually worse) even without low-rank gramians Passivity and stability preservation possible via post-processing Not beneficial if the model is symmetric