TOWARDS A CLASSIFICATION OF REDUCED ORDER MODELING TECHNIQUES. G.Lippens, L. Knockaert and D. De Zutter

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1 TOWARDS A CLASSIFICATION OF REDUCED ORDER MODELING TECHNIQUES G.Lippens, L. Knockaert and D. De Zutter May, 4th 2006 G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 1

2 OVERVIEW Introduction Oblique projections Idempotent projectors ROM based on orthogonal basis functions ROM based on Moebius transforms Classification scheme of Moebius related ROM Conclusion G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 1

3 MAJOR OBJECTIVES OF ROM METHODS Obtaining a smaller model Accuracy over a predefined bandwidth Preservation of structure Size of reduced model as small as possible At reasonable computational costs G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 2

4 MAJOR OBJECTIVES OF THIS TALK Relating ROM to oblique projections Oblique projections as a unifier Relating different Krylov subspace methods : the Möebius transform Discussing frequency properties of Krylov subspace methods Bandlimited reductions Reductions of resonant systems G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 3

5 PSfrag replacements OBLIQUE PROJECTIONS IN 2D w A a γ C B Q α β G H D u v G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 4

6 OBLIQUE PROJECTIONS IN 2D cont. b = u(v T u) 1 v T a (1) = u(v T u) 1 (v T a) = u AB cos 0 cos γ = cos γ AB u = DE u (2) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 5

7 3D PROJECTIONS The space R 3 Basis : (x = [100] T, y = [010] T, z = [001] T ). Projecting onto the 1D subspace S u = span(x) and parallel to the 2D subspace S w = span(y, z) Projector : P = x(x T x) 1 x T which is : Q x;//yz = (3) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 6

8 GENERALIZING : N-D OBLIQUE PROJECTIONS These observations also hold for a general R n space. If the columns of a matrix U span a space S U and the columns of V span the space S V, the projection operator projecting onto S U and parallel to SV ) is determined by : Q = U(V T U) 1 V T (4) If the matrix Q has nulspace N and range R, it can be proven that the spectral norm Q 2 satisfies : Q 2 = 1/ sin θ (5) where θ is the angle between S u and S w, defined by cos θ = max v T u, and where u and v are two unit vectors from the the range and the nullspace of Q respectively. G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 7

9 IDEMPOTENT PROJECTORS The projection operator which projects parallel to the yz plane, onto the x axis : ( ) 1 ( Q 2 ) 1 Qx 2 = sup x 0 x 2 = Q[100] T 2 = 1 = sin(θ) = sin( π 2 ) (6) It is readily seen that a matrix Q = U(V T U) 1 V T is idempotent, i.e. Q 2 = Q from : U(V T U) 1 V T U(V T U) 1 V T = U(V T U) 1 V T Q has as only eigenvalues one or zero. Thus the range and the nullspace of Q form complementary subspaces while dim(r(q)) + dim(n(q)) = n. (7) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 8

10 COMPLEX OPERATORS Now consider the following operator with A being a complex matrix and V and U real : Q A = U(V T AU) 1 V T A (8) This operator is also an idempotent, and the range and nullspaces of this operator also form complementary subspaces in C n, a complex vector space where the scalar product applies. a b = a H b (9) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 9

11 IDEMPOTENS RELATED TO ROM We suppose that V T AU is nonsingular. This idempotent can consequently be written as : Q A = U(X H U) 1 X H (10) where X H = V T A. Q A is the operator projecting onto colspan(u) and parallel to colspan(x). It is an oblique projector which we will use in order to perform a model order reduction. G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 10

12 HIGHER ORDER TRANSFER FUNCTIONS Linear systems of order n i P i( d dx )i x = Bu y = L T x P (s) is of order n and has a specific structure B and L represent the input and the output respectively. Preservation of structure : symmetry, positive definiteness,... F (s) = L T P (s) 1 B (11) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 11

13 INFORMATION LOSS Maintain the essential information of the original system N q matrices V, W which conserve the essential properties Some specific information will always be lost by projection Some information is much more essential than other information. Frequency dependent behavior is important Numerical issues : orthogonality G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 12

14 TWO MAJOR SUBCLASSES OF ROM METHODS (1) Expand H(s) in orthogonal basis functions Projection matrix is constructed from weighing H(s) with basis functions Nearly-exact moment matching determined by projection angle Higher order systems Structure can be preserved (2) Application of a Möbius transform Projection matrix is constructed from a Krylov series Exact moment matching can be proven First order systems G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 13

15 (1) ORTHOGONAL BASIS In order to obtain the matrices V and W we expand P (s) 1 B in a complete and orthonormal basis γ k (s). If we transpose the reduced and original transfer functions, a completely analogous reasoning follows. In that case, B and L are interchanged, as are V and W. P (s) 1 B = P (s) T L = r 1 k=0 r 1 k=0 k k γ k (s) + R r (s) (12) l k γ k (s) + R l (s) (13) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 14

16 PROJECTION VECTORS FROM SCALAR PRODUCT If the γ k (s) form an orthonormal basis with respect to the scalar product f g = 1 f(jω)g(jω)dω (14) 2π it is possible to calculate the coefficients T k by taking the L 2 scalar product of both sides of (12) with the γ k (s) : k k = 1 2π l k = 1 2π B B B P (iω) 1 B γ k (s)(iω) dω P (iω) T L γ k (s)(iω) dω (15) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 15

17 LEFT AND RIGHT SPACES Next, the N q matrices K r and K l are defined as K r = [k 0, k 1,..., k r 1 ] K l = [l 1, l 2,..., l r 1 ] (16) The columns of these two matrices K r and K l span the right subspace and the left subspace respectively. In specific cases, K r and K l can be Krylov matrices, i.e. they can be generated by stacking the columns A n R, (r = 0... q). G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 16

18 LEFT AND RIGHT SPACES cont. If K l, K r are such that det(kl T K r) 0, there exists an idempotent Q such that QK r = K r Q T K l = K l Q = V W T W T V = I q (17) Now if we suppose that K T l K r is nonsingular, let us observe Q I = V W T, with W T V = I with I the identity matrix, such that Q I K r = K r. G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 17

19 CLOSENESS OF PROJECTIONS This implies while Q P (s) Q I = Q I. Multiplication of the first equation of (12) with Q P (s) : Q I k k = k k k = 0,..., r 1 (18) V (W T P (s)v ) 1 W T P (s)p 1 (s)b = V (W T P (s)v ) 1 W T B = r 1 k=0 k k γ k (s) + Q P (s) R r (s) (19) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 18

20 CLOSE EXPANSIONS After left multiplying (19) with L T, we write down the following expansions for the transfer functions : F (s) = F R (s) = r 1 k=0 r 1 k=0 L T k k γ k (s) + L T R r (s) (20) L T k k γ k (s) + L T Q P (s)r r (s) (21) Conditions for closeness : F (s) and F R (s), have approximately the first r coefficients in their {γ k (s)} expansion in common: R r is small enough θ associated with the idempotent Q P (s) is close to π 2 G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 19

21 THE SYMMETRIC CASE In the symmetric case P (s) = P (s) T, L = B, it is clear that we have k k = l k and K = L, implying W = V and Q I = Q T I = V V T, which is then an orthogonal projector and θ = π 2. In that case the reduced order transfer function simplifies to F R (s) = L T V (V T P (s)v ) 1 V T B (22) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 20

22 NEARLY ORTHOGONAL The idempotents which can be written as Q 1 = G(H H P G) 1 H H P, are always close to orthogonal projections. Consider the orthogonal projector Q 2 = G(H H G) 1 H H. Q 2 1 = Q 1 Q 1 Q 2 = G(H H P G) 1 H H P G(H H G) 1 H H = Q 2 Q 2 2 = Q 2 Q 2 Q 1 = G(H H G) 1 H H G(H H P G) 1 H H P = Q 1 (23) from which we deduce : (Q 2 Q 1 ) 2 = Q Q 2 2 Q 2 Q 1 Q 1 Q 2 = 0 (24) or Q 2 Q 1 is nilpotent. G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 21

23 (2) FIRST ORDER SYSTEMS The K and L matrices are built from a matrix series Important collection of reduction algorithms. Krylov techniques interrelated by a Möbius transform. Exact moment matching. Reduction of first order systems : Consider the following system description where P (s) is first order : { Cẋ = Gx + Bu y = L T x (25) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 22

24 TRANSFER FUNCTION (FIRST ORDER) The transfer matrix belonging to (25) describing the input-output relation in the Laplace domain is given by : H(s) = y(s) u(s) = LT (G + sc) 1 B (26) and for the reduced system we are looking for, the transfer matrix function reads : H R (s) = y(s) u(s) = L R T (G R + sc R ) 1 B R (27) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 23

25 THE MOEBIUS TRANSFORM This subset of ROM techniques under scrutiny is based on the transform: s = aσ + b cσ + d σ = b sd a sc (28) where not all a, b, c, d such that the related determinant : = ad bc 0 (29) This transform is known as a bilinear or Möbius transformation, it has the property to map the imaginary axis onto the unit circle in the complex plane. G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 24

26 TRANSFER FUNCTION IN TRANSFORMED DOMAIN We apply this operation consequently to the original transfer function: and to the reduced transfer function : H(σ) = (cσ + d)l T (I σa) 1 R (30) H R (σ) = (cσ + d)l T R (I σa R ) 1 R R (31) where the new matrices A and R are related to C, G and B as: A = (dg + bc) 1 (cg + ac) (32) R = (dg + bc) 1 B (33) with similar expressions hold for A R and R R. G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 25

27 SPECTRAL RADIUS OF EXPANSION valid for σ < 1/ρ(A), with (I σa) 1 = i=0 σa i (34) ρ(a) = max{ λ(a) } (35) with λ(a) any eigenvalue of the matrix A. Expanding around σ = 0: ( ) H(σ) = (cσ + d)l T A i σ i R i=0 H(σ) = (cσ + d) L T A i Rσ i (36) i=0 G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 26

28 TRANSFORMED BASIS FUNCTIONS The basisfunctions γ k (s) : γ k (s) = ( c b sd ) ( a sc + d b sd ) k a sc = (ad bc) These expansion functions are not necessarily orthogonal. The expansion coefficients k k and l k read : ( b + sd)k (a sc) k+1 (37) l k = ( A T ) k L k k = A k R (38) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 27

29 LEFT AND RIGHT KRYLOV SPACES The right Krylov subspace is thus determined by : K(A, R, q) = colspan [ R, AR, A 2 R,..., A q 1 R ] (39) and the left Krylov subspace is defined as: K(A T, L, q) = colspan [L, A T L, ( A T ) 2 ( L,..., A T ) ] q 1 L (40) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 28

30 MOMENT MATCHING The traditional way of writing the expansion of the transfer function is : H = (cσ + d) M i σ i (41) i=0 The scalar coefficients M i are called the moments. The transfer matrix of the reduced system becomes : H R (σ) = (cσ + d) L T R A i R R R σ i (42) = (cσ + d) i=0 M R iσ i (43) i=0 G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 29

31 MOMENT MATCHING contd. Now we impose that the moments of the reduced system should match the moments of the original system: Numerical errors Frequency dependent errors M i = L T A i R = L T R A i R R R = M R i (44) H(σ) = H R (σ) + (cσ + d)o(σ q ) (45) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 30

32 THE REDUCED FIRST ORDER SYSTEM Stacking the column vectors v i and w i into matrices V and W : P R (s) = W T P (s)v B R = V T B L R = W T L (46) in the case one employs both the left and right Krylov spaces, and : P R (s) = V T P (s)v B R = V T B L R = V T L (47) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 31

33 THE FIRST ORDER PROJECTOR We obtain the transfer function : H R (s) = L T V (W T P (s)v ) 1 W T B = Q P (s)p (s) 1 B (48) Krylov subspace reduced order modeling algorithms constitute a subclass of methods The matching of the first q moments is exact, in contrast to the more general projection methods G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 32

34 LAGUERRE-SVD σ = s α s + α where α is real while the transformation parameters a = α, b = α, c = 1 and d = 1. The resulting A and R matrices then read : (49) A = (αc + G) 1 (αc G) R = (αc + G) 1 B (50) The change of variables maps the s-domain frequency variable onto the unit circle. The transfer function is written as : H(s) = L T (G + sc) 1 B = 2α s + α i=0 ( ) i s α M i (51) s + α G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 33

35 LAGUERRE BASIS We observe that the γ i (s) are in fact the Laguerre functions, which are written in the Laplace domain as : which read in the time domain: γ i (s) = 2α s + α ( ) i s α (52) s + α φ α i (t) = 2αe αt l i (2αt) (53) where α is the scaling parameter and l i (t) is the Laguerre polynomial l i (t) = et i! d i dt i ( e t t i) (54) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 34

36 LAGUERRE MOMENTS In terms of moments, we obtain : H(σ) = (1 σ)l T ((αc + G) + σ(αc G)) 1 B (55) = 1 σ 2α i=0 M i σ i (56) Optimal estimate for the Laguerre parameter α is motivated, this value being : α 2πf max (57) Note that γ i (s) is the product of the low pass filter ( ) n s α filter s+α. 2α s+α with 3dB bandwith f max and the all pass G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 35

37 LAGUERRE EXPANSION As a consequence, advantageous frequency properties emerge. We can also rewrite the transfer function as : H(s) = L T (s + C 1 G) 1 C 1 B = L T γ n (C 1 G)C 1 B γ n (s) (58) n=0 The summation in (58) is a consequence of the identity : 1 s + u = n=0 γ n (s)γ n (u) R(s), R(u) 0 (59) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 36

38 KRYLOV MATRIX Defining the column vectors k i as : k i = γ i (C 1 G)C 1 B = 1 2π we consequently construct the matrix (iωc + G) 1 B γ n (iω) dω i = 0,..., q 1 (60) K = [k 0, k 1,..., k q 1 ] (61) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 37

39 BANDLIMITED LAGUERRE Find a new set of basis functions, orthonormal over B = [ β, α] [α, β], with β > α > 0. Construct a frequency transform which maps the orthogonal Laguerre basis onto the desired new basis : k i = 1 2π where the γ i are given by : B (iωc + G) 1 B γ i (iω) dω n = 0,..., q 1 (62) γ i (s) = τ(s) φ i (η(s)) i = 0, 1,... (63) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 38

40 FREQUENCY TRANSFORM The φ i are the Laguerre expansion functions discussed in the previous section with η(s) = β2 s s 2 + α 2 s 2 + β 2 τ(s) = β s2 + s β 2 + 2αβ 3α 2 + αβ s(s 2 + β 2 ) (64) The equations (64) emerge as a result of a coordinate transform : ω = ζ(ν) def = β2 ν ν 2 α 2 β 2 ν 2 α ν β (65) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 39

41 BANDLIMITED PROJECTION MATRIX As these basis functions can not be derived from a particular Möbius transform, they yield a projection matrix which is not a Krylov matrix anymore : K = [k 0, k 1,..., k q 1 ] (66) The projection matrix U consists of a basis for colspan(k) and can e.g. obtained from an SVD which ultimately provides the bandlimited reduced order model K = UΣV T (67) F R (s) = L T U ( s U T CU + U T GU ) 1 U T B (68) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 40

42 EXAMPLE : A TRANSMISSION LINE Comparison of Laguerre-SVD and Bandlimited Laguerre reductions Re [Z(f)] (Ω) Laguerre SVD q = 90 Bandlimited q = 90, m = 10 Bandlimited q = 90, m = 20 Laguerre SVD q = 90 Bandlimited q = 90, m = 10 Bandlimited q = 90, m = 20 f (Hz) Simulation Bandwidth Simulation bandwidth f (Hz) x 10 9 G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 41

43 EXAMPLE : A PATCH ANTENNA Bandlimited reduction of a dedicated patch antenna system with Bandlimited Laguerre Direct inversion Bandlimited Magnitude Simulation bandwidth f (Hz) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 42

44 KAUTZ BASIS FUNCTIONS Generalizing the bandlimited laguerre functions : φ 2n (s) = 2τ (s + τ 2 + σ 2 ) φ 2n+1 (s) = 2τ (s τ 2 + σ 2 ) which is the two-parameter Kautz basis. This basis is suited for resonant systems : ((s τ) 2 + σ 2 ) n ((s + τ) 2 + σ 2 ) n+1 n = 0, 1,... ((s τ) 2 + σ 2 ) n ((s + τ) 2 + σ 2 ) n+1 n = 0, 1,... (69) φ n (iω) 2 def = M(ω) = 2τ(ω 2 + τ 2 + σ 2 ) (τ 2 + σ 2 ω 2 ) 2 + 4ω 2 τ 2 (70) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 43

45 BANDLIMITED KAUTZ Consider the functions where and γ n (s) = ρ(s) φ n (η(s)) n = 0, 1,... (71) η(s) = β2 s ρ(s) = β s2 + s β 2 + 2αβ 3α 2 + αβ s(s 2 + β 2 ) satisfying the narrowband orthonormality conditions. s 2 + α 2 s 2 + β 2 (72) (73) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 44

46 ERROR CURVES COMPARED Bandlimited Kautz compared against Multipoint Padé L1 error norms q = q = q = q = x 10 9 f f (GHz) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 45

47 CONVERGENCE COMPARED Bandlimited Laguerre (full) - Kautz-symmetric (dotted) - Kautz-resonance (dash-dotted) Relative error γ = 1e8, σ = 5e9 γ = σ = γ BL γ = γ BL, σ = q G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 46

48 HIGH FREQUENCY KRYLOV TYPE REDUCTIONS The Möbius transform σ = 1/s (a = 0, b = 1, c = 1, d = 0) can be used advantageously. The transfer function reads : H(σ) = 1 σ LT (I + σa) 1 R (74) and the associated matrices A and R are : A = C 1 G R = C 1 B (75) The related γ k (s) are γ k (s) = 1 s k+1 (76) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 47

49 PADE VIA LANCZOS the relevant Möbius transform is defined by : σ = s s 0 (77) which is obtained by substituting (a = 1, b = s 0, c = 0, d = 1) in (28). The associated matrices A and R become : A = (G + s 0 C) 1 C R = (G + s 0 C) 1 B (78) and we need a Taylor expansion about s = s 0. The related γ k (s) are : γ k (s) = (s s 0 ) k (79) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 48

50 LANCZOS PROJECTION The Lanczos process tridiagonalizes A according to: W T AV = DT (80) Here, T is a tridiagonal matrix and D = W T V (81) The matrices W and V are found to be biorthogonal. H(σ) = L T (I σa) 1 R (82) H R (σ) = L T V ( W T V σw T AV ) 1 W T R (83) = L T V (I σt ) 1 D 1 W T R (84) G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 49

51 ARNOLDI Krylov subspace method based on the Möbius transform σ = s s 0. The associated A matrix is reduced to an upper Hessenberg matrix H q while : AV = V H q V T V = I 1 (85) Modified Gram-Schmidt orthogonalization. Only one LU decomposition and q forward/backward substitutions are needed. Process may suffer loss of orthogonality in V while H q is constructed. Ways to circumvent this problem have been found, as e.g. the Implicit Restarted version. As an expansion point σ 0 = 2πf max is usually chosen. G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 50

52 PRIMA σ = s (86) which is equivalent to substituting (a = 1, b = 0, c = 0, d = 1) in (28). The related A and R matrices then read : A = G 1 C R = G 1 B (87) C 0. v. = R G v + Bu 0 L i G T 0 i (88) y = B T v i where R, C and L are describing the resistors, capacitors and inductors respectively. G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 51

53 CONCLUSIONS Similarities between different ROM methods have been highlighted. Systems are seen as being obliquely projected onto a specific subspace. Preservation of structure is an important issue. Frequency dependent convergence behavior has been discussed. An important subclass of first order reduction methods is determined by a Möbius transform. Bandlimited reductions are efficient for large systems. Resonant systems can be reduced faster with an appropriate choice of basis functions. G. Lippens 2nd Workshop Adv. Comp. Electromagnetics May 2006 p 52

54 Second Workshop on Advanced Computational Electromagnetics Wednesday, May 3, 2006 Ghent May 3-4, Program - 8h30 9h00 Welcome Andreas Cangellaris, University of Illinois at Urbana-Champaign, USA Model Order Reduction of Finite Element Models of Electromagnetic Systems Using Krylov Subspace Methods 10h30 Coffee Break 10h45 Dominique Lesselier, Supélec, France 3-D Electromagnetic Inverse Scattering Methodologies with Emphasis on the Retrieval of Small Objects 12h15 Lunch 14h00 Rob Remis, Delft University of Technology, The Netherlands Low- and High-Frequency Model-Order Reduction of Electromagnetic Fields 15h30 Coffee Break 15h45 Hendrik Rogier, Ghent University, Belgium State-of-the-art Antenna Systems in Mobile Communications Thursday, May 4, h30 9h00 Welcome Andreas Cangellaris, University of Illinois at Urbana-Champaign, USA Comprehensive Electromagnetic Modeling of On-Chip Noise Generation and Coupling During High Speed Switching 10h30 Coffee Break 10h45 Davy Pissoort, Ghent University, Belgium Fast and Accurate Modeling of Photonic Crystal Devices 12h15 Lunch 14h00 Tom Dhaene, University of Antwerp, Belgium Electromagnetic-Based Scalable Metamodeling 15h30 Coffee Break 15h45 Luc Knockaert, Gunther Lippens, and Daniël De Zutter, Ghent University, Belgium Towards a Classification of Projection-Based Model Order Reduction

55 Second Workshop On Advanced Computational Electromagnetics Organized by Dr. D. Pissoort Prof. D. De Zutter Prof. F. Olyslager Prof. A. Franchois