Model Order Reduction for Wave Equations Rob F. Remis and J orn T. - PowerPoint PPT Presentation

General solution and symmetry Krylov MOR Model Order Reduction for Wave Equations Rob F. Remis and J¨ orn T. Zimmerling DCSE Fall School, Delft, November 4 – 8, 2019 1

General solution and symmetry Krylov MOR General solution and symmetry Semidiscrete Maxwell/wave field system (D + S + M ∂ t ) f( t ) = − q ′ ( t ) , t > 0 Field and source vector vanish for t < 0 Time dependence of source can be factored out: q ′ ( t ) = w ( t )˜ q w ( t ) is called the source wavelet or source signature and vanishes for t < 0 2

General solution and symmetry Krylov MOR General solution and symmetry Semidiscrete Maxwell/wave field system (D + S + M ∂ t ) f( t ) = − w ( t )˜ q , t > 0 Symmetry properties of matrix D: Matrix D is skew-symmetric w.r.t. W: D T W = − WD Matrix D is symmetric w.r.t. W δ − : D T W δ − = W δ − D 3

General solution and symmetry Krylov MOR General solution and symmetry Multiply system by M − 1 to obtain (A + I ∂ t ) f( t ) = − w ( t )q , t > 0 with q = M − 1 ˜ A = M − 1 (D + S) q and solution can be written in terms of the matrix exponential function (evolution operator) f( t ) = − w ( t ) ∗ U ( t ) exp( − A t )q , t > 0 U ( t ): Heaviside unit step function ∗ : convolution in time 4

General solution and symmetry Krylov MOR General solution and symmetry Initial-value problem: (A + I ∂ t ) f( t ) = 0 , t > 0 with f(0) = f 0 . Solution: f( t ) = exp( − A t )f 0 t ≥ 0 5

General solution and symmetry Krylov MOR General solution and symmetry Lossless media: A = M − 1 D A. Matrix A is skew-symmetric w.r.t. WM A T WM = − WMA General case: A = M − 1 (D + S) B. Matrix A is symmetric w.r.t. WM δ − A T WM δ − = WM δ − A 6

General solution and symmetry Krylov MOR General solution and symmetry Symmetry property A is related to energy conservation Solution initial value problem: f( t ) = exp( − A t )f 0 , t ≥ 0 Stored energy in initial field is given by E 0 = 1 2f T 0 WMf 0 7

General solution and symmetry Krylov MOR General solution and symmetry Stored energy at time instant t : E ( t ) = 1 2f T ( t )WMf( t ) = 1 2f T 0 exp( − A T t )WM exp( − A t )f 0 = 1 2f T 0 WM exp(+A t ) exp( − A t )f 0 = E 0 8

General solution and symmetry Krylov MOR General solution and symmetry WM is diagonal positive definite Energy inner product � x , y � en = y T WMx Inner product induces the energy norm E ( t ) = 1 � f � en = � f , f � 1 / 2 2 � f � 2 en en 9

General solution and symmetry Krylov MOR General solution and symmetry Symmetry property B is related to reciprocity Introduce the matrices δ e = 1 δ h = 1 2(I + δ − ) 2(I − δ − ) and We have δ e δ − = δ − δ e = δ e and δ h δ − = δ − δ h = − δ h 10

General solution and symmetry Krylov MOR General solution and symmetry Electric-type vector: u = δ e u Magnetic-type vector: u = δ h u Source vector is of the electric-type: q = δ e q 11

General solution and symmetry Krylov MOR General solution and symmetry Solution: f q ( t ) = − w ( t ) ∗ U ( t ) exp( − A t )q Measurement: r T WMf q ( t ) for some receiver vector r Using symmetry property B, we have Electric-field measurement, r = δ e r: r T WMf q ( t ) = q T WMf r ( t ) Magnetic-field measurement, r = δ h r: r T WMf q ( t ) = − q T WMf r ( t ) 12

General solution and symmetry Krylov MOR General solution and symmetry Recall 1 2f T WMf = stored field energy in the domain 1 2f T WMf = sum of electric and magnetic field energy WM δ − is not positive definite 1 2f T WM δ − f = difference of electric and magnetic field energy Free-field Lagrangian: L ( t ) = 1 2f T ( t )WM δ − f( t ) 13

General solution and symmetry Krylov MOR General solution and symmetry Bilinear form � x , y � la = y T WM δ − x ( not an inner product) Energy and Lagrangian: E ( t ) = 1 L ( t ) = 1 2 � f , f � en and 2 � f , f � la 14

General solution and symmetry Krylov MOR General solution and symmetry Symmetry: Lossless case (S = 0): A is skew-symmetric w.r.t. �· , ·� en General case: A is symmetric w.r.t. �· , ·� la General solution (again): f( t ) = − w ( t ) ∗ U ( t ) exp( − A t )q , t > 0 15

General solution and symmetry Krylov MOR Krylov MOR Power series expansion matrix exponential function ∞ ( − A t ) k � exp( − A t ) = k ! k =0 Solution consists of a superposition of powers of A acting on q It makes sense to look for approximations that belong to the Krylov subspace K m = span { q , Aq , ..., A m − 1 q } 16

General solution and symmetry Krylov MOR Krylov MOR Lossless case: A = M − 1 D Let v 1 , v 2 , ...,v m be a basis of K m orthonormal w.r.t. WM � v i , v j � en = δ i , j Expand approximation f m ( t ) in terms of these basis vectors f m ( t ) = α 1 ( t )v 1 + α 2 ( t )v 2 + ... + α m ( t )v m = V m a m ( t ) with V m = [v 1 , v 2 , ..., v m ] and a m ( t ) = [ α 1 ( t ) , α 2 ( t ) , ..., α m ( t )] T 17

General solution and symmetry Krylov MOR Krylov MOR V m is a tall matrix having the basis vectors as its columns Residual of the field approximation r m ( t ) = − w ( t )q − Af m ( t ) − ∂ t f m ( t ) = − w ( t )q − AV m a m ( t ) − V m ∂ t a m ( t ) We determine the expansion coefficients from the Galerkin condition V T m WMr m ( t ) = 0 18

General solution and symmetry Krylov MOR Krylov MOR Using the orthonormality of the basis vectors w.r.t. WM, we obtain (T m + I m ∂ t ) a m ( t ) = − w ( t )V T m WMq , t > 0 with I m identity matrix order m , and T m = V T m WMAV m Note that T m is skew-symmetric 19

General solution and symmetry Krylov MOR Krylov MOR In addition, we take v 1 = � q � − 1 en q Note that v 1 is first column of matrix V m v 1 = V m e 1 Consequently, q = � q � en V m e 1 20

General solution and symmetry Krylov MOR Krylov MOR and we have (T m + I m ∂ t ) a m ( t ) = − w ( t ) � q � en e 1 , t > 0 Solution: a m ( t ) = −� q � en w ( t ) ∗ U ( t ) exp( − T m t )e 1 , t > 0 Field approximation or ROM: f m ( t ) = −� q � en w ( t ) ∗ U ( t )V m exp( − T m t )e 1 , t > 0 21

General solution and symmetry Krylov MOR Krylov MOR The basis vectors can be generated using the algorithm β i +1 v i +1 = Av i + β i v i − 1 , i = 1 , 2 , ..., m with v 0 = 0, β 1 = � q � en , and the β i , i ≥ 2, are determined from the condition � v i , v i � en = 1 , i = 1 , 2 , .. Proof is by induction This is the Lanczos algorithm for skew-symmetric matrices 22

General solution and symmetry Krylov MOR Krylov MOR After m steps of this algorithm we have the summarizing equation AV m = V m T m + β m +1 v m +1 e T m with   0 − β 2 β 2 0    ...  = V T T m = m WMAV m       − β m   β m 0 23

General solution and symmetry Krylov MOR Krylov MOR Compared with an explicit leap-frog time-stepping scheme 1 /β i act as a time step For the general case proceed in a similar manner Basis vectors are “orthonormal” w.r.t. �· , ·� la Matrix T m is tridiagonal and (complex/sign) symmetric in this case Remark: the resulting Lanczos algorithm may break down, since �· , ·� la is not an inner product 24

General solution and symmetry Krylov MOR Krylov MOR Frequency-domain modeling A Laplace transform gives ˆ w ( s )(A + s I) − 1 q f( s ) = − ˆ Matrix resolvent instead of matrix exponential needs to be evaluated ROM: ˆ w ( s )V m (T m + s I) − 1 e 1 f m ( s ) = −� q � en ˆ 25

General solution and symmetry Krylov MOR Krylov MOR Convergence, what to expect? High-frequency expansion of resolvent: ∞ (A + s I) − 1 = s − 1 (I + s − 1 A) − 1 = s − 1 � ( − s − 1 A) k k =0 Powers of A: Early times/high frequencies are approximated first 26

General solution and symmetry Krylov MOR Krylov MOR When it exists, the group inverse A # of A is uniquely defined by the conditions A # AA # = A , AA # = A # A AA # A = A , and AA # projector onto the range of A Source vector belongs to the range of A: q = AA # q ˆ w ( s )(A + s I) − 1 q f( s ) = − ˆ w ( s )(A + s I) − 1 AA # q = − ˆ w ( s )A # (I + s A # ) − 1 q = − ˆ ∞ � w ( s )A # ( − s A # ) k q = − ˆ k =0 Low-frequency expansion Inverse powers of A: late-times/low-frequencies are approximated first 27

General solution and symmetry Krylov MOR Krylov MOR Construct ROMs belonging to the Krylov space K m = span { q , A # q , ..., (A # ) m − 1 q } A # inherits symmetry properties of A Lanczos algorithms with A # A # can be determined explicitly Action of A # on a vector requires solution of Poisson equation(s) 28

General solution and symmetry Krylov MOR Krylov MOR Early- and late-time field approximations Construct a ROM that belongs to the extended Krylov space K k , m = span { (A # ) k − 1 q , (A # ) k − 2 q , ..., A # q , q , Aq , ..., A m − 1 q } By exploiting symmetry, a basis of this space can again be generated via short recurrence relations 29

General solution and symmetry Krylov MOR Krylov MOR Standard Krylov: field approximated by a polynomial in A Extended Krylov: field approximated by a Laurent polynomial in A Rational Krylov: field approximated by a rational function in A 30