Iterative factorization of the error system in Moment Matching and applications to error bounds Heiko Panzer, Thomas Wolf, Boris Lohmann GAMM-Workshop – Applied and Numerical Linear Algebra Bremen,

Heiko Panzer2 Error Decomposition in Krylov Subspace methods If V spans an Input Krylov Subspace: If W spans an Output Krylov Subspace: Wolf/P The following decompositions hold for Krylov Subspace methods: What can we use these factorizations for?!

Heiko Panzer3 Feature 1: Numerical advantage Compare the classical error system… …to its new formulation… contains twice the (almost) same dynamics subtraction is performed in the output signals small scale and easy to analyze invariant zeros at expansion points, poles equal to those of G r (s) → all-pass for IRKA/ISRK very similar to G(s) input vector/matrix contains subtraction

Heiko Panzer4 Numerical advantage II Example: Reduce ISS model [2] using RK-ICOP [Eid 2009] q = 14 ICOP: s opt = 0.94 Assume we want to perform a second reduction step starting from G(s)-G r (s) q = 14 ICOP: s opt = 0.77 Use GB┴ instead: ICOP: s opt = 37.7 →Detected frequency much better suited

Heiko Panzer5 Numerical advantage III Example: ISS model →Additional information is available and can be used to optimize RK-ICOP!

Heiko Panzer6 Feature 2: H 2 -error in SVD-Krylov-methods The Gramian Q is assumed to be known anyway in SVD-Krylov. → Cheap error bound! Special case ISRK [4]: All-pass in case of locally H 2 -optimal reduction small scale Wolf/P. 2011

Heiko Panzer7 Feature 3: Physical interpretability Main idea:Can we use the new error representation for a better understanding of the reduction result from a physical/engineering point of view? input u(t) error y e (t) input u(t) error y e (t) approx. y r (t) output y(t) ? We can easily compute this intermediate signal… … which enters our original model via a different input structure

Heiko Panzer8 Physical interpretability II input u(t) error y e (t) Example:Continuous heat equation [2]Rational Krylov. n=200, q=15 not all-pass, but amplitude is diminished at all frequencies →The error resulting from MOR is equivalent to the output caused by a minor additional heat source. An engineer might regard this as admissable.

Heiko Panzer9 Feature 4: Iterative decomposition qq2q2 nqq2q2 q q2q2 nqq2q2 q q q2q2 qq2q2 q q3q3 nq3q3 q q2q2 qq2q2 q q3q3 nq3q3 q q2q2 ∑q i n Let V form an Input Krylov Subspace 2 nd reduction 3 rd reduction

Heiko Panzer10 Iterative decomposition II qq2q2 nqq2q2 q q2q2 nqq2q2 q qq2q2 qq2q2 qq3q3 nq3q3 q q q2q2 q3q3 nq3q3 qq2q2 ∑q i n qq2q2 Let W form an Output Krylov Subspace 2 nd reduction 3 rd reduction

Heiko Panzer11 Iterative decomposition III q q2q2 qq2q2 q q3q3 n q3q3 q q2q2 ∑q i n∑q i V One can iteratively influence the input and output matrices of the remaining large scale model according to one‘s objectives! ∑q i W from Output Krylov decomposition from Input Krylov decomposition large scale

Heiko Panzer12 Feature 5: Dissipativity-based error bound A large class of systems fulfills → Positive definite E → Strictly dissipative A This is e.g. true for port-Hamiltonian systems with R > 0 or can be achieved for typical second order systems. Then: compute iteratively (inexpensive!) P./Wolf ACC2012

Heiko Panzer13 Dissipativity-based error bound II use bound for dissipative systems easy to compute Objectives: Iteratively lower bound on G*(s) by making B ┴ and C ┴ smaller → orthogonal projection with W=V Keep feedthrough-filters close to all-pass (avoid peaks that boost H ∞ -norm) → IRKA/ISRK or pole-placement algorithms Framework for computation and iterative sharpening of an H 2 error bound: goal conflict

Heiko Panzer14 Dissipativity-based error bound III Proposed algorithm: 1.Reduce original model by RK method of your choice 2.Decompose error system (compute new B ┴ or C ┴ and feedthrough) 3.Evaluate bound 4.Reduce G*(s) using a modified one-sided IRKA (with W:=V or V:=W) → guaranteed preservation of stability → orthogonal projection makes terms smaller and smaller → in case of convergence the feedthrough-models are all-pass 5.Return to step 2. Framework for computation and iterative sharpening of an H 2 error bound: SISO:

Heiko Panzer15 Feature 5: Dissipativity-based error bound Example:CD Player [2]

Heiko Panzer16 Feature 5: Dissipativity-based error bound Example:Butterfly Gyroscope [2]

Heiko Panzer17 Conclusions The new factorization… is most inexpensive to compute exhibits nice numerical behaviour offers cheap H 2 error expressions in SVD-Krylov-methods like ISRK makes the error physically interpretable can be iteratively applied provides a purely Krylov-based H 2 error bound for strictly dissipative systems But… we need a strategy how the iterative reductions must be performed we have no experience with MIMO, so far

Heiko Panzer18 Bibliography [1]R. Eid: Time Domain Moment Matching. PhD Thesis. TU München, [2]Oberwolfach Model Reduction Benchmark Collection. Available online at [3]H. Panzer, J. Hubele, R. Eid and B. Lohmann: Generating a Parametric Finite Element Model of a 3D Cantilever Timoshenko Beam Using Matlab. TRAC-4 Number Available online at [4]S. Gugercin, An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems. Linear Algebra and its Applications, 2008.

Heiko Panzer19 Feature 2: Physical interpretability How small scale and easy to analyze Example:Continuous heat equation [2] FEM discretization

Heiko Panzer20 Feature 2: Physical interpretability Main idea:Can we use the new error representation for a better understanding of the reduction result from a physical/engineering point of view? input u(t) error y e (t) Perform MOR using IRKA to q=12. Example:Continuous heat equation [2] approx. y r (t) output y(t)