Krylov Subspace Methods for Nonlinear Model Reduction

Transcription

1 MAX PLANCK INSTITUT Conference in honour of Nancy Nichols 70th birthday Reading, 2 3 July 2012 Krylov Subspace Methods for Nonlinear Model Reduction Peter Benner and Tobias Breiten Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg, Germany DYNAMIK KOMPLEXER TECHNISCHER SYSTEME MAGDEBURG Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 1/22

2 Outline 1 Motivation 2 Quadratic-Bilinear Control Systems 3 Nonlinear Model Reduction by Generalized Moment-Matching 4 Numerical Examples 5 Conclusions and Outlook Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 2/22

3 Motivation Nonlinear Model Reduction Given a large-scale state-nonlinear control system of the form { ẋ(t) = f (x(t)) + bu(t), Σ : y(t) = c T x(t), x(0) = x 0, with f : R n R n nonlinear and b, c R n, x R n, u, y R. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 3/22

4 Motivation Nonlinear Model Reduction Given a large-scale state-nonlinear control system of the form { ẋ(t) = f (x(t)) + bu(t), Σ : y(t) = c T x(t), x(0) = x 0, with f : R n R n nonlinear and b, c R n, x R n, u, y R. Optimization, control and simulation cannot be done efficiently! Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 3/22

5 Motivation Nonlinear Model Reduction Given a large-scale state-nonlinear control system of the form { ẋ(t) = f (x(t)) + bu(t), Σ : y(t) = c T x(t), x(0) = x 0, with f : R n R n nonlinear and b, c R n, x R n, u, y R. Optimization, control and simulation cannot be done efficiently! MOR { ˆx(t) = ˆf (ˆx(t)) + ˆbu(t), ˆΣ : ŷ(t) = ĉ T ˆx(t), ˆx(0) = ˆx 0, with ˆf : Rˆn Rˆn and ˆb, ĉ Rˆn, x Rˆn, u R and Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 3/22

6 Motivation Nonlinear Model Reduction Given a large-scale state-nonlinear control system of the form { ẋ(t) = f (x(t)) + bu(t), Σ : y(t) = c T x(t), x(0) = x 0, with f : R n R n nonlinear and b, c R n, x R n, u, y R. Optimization, control and simulation cannot be done efficiently! MOR { ˆx(t) = ˆf (ˆx(t)) + ˆbu(t), ˆΣ : ŷ(t) = ĉ T ˆx(t), ˆx(0) = ˆx 0, with ˆf : Rˆn Rˆn and ˆb, ĉ Rˆn, x Rˆn, u R and ŷ y R, ˆn n. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 3/22

7 Motivation Common Reduction Techniques Proper Orthogonal Decomposition (POD) Take computed or experimental snapshots of full model: [x(t 1 ), x(t 2 ),..., x(t N )] =: X, perform SVD of snapshot matrix: X = VSW T Vˆn Sˆn W Ṱ n. Reduction by POD-Galerkin projection: ˆx = V Ṱ n f (Vˆnˆx) + V Ṱ n Bu. Requires evaluation of f discrete empirical interpolation [Sorensen/Chaturantabut 09]. Input dependency due to snapshots! Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 4/22

8 Motivation Common Reduction Techniques Proper Orthogonal Decomposition (POD) Take computed or experimental snapshots of full model: [x(t 1 ), x(t 2 ),..., x(t N )] =: X, perform SVD of snapshot matrix: X = VSW T Vˆn Sˆn W Ṱ n. Reduction by POD-Galerkin projection: ˆx = V Ṱ n f (Vˆnˆx) + V Ṱ n Bu. Requires evaluation of f discrete empirical interpolation [Sorensen/Chaturantabut 09]. Input dependency due to snapshots! Trajectory Piecewise Linear (TPWL) Linearize f along trajectory, reduce resulting linear systems, construct reduced model by weighted sum of linear systems. Requires simulation of original model and several linear reduction steps, many heuristics. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 4/22

9 Quadratic-Bilinear Control Systems Quadratic-Bilinear Differential Algebraic Equations (QBDAEs) Let us consider the class of quadratic-bilinear differential algebraic equations Σ : { Eẋ(t) = Ax(t) + Hx(t) x(t) + Nx(t)u(t) + bu(t), y(t) = c T x(t), x(0) = x 0, where E, A, N R n n, H R n n2 (Hessian tensor), b, c R n. A large class of smooth nonlinear control-affine systems can be transformed into the above type of control system. The transformation is exact, but a slight increase of the state dimension has to be accepted. Input-output behavior can be characterized by generalized transfer functions enables us to use Krylov-based reduction techniques. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 5/22

10 Quadratic-Bilinear Control Systems Transformation via McCormick Relaxation [McCormick 76] Theorem [Gu 09] Assume that the state equation of a nonlinear system Σ is given by ẋ = a 0 x + a 1 g 1 (x) a k g k (x) + Bu, where g i (x) : R n R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 6/22

11 Quadratic-Bilinear Control Systems Transformation via McCormick Relaxation [McCormick 76] Theorem [Gu 09] Assume that the state equation of a nonlinear system Σ is given by ẋ = a 0 x + a 1 g 1 (x) a k g k (x) + Bu, where g i (x) : R n R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example ẋ 1 = exp( x 2 ) x , ẋ 2 = x 2 + u. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 6/22

12 Quadratic-Bilinear Control Systems Transformation via McCormick Relaxation [McCormick 76] Theorem [Gu 09] Assume that the state equation of a nonlinear system Σ is given by ẋ = a 0 x + a 1 g 1 (x) a k g k (x) + Bu, where g i (x) : R n R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example ẋ 1 = exp( x 2 ) x , ẋ 2 = x 2 + u. z 1 := exp( x 2 ), Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 6/22

13 Quadratic-Bilinear Control Systems Transformation via McCormick Relaxation [McCormick 76] Theorem [Gu 09] Assume that the state equation of a nonlinear system Σ is given by ẋ = a 0 x + a 1 g 1 (x) a k g k (x) + Bu, where g i (x) : R n R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example ẋ 1 = exp( x 2 ) x , ẋ 2 = x 2 + u. z 1 := exp( x 2 ), z 2 := x Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 6/22

14 Quadratic-Bilinear Control Systems Transformation via McCormick Relaxation [McCormick 76] Theorem [Gu 09] Assume that the state equation of a nonlinear system Σ is given by ẋ = a 0 x + a 1 g 1 (x) a k g k (x) + Bu, where g i (x) : R n R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example ẋ 1 = exp( x 2 ) x , ẋ 2 = x 2 + u. z 1 := exp( x 2 ), z 2 := x ẋ 1 = z 1 z 2, Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 6/22

15 Quadratic-Bilinear Control Systems Transformation via McCormick Relaxation [McCormick 76] Theorem [Gu 09] Assume that the state equation of a nonlinear system Σ is given by ẋ = a 0 x + a 1 g 1 (x) a k g k (x) + Bu, where g i (x) : R n R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example ẋ 1 = exp( x 2 ) x , ẋ 2 = x 2 + u. z 1 := exp( x 2 ), z 2 := x ẋ 1 = z 1 z 2, ẋ 2 = x 2 + u, Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 6/22

16 Quadratic-Bilinear Control Systems Transformation via McCormick Relaxation [McCormick 76] Theorem [Gu 09] Assume that the state equation of a nonlinear system Σ is given by ẋ = a 0 x + a 1 g 1 (x) a k g k (x) + Bu, where g i (x) : R n R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example ẋ 1 = exp( x 2 ) x , ẋ 2 = x 2 + u. z 1 := exp( x 2 ), z 2 := x ẋ 1 = z 1 z 2, ẋ 2 = x 2 + u, ż 1 = z 1 ( x 2 + u) = z 1 x 2 z 1 u, Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 6/22

17 Quadratic-Bilinear Control Systems Transformation via McCormick Relaxation [McCormick 76] Theorem [Gu 09] Assume that the state equation of a nonlinear system Σ is given by ẋ = a 0 x + a 1 g 1 (x) a k g k (x) + Bu, where g i (x) : R n R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example ẋ 1 = exp( x 2 ) x , ẋ 2 = x 2 + u. z 1 := exp( x 2 ), z 2 := x ẋ 1 = z 1 z 2, ẋ 2 = x 2 + u, ż 1 = z 1 ( x 2 + u) = z 1 x 2 z 1 u, ż 2 = 2 x1 z1 z2 2 z 2 = x 1 z 1. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 6/22

18 Nonlinear Model Reduction Variational Analysis and Linear Subsystems Analysis of nonlinear systems by variational equation approach: Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 7/22

19 Nonlinear Model Reduction Variational Analysis and Linear Subsystems Analysis of nonlinear systems by variational equation approach: consider input of the form αu(t), Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 7/22

20 Nonlinear Model Reduction Variational Analysis and Linear Subsystems Analysis of nonlinear systems by variational equation approach: consider input of the form αu(t), nonlinear system is assumed to be a series of homogeneous nonlinear subsystems, i.e. response should be of the form x(t) = αx 1 (t) + α 2 x 2 (t) + α 3 x 3 (t) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 7/22

21 Nonlinear Model Reduction Variational Analysis and Linear Subsystems Analysis of nonlinear systems by variational equation approach: consider input of the form αu(t), nonlinear system is assumed to be a series of homogeneous nonlinear subsystems, i.e. response should be of the form x(t) = αx 1 (t) + α 2 x 2 (t) + α 3 x 3 (t) comparing coefficients of α i, i = 1, 2,..., leads to series of systems Eẋ 1 = Ax 1 + bu, Eẋ 2 = Ax 2 + Hx 1 x 1 + Nx 1 u, Eẋ 3 = Ax 3 + H (x 1 x 2 + x 2 x 1 ) + Nx 2 u. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 7/22

22 Nonlinear Model Reduction Variational Analysis and Linear Subsystems Analysis of nonlinear systems by variational equation approach: consider input of the form αu(t), nonlinear system is assumed to be a series of homogeneous nonlinear subsystems, i.e. response should be of the form x(t) = αx 1 (t) + α 2 x 2 (t) + α 3 x 3 (t) comparing coefficients of α i, i = 1, 2,..., leads to series of systems Eẋ 1 = Ax 1 + bu, Eẋ 2 = Ax 2 + Hx 1 x 1 + Nx 1 u, Eẋ 3 = Ax 3 + H (x 1 x 2 + x 2 x 1 ) + Nx 2 u. although i-th subsystem is coupled nonlinearly to preceding systems, linear systems are obtained if terms x j, j < i, are interpreted as pseudo-inputs. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 7/22

23 Nonlinear Model Reduction Generalized Transfer Functions In a similar way, a series of generalized symmetric transfer functions can be obtained via the growing exponential approach: Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 8/22

24 Nonlinear Model Reduction Generalized Transfer Functions In a similar way, a series of generalized symmetric transfer functions can be obtained via the growing exponential approach: H 1 (s 1 ) = c T (s 1 E A) 1 b, }{{} G 1(s 1) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 8/22

25 Nonlinear Model Reduction Generalized Transfer Functions In a similar way, a series of generalized symmetric transfer functions can be obtained via the growing exponential approach: H 1 (s 1 ) = c T (s 1 E A) 1 b, }{{} G 1(s 1) H 2 (s 1, s 2 ) = 1 2! ct ((s 1 + s 2 )E A) 1 [N (G 1 (s 1 ) + G 1 (s 2 )) +H (G 1 (s 1 ) G 1 (s 2 ) + G 1 (s 2 ) G 1 (s 1 ))], Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 8/22

26 Nonlinear Model Reduction Generalized Transfer Functions In a similar way, a series of generalized symmetric transfer functions can be obtained via the growing exponential approach: H 1 (s 1 ) = c T (s 1 E A) 1 b, }{{} G 1(s 1) H 2 (s 1, s 2 ) = 1 2! ct ((s 1 + s 2 )E A) 1 [N (G 1 (s 1 ) + G 1 (s 2 )) +H (G 1 (s 1 ) G 1 (s 2 ) + G 1 (s 2 ) G 1 (s 1 ))], H 3 (s 1, s 2, s 3 ) = 1 3! ct ((s 1 + s 2 + s 3 )E A) 1 [ N(G 2 (s 1, s 2 ) + G 2 (s 2, s 3 ) + G 2 (s 1, s 3 )) + H ( G 1 (s 1 ) G 2 (s 2, s 3 ) + G 1 (s 2 ) G 2 (s 1, s 3 ) + G 1 (s 3 ) G 2 (s 1, s 3 ) + G 2 (s 2, s 3 ) G 1 (s 1 ) + G 2 (s 1, s 3 ) G 1 (s 2 ) + G 2 (s 1, s 2 ) G 1 (s 3 ) )]. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 8/22

27 Nonlinear Model Reduction Constructing the Projection Matrix Goal: s q 1 H 1 (σ) = 1 s q 1 1 Ĥ 1 (σ), H s1 l 2 (σ, σ) = Ĥ sm 2 s1 l 2 (σ, σ), l + m q 1 sm 2 Construct the following sequence of nested Krylov subspaces Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 9/22

28 Nonlinear Model Reduction Constructing the Projection Matrix Goal: s q 1 1 H 1 (σ) = s q 1 1 Ĥ 1 (σ), H s1 l 2 (σ, σ) = Ĥ sm 2 s1 l 2 (σ, σ), l + m q 1 sm 2 Construct the following sequence of nested Krylov subspaces V 1 =K q ( (A σe) 1 E, (A 1 σe) 1 b ) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 9/22

29 Nonlinear Model Reduction Constructing the Projection Matrix Goal: s q 1 1 H 1 (σ) = s q 1 1 Ĥ 1 (σ), H s1 l 2 (σ, σ) = Ĥ sm 2 s1 l 2 (σ, σ), l + m q 1 sm 2 Construct the following sequence of nested Krylov subspaces V 1 =K q ( (A σe) 1 E, (A 1 σe) 1 b ) for i = 1 : q V i 2 = K q i+1 ( (A 2σE) 1 E, (A 2σE) 1 NV 1 (:, i) ), Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 9/22

30 Nonlinear Model Reduction Constructing the Projection Matrix Goal: s q 1 1 H 1 (σ) = s q 1 1 Ĥ 1 (σ), H s1 l 2 (σ, σ) = Ĥ sm 2 s1 l 2 (σ, σ), l + m q 1 sm 2 Construct the following sequence of nested Krylov subspaces V 1 =K q ( (A σe) 1 E, (A 1 σe) 1 b ) for i = 1 : q V i 2 = K q i+1 ( (A 2σE) 1 E, (A 2σE) 1 NV 1 (:, i) ), for j = 1 : min(q i + 1, i) V i,j 3 = K q i j+2 ( (A 2σE) 1 E, (A 2σE) 1 HV 1 (:, i) V 1 (:, j) ), V 1 (:, i) denoting the i-th column of V 1. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 9/22

31 Nonlinear Model Reduction Constructing the Projection Matrix Goal: s q 1 1 H 1 (σ) = s q 1 1 Ĥ 1 (σ), H s1 l 2 (σ, σ) = Ĥ sm 2 s1 l 2 (σ, σ), l + m q 1 sm 2 Construct the following sequence of nested Krylov subspaces V 1 =K q ( (A σe) 1 E, (A 1 σe) 1 b ) for i = 1 : q V i 2 = K q i+1 ( (A 2σE) 1 E, (A 2σE) 1 NV 1 (:, i) ), for j = 1 : min(q i + 1, i) V i,j ( 3 = K q i j+2 (A 2σE) 1 E, (A 2σE) 1 HV 1 (:, i) V 1 (:, j) ), ( ) V 1 (:, i) denoting the i-th column of V 1. Set V = orth [V 1, V2 i, V i,j 3 ] and construct ˆΣ by the Galerkin-Projection P = VV T : Â = V T AV Rˆn ˆn, Ĥ = V T H(V V) Rˆn ˆn2, ˆN = V T NV Rˆn ˆn, ˆb = V T b Rˆn, ĉ T = c T V Rˆn. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 9/22

32 Nonlinear Model Reduction Tensors and Matricizations: A Short Excursion [Kolda/Bader 09, Grasedyck 10] A tensor is a vector (A i ) i I R I indexed by a product index set I = I 1 I d, I j = n j. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 10/22

33 Nonlinear Model Reduction Tensors and Matricizations: A Short Excursion [Kolda/Bader 09, Grasedyck 10] A tensor is a vector (A i ) i I R I indexed by a product index set I = I 1 I d, I j = n j. For a given tensor A, the t-matricization A (t) is defined as A (t) R It I t, A (t) (i µ)µ t, (i µ)µ t := A (i1,...,i d ), t := {1,..., d}\t. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 10/22

34 Nonlinear Model Reduction Tensors and Matricizations: A Short Excursion [Kolda/Bader 09, Grasedyck 10] A tensor is a vector (A i ) i I R I indexed by a product index set I = I 1 I d, I j = n j. For a given tensor A, the t-matricization A (t) is defined as A (t) R It I t, A (t) (i µ)µ t, (i µ)µ t := A (i1,...,i d ), t := {1,..., d}\t. Example: For a given 3-tensor A (i1,i 2,i 3) with i 1, i 2, i 3 {1, 2}, we have: [ ] A (1) A(1,1,1) A = (1,2,1) A (1,1,2) A (1,2,2), A (2,1,1) A (2,2,1) A (2,1,2) A (2,2,2) A (2) = [ ] A(1,1,1) A (2,1,1) A (1,1,2) A (2,1,2). A (1,2,1) A (2,2,1) A (1,2,2) A (2,2,2) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 10/22

35 Nonlinear Model Reduction Tensors and Matricizations: A Short Excursion [Kolda/Bader 09, Grasedyck 10] A tensor is a vector (A i ) i I R I indexed by a product index set I = I 1 I d, I j = n j. For a given tensor A, the t-matricization A (t) is defined as A (t) R It I t, A (t) (i µ)µ t, (i µ)µ t := A (i1,...,i d ), t := {1,..., d}\t. Allows to compute matrix products more efficiently. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 10/22

36 Nonlinear Model Reduction Two-Sided Projection Methods Similarly to the linear case, one can exploit duality concepts, in order to construct two-sided projection methods. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 11/22

37 Nonlinear Model Reduction Two-Sided Projection Methods Similarly to the linear case, one can exploit duality concepts, in order to construct two-sided projection methods. Interpreting H (2) now as the 2-matricization of the Hessian 3-tensor from R n n n, one can show that the dual Krylov spaces have to be constructed as follows: ) W 1 =K q ((A 2σE) T E T, (A 2σE) T c for i = 1 : q W i 2 = K q i+1 ( ) (A σe) T E T, (A σe) T N T W 1 (:, i), for j = 1 : min(q i + 1, i) ) W i,j 3 = K q i j+2 ((A σe) T E T, (A σe) T H (2) V 1 (:, i) W 1 (:, j), Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 11/22

38 Nonlinear Model Reduction Two-Sided Projection Methods Similarly to the linear case, one can exploit duality concepts, in order to construct two-sided projection methods. Interpreting H (2) now as the 2-matricization of the Hessian 3-tensor from R n n n, one can show that the dual Krylov spaces have to be constructed as follows: ) W 1 =K q ((A 2σE) T E T, (A 2σE) T c for i = 1 : q W i 2 = K q i+1 ( ) (A σe) T E T, (A σe) T N T W 1 (:, i), for j = 1 : min(q i + 1, i) ) W i,j 3 = K q i j+2 ((A σe) T E T, (A σe) T H (2) V 1 (:, i) W 1 (:, j), Note: Due to the symmetry of the Hessian tensor, the 3-matricization H (3) coincides with H (2). Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 11/22

39 Nonlinear Model Reduction Multimoment matching Theorem Σ = (E, A, H, N, b, c) original QBDAE system. Reduced system by Petrov-Galerkin projection P = VW T with ) V 1 = K q1 (E, A, b, σ), W 1 = K q1 (E T, A T, c, 2σ Then, it holds: i+j s i 1 sj 2 for i = 1 : q 2 i H 1 s i 1 V 2 = K q2 i+1 (E, A, NV 1 (:, i), 2σ) ( ) W 2 = K q2 i+1 E T, A T, N T W 1 (:, i), σ for j = 1 : min(q 2 i + 1, i) V 3 = K q2 i j+2 (E, A, HV 1 (:, i) V 1 (:, j), 2σ) ( ) W 3 = K q2 i j+2 E T, A T, H (2) V 1 (:, i) W 1 (:, j), σ. (σ) = i Ĥ 1 s1 i (σ), i H 1 s i 1 (2σ) = i Ĥ 1 s1 i (2σ), i = 0,..., q 1 1, H 2 (σ, σ) = i+j Ĥ 2 (σ, σ), i + j 2q 2 1. s1 i sj 2 Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 12/22

40 Numerical Examples A nonlinear RC Circuit v 1 v 2 v ξ 1 v ξ g(v) g(v) g(v) g(v) i = u(t) g(v) C C C C C g(v) = e 40v + v 1, C = 1, v(t) = f (v(t), g(v(t))) + Bu(t) y(t) = v 1 (t) state-nonlinear control system clever transformation only doubles the state dimension of the resulting quadratic-bilinear system Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 13/22

41 Numerical Examples A nonlinear RC Circuit Comparison between moment-matching and POD (u(t) = e t ) Original system, n = sided proj., ˆn = 11 2-sided proj., ˆn = 11 POD, ˆn = Transient response Relative error y(t) Time (t) Time (t) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 13/22

42 Numerical Examples A nonlinear RC Circuit Comp. between moment-matching and POD (u(t) = (cos(2π t 10 ) + 1)/2) Original system, n = sided proj., ˆn = 11 2-sided proj., ˆn = 11 POD, ˆn = Transient response Relative error y(t) Time (t) Time (t) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 13/22

43 Numerical Examples The Chafee-Infante equation Let us consider a cubic nonlinear PDE of the form v t + v 3 = v xx + v, in (0, 1) (0, T ), v(0, ) = u(t), in (0, T ), v x (1, ) = 0, in (0, T ), v(x, 0) = v 0 (x), in (0, 1) original state dimension n = 500, QBDAE dimension N = 2 500, reduced QBDAE dimension r = 9 Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 14/22

44 Numerical Examples The Chafee-Infante equation Comparison between moment-matching and POD (u(t) = 5 cos (t)) 2 1 FOM, n = 500 POD, ˆn = 9 1-sided MM, ˆn = 9 2-sided MM, ˆn = 9 y(t) Time (t) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 15/22

45 Numerical Examples The Chafee-Infante equation Comp. between moment-matching and POD (u(t) = 50 sin (t)) 4 2 FOM, n = 500 POD, ˆn = 9 1-sided MM, ˆn = 9 2-sided MM, ˆn = 9 y(t) Time (t) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 16/22

46 Numerical Examples The FitzHugh-Nagumo System FitzHugh-Nagumo system modeling a neuron [Chaturantabut, Sorensen 09] ɛv t (x, t) = ɛ 2 v xx (x, t) + f (v(x, t)) w(x, t) + g, w t (x, t) = hv(x, t) γw(x, t) + g, with f (v) = v(v 0.1)(1 v) and initial and boundary conditions v(x, 0) = 0, w(x, 0) = 0, x [0, 1], v x (0, t) = i 0 (t), v x (1, t) = 0, t 0, where ɛ = 0.015, h = 0.5, γ = 2, g = 0.05, i 0 (t) = t 3 exp( 15t) original state dimension n = , QBDAE dimension N = , reduced QBDAE dimension r = 20 Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 17/22

47 Numerical Examples The FitzHugh-Nagumo System Limit cycle behavior for 1-sided proj. (ROM, ˆn = 20, σ = 4) 0.2 Orig. system, n = sided proj., ˆn = 20, σ = 4 w(x, t) v(x, t) x 0.4 Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 18/22

48 Numerical Examples The FitzHugh-Nagumo System POD via moment-matching (training input) FOM, n = 2000 POD, ˆn = 22 POD-MM, ˆn = 22 w(t) v(t) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 19/22

49 Numerical Examples The FitzHugh-Nagumo System POD via moment-matching (varying input) FOM, n = 2000 POD, ˆn = 22 POD-MM, ˆn = 22 w(t) v(t) Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 20/22

50 Conclusions and Outlook Many nonlinear dynamics can be expressed by a system of quadratic-bilinear differential algebraic equations. For these type of systems, a frequency domain analysis leads to certain generalized transfer functions. There exist Krylov subspace methods that extend the concept of moment-matching using basic tools from tensor theory allows for better approximations. In contrast to other methods like TPWL and POD, the reduction process is independent of the control input. Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 21/22

51 Conclusions and Outlook Many nonlinear dynamics can be expressed by a system of quadratic-bilinear differential algebraic equations. For these type of systems, a frequency domain analysis leads to certain generalized transfer functions. There exist Krylov subspace methods that extend the concept of moment-matching using basic tools from tensor theory allows for better approximations. In contrast to other methods like TPWL and POD, the reduction process is independent of the control input. Optimal choice of interpolation points? Stability-preserving reduction possible? Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 21/22

52 Last but not least... [image source: Heike Faßbender] Happy birthday, Nancy! Max-Planck-Institut Magdeburg P. Benner, Krylov Subspace Methods for Nonlinear Model Reduction 22/22