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Combined State and Parameter Reduction Christian Himpe ([email protected]) Mario Ohlberger ([email protected]) WWU Münster Institute for Computational and Applied Mathematics GAMM 2014 π-Day
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held at 85th GAMM Annual Meeting Erlangen - march 2014

### Transcript of Combined State and Parameter Reduction

• 1. Combined State and Parameter Reduction Christian Himpe ([email protected]) Mario Ohlberger ([email protected]) WWU Mnster Institute for Computational and Applied Mathematics GAMM 2014 -Day

2. Overview 1 Control Systems 2 Application 3 Model Reduction 1 Gramian-Based Combined Reduction 2 Optimization-Based Combined Reduction 4 Numerical Results 3. Control Systems General Control System: x(t) = f (t, x(t), u(t), ) y(t) = g(t, x(t), u(t), ) x(0) = x0 State x(t) RN Input u(t) RM Output y(t) RO Paramter RP Vector Field f : R RN RM RP RN Functional g : R RN RM RP RO 4. Control Systems General Control System: x(t) = f (t, x(t), u(t), ) y(t) = g(t, x(t), u(t), ) x(0) = x0 Example: x(t) = A()x(t) + Bu(t) y(t) = Cx(t) x(0) = x0 State x(t) RN Input u(t) RM Output y(t) RO Paramter RP Vector Field f : R RN RM RP RN Functional g : R RN RM RP RO 5. Brain as a Control System1 Inputs: Sensory Input States: Brain Regions Outputs: Measurement of Regions Activity Parameters: Connectivity of Regions 1 Friston et al 2003 6. Neuronal Network Models fMRI / fNIRS2 x = A()x + Bu z = f (z, x) y = g(z) EEG / MEG3 x = ALx + AN()f (x) + Bu y = Cx 2 Friston et al 2003, Stephan et al 2008 3 David et al 2004, David et al 2006 7. Model Order Reduction Model: x = f (x, u) y = g(x, u) Model Order: dim(x) 1 dim(u) dim(x) dim(y) dim(x) Reduced Order Model: xr = fr (xr , u) yr = gr (xr , u) Reduced Order: dim(xr ) dim(x), y yr 1 8. Model Reduction for Parametrized Systems Parametrized Model: x = f (x, u, ) y = g(x, u, ) Model Order: dim(x) 1 dim(u) dim(x) dim(y) dim(x) Reduced Order Model: xr = fr (xr , u, ) yr = gr (xr , u, ) Reduced Order: dim(xr ) dim(x), y yr 1, 9. Combined State and Parameter Reduction Parametrized Model: x = f (x, u, ) y = g(x, u, ) Model Order: dim() 1 dim(x) 1 dim(u) dim(x) dim(y) dim(x) Reduced Order Model: xr = fr (xr , u, r ) yr = gr (xr , u, r ) Reduced Order: dim(xr ) dim(x), y yr 1, 10. Projection-Based Model Reduction State Projection: V Rdim(x)r , r < dim(x), VV T = 1 State Reduced Model: xr = fr (xr , u, ) = V T f (Vxr , u, ) yr = gr (xr , u, ) = g(Vxr , u, ) 11. Projection-Based Model Reduction Parameter Projection: V Rdim()s , s < dim(), VVT = 1 Parameter Reduced Model: xr = fr (x, u, r ) = f (x, u, Vr ) yr = gr (x, u, r ) = g(x, u, Vr ) 12. Projection-Based Model Reduction State and Parameter Projections: V Rdim(x)r , r < dim(x), VV T = 1 V Rdim()s , s < dim(), VVT = 1 State and Parameter Reduced Model: xr = fr (xr , u, r ) = V T f (Vxr , u, Vr ) yr = gr (xr , u, r ) = g(Vxr , u, Vr ) 13. Dual Approach Gramian-Based Approximately balance system, based on controllability and observability. Truncate least controllable and observable states. Treat parameters as constant states. Optimization-Based Iteratively assemble projections, utilizing greedy algorithm to nd reduced parameters. Build parameter projection base vector by vector. Parameter projection then induces the state projection. 14. Gramian-Based Model Reduction4 x(t) = Ax(t) + Bu(t) y(t) = Cx(t) Controllability How well can x be driven by u? Controllability Gramian: WC := 0 eAt BBT eAT t dt Observability How well is change in x visible in y? Observability Gramian: WO := 0 eAT t CT CeAt dt Cross Gramian: WX := 0 eAt BCeAt dt dim(u) = dim(y) 4 Moore 1981, Fernando and Nicholson 1983 15. Empirical Gramians5 WC := 0 eAt BBT eAT t dt = 0 x(t)x(t)T dt WO := 0 eAT t CT CeAt dt = 0 (t)(t)T dt Empirical Gramians equal Classic Gramians Extends to nonlinear systems POD-related method 5 Lall et al 1999, Hahn et al 2002 16. Empirical Cross Gramian7 WX := 0 eAt BCeAt dt = 0 x(t)(t)T dt Approximate Balanced Truncation6: WX SVD = VDU Vr = PV , P : RNN RNr 6 Sorensen and Antoulas 2002 7 Streif et al 2006 17. Empirical Joint Gramian8 Augmented System: x = x = f (x, u, ) 0 y = g(x, u, ) x0 = x0 Joint Gramian: WJ := WX = WX w 0 0 Cross-Identiability Gramian: WI := 1 2 wT (WX + W T X )1 w 8 Himpe and Ohlberger 2013 (Preprint) 18. Optimization-Based9 Purpose: Developed for large Bayesian inverse problems, to reduce model order before parameter optimization. Core Idea: Find optimal reduced order parameter, by minimizing error between full and proposed reduced model. 9 Bashir et al 2007, Lieberman et al 2010 19. Greedy-Algorithm10 1 Initialization: 0 RP , P0 = 0, V0 = x(0) 2 Maximize error between original and (current) reduced model: I+1 = argmax J() = argmax y() yr () 3 Snapshot of current proposed reduced model: xI+1 = x(I+1) 4 Update parameter projection: VI+1 = orth VI I+1 5 Update state projection: VI+1 = orth VI xI+1 10 Bui-Thanh et al 2008 20. Trust-Region11 1 Initialization: 0 = 1, P0 = 0, V0 = x(0) 2 Maximize error between original and (current) reduced model: I+1 = argmax y(PI ) yr (PI ) PI 3 Snapshot of current proposed reduced model: xI+1 = x(PI I+1) 4 Update parameter projection: VI+1 = orth VI Q I+1 5 Update state projection: VI+1 = orth VI xI+1 6 Enlarge parameter vector: I+1 = I+1 0 11 Himpe and Ohlberger 2014 (Preprint) 21. Data-Driven12 Given some output data yd : J() = y() yr () 12 Lieberman et al 2012 22. Data-Driven12 Given some output data yd : J() = y() yr () J() = y() yr () yd yr () 12 Lieberman et al 2012 23. Data-Driven12 Given some output data yd : J() = y() yr () J() = y() yr () yd yr () J() = yd yr () 12 Lieberman et al 2012 24. Software emgr Empirical Controllability Gramian Empirical Observability Gramian Empirical Cross Gramian Empirical Approximate Cross Gramian Empirical Sensitivity Gramian Empirical Identiability Gramian Empirical Joint Gramian optmor Optional Data-Driven Optional Trust-Region Congurable Objective Function Congurable State Selection Congurable Orthogonalization Arbitrary Parametrizations Lazy Arguments Default and Custom Integrators (Somewhat) Multi-Core Ready Released under Open-Source License Compatible with OCTAVE and MATLAB http://gramian.de http://j.mp/opt-mor 25. Numerical Results fMRI / fNIRS x = A()x + Bu z = f (z, x) y = g(z) EEG / MEG x = ALx + AN()f (x) + Bu y = Cx Linearization13: x = A()x + Bu y = Cx x = ALx + AN()x + Bu y = C x 13 Kamrani et al 2012 (fMRI), Moran et al 2007 (EEG) 26. fMRI 5 10 15 20 25 5 10 15 20 25 10 3 10 2 10 1 10 0 States Relative L2 Output Error (emgr) Parameters 5 10 15 20 25 5 10 15 20 25 10 3 10 2 10 1 10 0 States Relative L2 Output Error (optmor) Parameters 27. EEG 5 10 15 20 25 30 5 10 15 20 25 10 3 10 2 10 1 10 0 States Relative L2 Output Error (emgr) Parameters 5 10 15 20 25 30 5 10 15 20 25 10 3 10 2 10 1 10 0 States Relative L2 Output Error (optmor) Parameters 28. tl;dl Summary: Empirical Cross Gramian works in non-symmmetric setting Trust-Region and Data-Driven extensions prevent snapshots Empirical Gramians: parameter reduction error dominates Optimization-Based: state reduction error dominates Source Code: http://j.mp/gamm14 Thanks!