Manifold Construction and Parameterization for Nonlinear - PowerPoint PPT Presentation

Manifold Construction and Parameterization for Nonlinear Manifold-Based Model Reduction Chenjie Gu and Jaijeet Roychowdhury {gcj,jr}@eecs.berkeley.edu University of California, Berkeley ASPDAC 2010 Slide 1

Outline Background ● Background ● Introduction to MOR and maniMOR ● Manifold construction and parameterization Manifold construction using integral curves ● Manifold construction using integral curves ● DC manifold and the normalized integral curve equation ● Ideal and almost-ideal manifold ● Algorithm Experimental results ● Experimental results Conclusion ● Conclusion ASPDAC 2010 Slide 2

Background ASPDAC 2010 Slide 3

Model Order Reduction Original system (size n) d~ x dt = f ( ~ x ) + B~ u ( t ) ~ y = C~ x ~ u ( t ) x 2 R n ;~ z 2 R q ; x 7! ~ q ¿ n v : ~ z; ~ Reduced system (size q) d~ z dt = f r ( ~ z ) + B r ~ u ( t ) y = C r ~ ~ z u ( t ) ~ ASPDAC 2010 Slide 4

Low-order Linear Subspace 2 3 2 3 2 3 2 3 ¡ 10 x 1 1 1 x 1 1 d 4 5 = 4 5 4 5 + 4 5 u ( t ) ¡ 1 x 2 1 0 x 2 0 dt ¡ 1 x 3 1 0 x 3 0 Low-order linear subspace Defined by x = V z ASPDAC 2010 Slide 5

Low-order Nonlinear Manifold 2 3 2 3 2 3 2 3 2 3 ¡ 10 x 1 1 1 x 1 0 1 d 4 5 = 4 5 4 5 ¡ 4 5 + 4 5 u ( t ) ¡ 1 x 2 1 0 x 2 0 0 dt x 2 ¡ 1 x 3 1 0 x 3 0 1 Low-order nonlinear manifold ManiMOR: MOR Based on Nonlinear Projection on Nonlinear Manifolds ASPDAC 2010 Slide 6

Key Steps in ManiMOR “Find” the nonlinear manifold Find” the nonlinear manifold ● “ ● Capture important dynamics “Parameterize” the manifold Parameterize” the manifold ● “ ● Build up the coordinate system ASPDAC 2010 Slide 7

Manifold and Its Parameterization 8 < x = cos( t ) y = sin( t ) : z = t ASPDAC 2010 Slide 8

Manifold and Its Parameterization M Parameterization of the manifold Tangent space T x M ½ R n R q ~ U Ã x v z U System of coordinates Manifold defined by No explicit mapping may be derived. pairs of f x; T x M g Instead, use piecewise linear approximation. ASPDAC 2010 Slide 9

Manifold and Its Parameterization 1. Identify the manifold that capture important dynamics 2. Compute and store pairs of f x; T x M g = f z; T z M g ASPDAC 2010 Slide 10

DC Manifold DC operating points constitute a DC manifold. d~ x dt = f ( ~ x ) + B~ u ( t ) = 0 How to compute and parameterize the DC manifold? ASPDAC 2010 Slide 11

DC Manifold f ( ~ x ) + B~ u ( t ) = 0 A straight-forward solution: Computation: Perform DC sweep analysis u Parameterization: Define coordinates using values of z Problems: Hard to choose step size in DC sweep analysis Not generalizable to higher dimensions ASPDAC 2010 Slide 12

Introduction to Integral Curve Given a vector field , v ( x ) dx its integral curve is the curve such that ° ´ x ( t ) dt = v ( x ) ASPDAC 2010 Slide 13

DC Manifold as an Integral Curve Need to derive the relationship between and dx du @f dx f ( ~ x ) + B~ u ( t ) = 0 du + B = 0 @x dx The first du = ¡ [ G ( x )] ¡ 1 B Krylov basis. Initial condition: x ( u = 0) = x DC j u =0 Solutions are DC operating points. Any numerical integration / transient analysis code can be applied. ASPDAC 2010 Slide 14

Parameterization using Euclidean Distance Parameterization using values of u ( x 3 ; y 3 ) ( x 2 ; y 2 ) u 0 + 2 h ( x 1 ; y 1 ) u 0 + h u 0 Parameterization using Euclidean Distance ( x 3 ; y 3 ) ( x 2 ; y 2 ) u 0 + 2 h u 0 + h ( x 1 ; y 1 ) u 0 Sample points equally spaced on the DC manifold ASPDAC 2010 Slide 15

Parameterization using Euclidean Distance ASPDAC 2010 Slide 16

Normalized Integral Curve Equation Local Euclidean distance is dx du = ¡ [ G ( x )] ¡ 1 B jj dx jj 2 = j du j ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ dx ¯ ¯ ¯ ¯ = jj [ G ( x )] ¡ 1 B jj 2 = 1 Generally not satisfied ¯ ¯ ¯ ¯ du 2 Normalize RHS Normalized Integral [ G ( x )] ¡ 1 B dx du = Curve Equation jj [ G ( x )] ¡ 1 B jj 2 Does it define the same integral curve? ASPDAC 2010 Slide 17

Validation ASPDAC 2010 Slide 18

Normalized Integral Curve Equation Theorem: Suppose ; and satisfy t = ¾ ( ¿ ) x ( t ) x ( ¿ ) ^ d d x ( ¿ ) = ¾ 0 ( ¿ ) g (^ and , respectively. dtx ( t ) = g ( x ( t )) d¿ ^ x ( ¿ )) Then and span the same state space. x ( t ) x ( ¿ ) ^ Sketch of proof: dt = ¾ 0 ( ¿ ) d¿ Since , we have . t = ¾ ( ¿ ) x ( ¿ ) ´ x ( t ) = ^ Define , then ^ x ( ¾ ( t )) d x ( ¿ ) = d ^ x ( ¿ ) dt d¿ = ¾ 0 ( ¿ ) g ( x ( t )) = ¾ 0 ( ¿ ) g (^ d¿ ^ x ( ¿ )) dt ASPDAC 2010 Slide 19

Normalized Integral Curve Equation [ G ( x )] ¡ 1 B dx dx du = ¡ [ G ( x )] ¡ 1 B du = jj [ G ( x )] ¡ 1 B jj 2 Solution: Solution: ^ x ( u ) x (^ u ) Z ^ u 1 Define u = ¾ (^ u ) = d¹ x ( ¹ ))] ¡ 1 B jj 2 jj [ G (^ 0 From the theorem, and define the same integral curve. x ( u ) x (^ ^ u ) ASPDAC 2010 Slide 20

Normalized Integral Curve Equation [ G ( x )] ¡ 1 B dx The first normalized du = Krylov basis. jj [ G ( x )] ¡ 1 B jj 2 Directly available from Krylov subspace methods. Generalizable to higher dimensions. ASPDAC 2010 Slide 21

Ideal Nonlinear Manifold @x @x @x ¢ ¢ ¢ ; = v 1 ( x ) ; = v 2 ( x ) ; = v q ( x ) : @z 1 @z 2 @z q is the projection matrix V ( x ) = [ v 1 ( x ) ; ¢ ¢ ¢ ; v q ( x )] for the reduced linearized system (at ). x For example, Arnoldi algorithm generates a basis for K q ([ G ( x )] ¡ 1 ; B ) = f [ G ( x )] ¡ 1 B; [ G ( x )] ¡ 2 B; ¢ ¢ ¢ ; [ G ( x )] ¡ q B g However, this set of PDEs is over-determined. ASPDAC 2010 Slide 22

Almost-Ideal Manifold Construction @x = v 1 ( x ) @z 1 x DC @x = v 2 ( x ) @x @z 2 = v 3 ( x ) @z 3 ASPDAC 2010 Slide 23

Almost-Ideal Manifold Construction ASPDAC 2010 Slide 24

Experimental Results ASPDAC 2010 Slide 25

A Hand-Calculable Example d dtx 1 = ¡ x 1 + x 2 ¡ u ( t ) d dtx 2 = x 2 1 ¡ x 2 · ¡ x 1 + x 2 · ¡ 1 ¸ ¸ f ( x ) = ; B = x 2 1 ¡ x 2 0 · ¡ 1 ¸ · ¸ 1 1 1 1 [ G ( x )] ¡ 1 = G ( x ) = ; ¡ 1 2 x 1 2 x 1 ¡ 1 2 x 1 1 ASPDAC 2010 Slide 26

DC and AC Manifold · ¡ 1 · ¸ ¸ 1 1 1 [ G ( x )] ¡ 1 = ; B = 2 x 1 ¡ 1 2 x 1 1 0 · ¸ 1 1 w 1 ( x ) = [ G ( x )] ¡ 1 B = 2 x 1 ¡ 1 2 x 1 · ¡ 1 ¡ 2 x 1 ¸ 1 w 2 ( x ) = [ G ( x )] ¡ 2 B = ¡ 4 x 1 (2 x 1 ¡ 1) 2 @x w 1 ( x ) DC manifold: = v 1 ( x ) = jj w 1 ( x ) jj 2 @z 1 w 2 ¡ < w 2 ; v 1 > v 1 @x AC manifold: = v 2 ( x ) = jj w 2 ¡ < w 2 ; v 1 > v 1 jj 2 @z 1 ASPDAC 2010 Slide 27

DC and AC Manifold ASPDAC 2010 Slide 28

Application to MOR 2 3 2 3 2 3 2 3 2 3 ¡ 10 x 1 1 1 x 1 0 1 d 5 ¡ 4 5 = 4 5 4 4 5 + 4 5 u ( t ) ¡ 1 x 2 1 0 x 2 0 0 dt x 2 ¡ 1 x 3 1 0 x 3 0 1 Trajectory of the full system stays close to the manifold ASPDAC 2010 Slide 29

Simulation of the Reduced Order Model Response to a step input Response to a sinusoidal input ASPDAC 2010 Slide 30

Conclusion Presented a manifold construction and ● Presented a manifold construction and parameterization procedure parameterization procedure ● Based on computing integral curves ● Preserves local distance ● Captures important system responses ● Such as DC and AC responses Application to manifold-based MOR ● Application to manifold-based MOR ● Validated against several examples ASPDAC 2010 Slide 31