Model reduction of partially-observed Motivation stochastic - PowerPoint PPT Presentation

Model reduction of partially-observed Motivation stochastic differential equations A control problem Partially- observed Langevin Carsten Hartmann (Matheon Research Center, FU Berlin) equation Examples joint work with Christof Sch¨ utte (FU Berlin) Numerical methods in molecular simulation (Hausdorff Research Institute Bonn, April 7-11, 2008)

Outline 1 Motivation Motivation Conformational flexibility A control problem 2 A control problem Partially- observed Balanced truncation Langevin equation Port-controlled Hamiltonian systems Examples 3 Partially-observed Langevin equation Controllability and observability Model reduction by balancing 4 Examples

Motivation Biological function depends on conformation’s flexibility. Motivation A control problem Partially- observed Langevin equation Examples 1 . 3 µ s simulation of dodeca-alanine at T = 300 K (implicit solvent, GROMOS96 force field)

Conformational flexibility The microscopic dynamics (molecule & solvent) is generated by the nonlinear Hamiltonian Motivation H : T ∗ Q → R , Q ⊆ R 3 n , H = p T M − 1 p + V ( q ) , A control problem Partially- with initial conditions distributed according to exp( − β H ). observed Langevin equation We suppose that the dynamics within a conformation Examples can be approximated by the linear Langevin equation x ( t ) = ( J − D ) ∇ H lin ( x ( t )) + S ˙ ˙ W ( t ) 2 ¯ 1 ¯ where H lin = 1 M − 1 x 2 + 1 2 x T 2 x T Lx 1 and � 0 � 0 � � � � 0 1 0 J = , D = , S = − 1 0 0 γ σ

Conformational flexibility, cont’d For γ s.p.d, the system is stable , i.e., all eigenvalues of Motivation A = ( J − D ) ∇ 2 H lin have strictly negative real part. A control The system satisfies H¨ ormander’s condition . If further problem 2 γ = βσσ T , this entails ergodicity with respect to Partially- observed Langevin equation d µ ( x ) ∝ exp( − β H lin ( x )) dx . Examples The Gaussian distribution exp( − β H lin ) indicates that all modes are flexible. Which are the most flexible ones? Often the most flexible modes are thought of as having the largest variance. But: variance is not always most important to the dynamics.

1 Motivation Motivation Conformational flexibility A control problem 2 A control problem Partially- observed Balanced truncation Langevin equation Port-controlled Hamiltonian systems Examples 3 Partially-observed Langevin equation Controllability and observability Model reduction by balancing 4 Examples

Conformational flexibility as a control problem Key observations 1 Not all modes are equally stiff; moreover noise and friction Motivation may not be spatially isotropic. A control problem 2 Not all modes are observed (e.g., generalized momenta). Partially- observed Langevin equation We may consider flexibility as the property of being Examples sensitive to excitations due to the noise (controllability). A sensible notion of flexibility should take into account what can be measured experimentally (observability). Determining the flexibility of a conformation therefore amounts to identifying a low-dimensional subspace of easily controllable and highly observable modes

Linear control systems For the moment let us replace the white noise by a smooth control function u ∈ L 2 ([0 , T ]), i.e., Motivation A control problem x ( t ) ˙ = Ax ( t ) + Su ( t ) , x (0) = 0 Partially- observed y ( t ) = Cx ( t ) , Langevin equation where A = ( J − D ) ∇ 2 H lin ∈ R 2 d × 2 d and y ∈ R k is a Examples linear observable (e.g., all configurations y = x 1 ). VoP yields the transfer function (input-output relation), � t e A ( t − s ) Su ( s ) ds . G : L 2 ([0 , T ]) → R k , y ( t ) = C 0

Model reduction by balanced truncation For the stable linear system ˙ x = Ax + Su , y = Cx compute controllability and observability Gramians Motivation A control � ∞ problem exp( At ) SS T exp( A T t ) dt Q = Partially- 0 observed � ∞ Langevin exp( A T t ) C T C exp( At ) dt . equation P = 0 Examples Balancing: find a transformation x �→ Tx , such that T − 1 QT − T = T T PT = diag( σ 1 , . . . , σ 2 d ) . Truncation: Project onto the first m columns of T . Moore 1981

Properties of balanced truncation Interpretation of the controllability Gramian Q : x ∈ R 2 d is “more controllable” than x ′ ∈ R 2 d if Motivation A control x T Qx > x ′ T Qx ′ ( | x | = | x ′ | = 1) . problem Partially- observed Interpretation of the observability Gramian P : given an Langevin equation initial state x (0) = x and zero input, u = 0, we have Examples � y � 2 L 2 = x T Px . Approximation error ( H ∞ error bound): � ( G − G trc ) u � L 2 σ m +1 < max < 2( σ m +1 + . . . + σ 2 d ) . � u � L 2 u Glover 1984, Rowley 2005

Port-controlled Hamiltonian systems Let’s go back to our second-order Langevin problem and consider the stable system Motivation x ( t ) ˙ = ( J − D ) ∇ H lin ( x ( t )) + Su ( t ) A control problem y ( t ) = Cx ( t ) . Partially- observed Langevin Balancing mixes configurations and momenta. Truncation equation (e.g., by projection) does not preserve structure . Examples Preserving structure requires to impose constraints on the Hamiltonian part (energy & structure matrix). Then ˙ ξ ( t ) = ( J trc − D trc ) ∇ H trc ( ξ ( t )) + S trc u ( t ) y ( t ) = C trc ξ ( t ) is stable with ξ ∈ R m (odd or even dim.) and J trc = − J T trc . Hartmann et al. 2007

1 Motivation Motivation Conformational flexibility A control problem 2 A control problem Partially- observed Balanced truncation Langevin equation Port-controlled Hamiltonian systems Examples 3 Partially-observed Langevin equation Controllability and observability Model reduction by balancing 4 Examples

Partially-observed Langevin equation Consider the family of Langevin equations for ǫ > 0 Motivation ( J − D ) ∇ H lin ( x ǫ ( t )) + √ ǫ S ˙ x ǫ ( t ) ˙ = W ( t ) A control problem y ǫ ( t ) Cx ǫ ( t ) . = Partially- observed Langevin Using again the shorthand A = ( J − D ) ∇ 2 H lin , we have equation Y ǫ t = CX ǫ t , where X ǫ t , X ǫ 0 = 0 is the family of solutions Examples � t t = √ ǫ e A ( t − s ) S dW ( s ) . X ǫ 0 The system is stable for all ǫ > 0. If 2 D = SS T it admits an ergodic invariant measure d µ ǫ ∝ exp( − H lin /ǫ ).

Controllability of the Langevin equation Consider the map F : H 1 ([0 , T ]) → C ([0 , T ]) and f = F ( u ) , u (0) = 0 that is defined point-wise by Motivation � t A control e A ( t − s ) S ˙ problem ( Fu )( t ) = u ( s ) ds . Partially- 0 observed Langevin We introduce the controllability function (rate function) equation Examples � T u ( t ) | 2 dt I x ( f ) = inf | ˙ u ∈ H 1 , f ( T )= x 0 measuring the minimum “energy” that is needed to steer the system from f (0) = 0 to f ( T ) = x within time T . We declare I x ( f ) = ∞ if no such u ∈ H 1 exists. cf. Dembo & Zeitsouni 1998

Controllability of the Langevin equation, cont’d Theorem (Hartmann,Sch¨ utte 2008) The controllability function L con ( x ) = I x ( f ) is given by Motivation A control L con ( x ) = x T Q ( T ) − 1 x problem Partially- observed with Q ( T ) = cov( X ǫ ( T )) for ǫ = 1 . Langevin equation Examples Proof: Minimize � u � 2 H 1 subject to ( Fu )( T ) = x . The idea of replacing the Brownian motion W ( t ) by its polygonal approximation u ∈ H 1 is to make sense of I x ( f ). If ǫ is small, LDT guarantees that f ( t ) is “close” to X ǫ ( t ). For T → ∞ , the controllability Gramian Q is the unique s.p.d solution of the Lyapunov equation AQ + QA T = − SS T .

Observability of the Langevin equation The observability function � T Motivation | Y 0 t | 2 dt , X 0 L obs ( x ) = 0 = x A control 0 problem Partially- measures the output energy up to time T in the absence observed of noise (i.e., ǫ = 0), if X 0 0 = x at time t = 0. Langevin equation For T → ∞ , it follows immediately that Examples L obs ( x ) = x T Px , where the observability Gramian P is the unique s.p.d solution of the Lyapunov equation A T P + PA = − C T C . Hartmann & Sch¨ utte 2008, cf. Moore 1981

Model reduction by balancing Compute the Gramians Q , P of the stable Langevin system ˙ x = ( J − D ) ∇ H ( x ) + SW , y = Cx . Motivation Find the balancing transformation x �→ Tx , such that A control problem T − 1 QT − T = T T PT = diag( σ 1 , . . . , σ 2 d ) . Partially- observed Langevin Notice: T − 1 QPT = diag( σ 2 1 , . . . , σ 2 equation 2 d ). Examples Constrain the system to the subspace of the largest singular values σ 1 , . . . , σ m or, alternatively, scale the smallest Hankel singular values according to ( σ m +1 , . . . , σ 2 d ) �→ δ ( σ m +1 , . . . , σ 2 d ) , δ > 0 . and balance the Langevin equation by x �→ T δ x . In the resulting perturbed system, let δ go to zero.