Summer Computer Simulation Conference (SCSC’03) Montréal, july 2003 Mapping ODEs to DEVS: Adaptive Quantization Jean-Sébastien Bolduc Hans Vangheluwe Modelling, Simulation and Design Lab (MSDL) School of Computer Science McGill University Montréal, Canada http://msdl.cs.mcgill.ca/ SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.1/16
Overview Apply Quantization to IVPs: � ˙ = f ( t, x ) x x (0) = x 0 . Why? How? Numerical analysis = ⇒ limitations of the approach. Presentation of an Adaptive Algorithm. SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.2/16
Motivation DEVS formalism (Zeigler a ): rigourous common basis for discrete-event modelling and simulation. Continuous-Time Discrete Formalisms Formalisms Quantization DESS DTSS DEVS Discretization a Zeigler, Praehofer and Kim, Theory of modeling and simulation. SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.3/16
Idea Forward Euler approximation: x i +1 = x i + h i · f ( t i , x i ) Discretization: h i known (fixed) in which state is the system going to be at a given future time? Quantization: h i unknown when will the state of the system reach a given value? h i = x i +1 − x i D f ( t i , x i ) ≤ | f ( t i , x i ) | SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.4/16
Quantization of R x Quanta Size: D > 0 Quanta Phase: 0 ≤ ρ < D d 1 Quanta: d k , k ∈ Z Index function: t d 0 � x � = k ⇐ ⇒ x ∈ d k ρ Adaptive Quantization: d − 1 D D ω = D B ω ∈ N 0 2 ω , SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.5/16
Non-Adaptive Quantization Algorithm state transition: QI above x i if ˙ x i > 0 x i +1 = σ ( x i , ˙ x i ) = QI below x i if ˙ x i < 0 if ˙ x i = 0 x i time-advance: σ ( x i , ˙ x i ) − x i if ˙ x i � = 0 , ˙ x i = h i if ˙ x i = 0 . ∞ D ≤ | ˙ x i | SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.6/16
Sample Result x = x 2 − 4 , x (0) = 1 . 95 ˙ with D = 0 . 05 , ρ = 0 Exact Numerical 10*h 2 1 0 x -1 -2 0 0.5 1 1.5 2 2.5 3 t SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.7/16
Consistency Local Truncation Error: = | x ( t 1 ) − x 1 | τ h 2 � � � ˙ 0 = f ( ξ, x ( ξ )) t 0 ≤ ξ ≤ t 1 . � , � � 2 for Discretization, we have τ = O ( h 2 ) . for Quantization, we would like τ = O ( D p ) , p > 0 . But D h 0 ≤ | f ( t 0 , x 0 ) | − → ∞ as f ( t 0 , x 0 ) → 0 . SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.8/16
Can only guarantee consistency, i.e. τ = O ( D 2 ) , if ∀ t and x, | f ( t, x ) | ≥ ε > 0 . (1) For an autonomous system, observe (fixed point) f ( x ( t 1 )) = 0 ⇒ f ( x ( t 2 )) = 0 , ∀ t 2 ≥ t 1 . But we still have ε → 0 τ = ∞ lim Conclusion: small slopes are a problem. SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.9/16
Convergence Global Error: � t � � x ( s ) − ˙ � � � � E ( t ) = | x ( t ) − ˜ x ( t ) | = ˙ x ( s ) ˜ ds � � � � 0 � t � � � �� � � � � � = s, x ( s ) − f s − µ ( s ) , ˜ x ( s ) − δ ( s ) f ds � . � � � 0 µ ( s ) : time elapsed since last transition δ ( s ) : distance to previous QI � t � t � � � � E ( t ) ≤ � x ( s ) − ˜ x ( s ) � ds + L � δ ( s ) � ds + L � � � � 0 0 � t � � + C � µ ( s ) � ds. � � 0 SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.10/16
Can further simplify as � t � t � � E ( t ) ≤ L E ( s ) ds + L Dt + C � µ ( s ) � ds. � � 0 0 But µ ( s ) is not bounded. For an autonomous system a , the last term is absent. We get ( Gronwall-Bellman inequality ): � e Lt − 1 � E ( t ) ≤ D ∈ O ( D ) . a Kofman and Junco, Quantized-State Systems: A DEVS Approach for Continuous System Simulation. SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.11/16
If condition (1) holds, then | µ ( s ) | ≤ D ε , and result above holds for Nonautonomous Systems. Conclusion: small slopes are a problem for Nonautonomous Systems. SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.12/16
Adaptive Algorithm Want to guarantee τ i ≤ TOL at each step, by modifying the quanta size D . Use Richardson extrapolation to approximate ∆ i ≈ τ i : ∆ i +1 = | ˆ x i +1 − x i +1 | h i � � = � f (ˆ 2 ) − f ( x i ) x i + 1 � . � � 2 x i +1 ˆ ∆ i +1 x x i +1 x i + 1 ˆ 2 x i t t i t i +1 SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.13/16
More tricky than in discretization: Does not generalize well to higher-order systems. Use lowest-order approximation x i +1 instead of higher-order ˆ x i +1 . Limited control through Adaptive Quantization. Simple algorithm: Recursively halve D until ∆ ≤ TOL After a transition, tentatively double D . SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.14/16
Sample Result x = x 2 − 4 , x (0) = 1 . 95 ˙ with D B = 1 , ρ B = 0 . Exact TOL = 1.0 e -2 TOL = 5.0 e-3 2 TOL = 2.5 e-3 1 0 x -1 -2 0 0.5 1 1.5 2 2.5 3 t SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.15/16
Conclusions Quantization limited by zero-slope problem: Local Truncation Error τ is unbounded. Global Error E ( t ) = O ( D ) for Autonomous Systems only. Can we do something? Early experiments in adaptation: not as trivial as in discretization. SCSC’03 Mapping ODEs to DEVS: Adaptive Quantization – p.16/16
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