A Geometric Index Reduction Method for DAE Systems Gabriela Jeronimo (1) Joint work with L. D’Alfonso (1) , F. Ollivier (2) , A. Sedoglavic (3) and P. Solern´ o (1) (1) Universidad de Buenos Aires, Argentina (2) ´ Ecole Polytechnique, France (3) Universit´ e de Lille I, France DART IV – October 2010 Gabriela Jeronimo Index reduction for DAE systems
DAE systems Consider a Differential Algebraic Equation (DAE) system f 1 ( X , . . . , X ( e ) ) = 0 . . (Σ) = . f n ( X , . . . , X ( e ) ) = 0 X ( k ) := { x ( k ) 1 , . . . , x ( k ) n } for every k ∈ Z ≥ 0 , F := f 1 , . . . , f n polynomials with coefficients in C or C ( t ). Gabriela Jeronimo Index reduction for DAE systems
DAE systems Consider a Differential Algebraic Equation (DAE) system f 1 ( X , . . . , X ( e ) ) = 0 . . (Σ) = . f n ( X , . . . , X ( e ) ) = 0 X ( k ) := { x ( k ) 1 , . . . , x ( k ) n } for every k ∈ Z ≥ 0 , F := f 1 , . . . , f n polynomials with coefficients in C or C ( t ). � � ∂ F If det � = 0, (Σ) is equivalent to an ODE system: ∂ X ( e ) � X ( e ) = G ( X , . . . , X ( e − 1) ) . ( � Σ) = Gabriela Jeronimo Index reduction for DAE systems
DAE systems Consider a Differential Algebraic Equation (DAE) system f 1 ( X , . . . , X ( e ) ) = 0 . . (Σ) = . f n ( X , . . . , X ( e ) ) = 0 X ( k ) := { x ( k ) 1 , . . . , x ( k ) n } for every k ∈ Z ≥ 0 , F := f 1 , . . . , f n polynomials with coefficients in C or C ( t ). � � ∂ F If det � = 0, (Σ) is equivalent to an ODE system: ∂ X ( e ) � X ( e ) = G ( X , . . . , X ( e − 1) ) . ( � Σ) = � � ∂ F What can be done when det = 0? ∂ X ( e ) Gabriela Jeronimo Index reduction for DAE systems
Semi-explicit systems x 1 ˙ = g 1 ( X ) . . . X = ( x 1 , . . . , x n ) (Σ 0 ) := ˙ = g n − 1 ( X ) x n − 1 g ( X ) = 0 Gabriela Jeronimo Index reduction for DAE systems
Semi-explicit systems x 1 ˙ = g 1 ( X ) . . . X = ( x 1 , . . . , x n ) (Σ 0 ) := ˙ = g n − 1 ( X ) x n − 1 g ( X ) = 0 From the last equation: n − 1 � ∂ g ∂ g ( X ) g i ( X ) + ∂ g ∂ t ( X ) + ( X ) ˙ x n = 0 . ∂ x i ∂ x n i =1 Gabriela Jeronimo Index reduction for DAE systems
Semi-explicit systems x 1 ˙ = g 1 ( X ) . . . X = ( x 1 , . . . , x n ) (Σ 0 ) := ˙ = g n − 1 ( X ) x n − 1 g ( X ) = 0 From the last equation: n − 1 � ∂ g ∂ g ( X ) g i ( X ) + ∂ g ∂ t ( X ) + ( X ) ˙ x n = 0 . ∂ x i ∂ x n i =1 If ∂ g � = 0, we can solve ˙ x n = g n ( X ). ∂ x n Gabriela Jeronimo Index reduction for DAE systems
Semi-explicit systems x 1 ˙ = g 1 ( X ) . . . X = ( x 1 , . . . , x n ) (Σ 0 ) := ˙ = g n − 1 ( X ) x n − 1 g ( X ) = 0 From the last equation: n − 1 � ∂ g ∂ g ( X ) g i ( X ) + ∂ g ∂ t ( X ) + ( X ) ˙ x n = 0 . ∂ x i ∂ x n i =1 If ∂ g � = 0, we can solve ˙ x n = g n ( X ). ∂ x n The first n − 1 equations plus this one form an ODE system. Gabriela Jeronimo Index reduction for DAE systems
Differentiation index of DAE systems The differentiation index of a DAE system is an integer σ ∈ Z ≥ 0 which measures the implicitness of the given system. Gabriela Jeronimo Index reduction for DAE systems
Differentiation index of DAE systems The differentiation index of a DAE system is an integer σ ∈ Z ≥ 0 which measures the implicitness of the given system. σ = minimum number of differentiations of the system required in order to obtain an ODE. Gabriela Jeronimo Index reduction for DAE systems
Differentiation index of DAE systems The differentiation index of a DAE system is an integer σ ∈ Z ≥ 0 which measures the implicitness of the given system. σ = minimum number of differentiations of the system required in order to obtain an ODE. Examples. For a DAE system F ( X , ˙ X , . . . , X ( e ) ) = 0 such that � � ∂ F det � = 0, we have σ = 0. ∂ X ( e ) Gabriela Jeronimo Index reduction for DAE systems
Differentiation index of DAE systems The differentiation index of a DAE system is an integer σ ∈ Z ≥ 0 which measures the implicitness of the given system. σ = minimum number of differentiations of the system required in order to obtain an ODE. Examples. For a DAE system F ( X , ˙ X , . . . , X ( e ) ) = 0 such that � � ∂ F det � = 0, we have σ = 0. ∂ X ( e ) For a semi-explicit DAE system, σ = 1. Gabriela Jeronimo Index reduction for DAE systems
Index reduction problem Given a high-index DAE system (Σ), obtain an equivalent DAE system (Σ 0 ) with low index (preferably 0 or 1, and semi-explicit). Gabriela Jeronimo Index reduction for DAE systems
Index reduction problem Given a high-index DAE system (Σ), obtain an equivalent DAE system (Σ 0 ) with low index (preferably 0 or 1, and semi-explicit). Motivation: low-index DAE systems are easier to solve than high-index DAE systems. Gabriela Jeronimo Index reduction for DAE systems
Index reduction problem Given a high-index DAE system (Σ), obtain an equivalent DAE system (Σ 0 ) with low index (preferably 0 or 1, and semi-explicit). Motivation: low-index DAE systems are easier to solve than high-index DAE systems. Previous work: [Gear 1988, 1989], [Brenan-Campbell-Petzold 1996], [Kunkel-Mehrmann 2006]. Gabriela Jeronimo Index reduction for DAE systems
Index reduction problem Given a high-index DAE system (Σ), obtain an equivalent DAE system (Σ 0 ) with low index (preferably 0 or 1, and semi-explicit). Motivation: low-index DAE systems are easier to solve than high-index DAE systems. Previous work: [Gear 1988, 1989], [Brenan-Campbell-Petzold 1996], [Kunkel-Mehrmann 2006]. Tools: Computation of successive derivatives of the equations, rewriting techniques relying on the Implicit Function Theorem, etc. Gabriela Jeronimo Index reduction for DAE systems
Our main results f 1 ( X , X (1) , . . . , X ( e ) ) = 0 . . (Σ) = with certain assumptions . . f n ( X , X (1) , . . . , X ( e ) ) = 0 Gabriela Jeronimo Index reduction for DAE systems
Our main results f 1 ( X , X (1) , . . . , X ( e ) ) = 0 . . (Σ) = with certain assumptions . . f n ( X , X (1) , . . . , X ( e ) ) = 0 (Σ) is generically equivalent to a first order semi-explicit DAE system with differentiation index 1 ˙ = g 1 ( U , v ) u 1 . . . (Σ 0 ) := U = ( u 1 , . . . , u r ) ˙ = g r ( U , v ) u r q ( U , v ) = 0 Gabriela Jeronimo Index reduction for DAE systems
Our main results f 1 ( X , X (1) , . . . , X ( e ) ) = 0 . . (Σ) = with certain assumptions . . f n ( X , X (1) , . . . , X ( e ) ) = 0 (Σ) is generically equivalent to a first order semi-explicit DAE system with differentiation index 1 ˙ = g 1 ( U , v ) u 1 . . . (Σ 0 ) := U = ( u 1 , . . . , u r ) ˙ = g r ( U , v ) u r q ( U , v ) = 0 a probabilistic algorithm to compute the differentiation index of (Σ) and the associated system (Σ 0 ). Gabriela Jeronimo Index reduction for DAE systems
Remarks (Σ) and (Σ 0 ) are equivalent in the sense that almost all analytic solution of (Σ) can be obtained from an analytic solution of (Σ 0 ) and conversely. Gabriela Jeronimo Index reduction for DAE systems
Remarks (Σ) and (Σ 0 ) are equivalent in the sense that almost all analytic solution of (Σ) can be obtained from an analytic solution of (Σ 0 ) and conversely. Our algorithms rely on: Gabriela Jeronimo Index reduction for DAE systems
Remarks (Σ) and (Σ 0 ) are equivalent in the sense that almost all analytic solution of (Σ) can be obtained from an analytic solution of (Σ 0 ) and conversely. Our algorithms rely on: an alternative characterization of the differentiation index ([DAlJeSo08]), Gabriela Jeronimo Index reduction for DAE systems
Remarks (Σ) and (Σ 0 ) are equivalent in the sense that almost all analytic solution of (Σ) can be obtained from an analytic solution of (Σ 0 ) and conversely. Our algorithms rely on: an alternative characterization of the differentiation index ([DAlJeSo08]), the polynomial time Kronecker algorithm for the computation of geometric resolutions of algebraic polynomial systems ([GiLeSa01], [Schost03]). Gabriela Jeronimo Index reduction for DAE systems
Assumptions on the system Gabriela Jeronimo Index reduction for DAE systems
Assumptions on the system K { X } = K [ x ( j ) : 1 ≤ i ≤ n , j ∈ N 0 ] with the derivation δ induced i by δ ( x ( j ) ) = x ( j +1) . i i Gabriela Jeronimo Index reduction for DAE systems
Assumptions on the system K { X } = K [ x ( j ) : 1 ≤ i ≤ n , j ∈ N 0 ] with the derivation δ induced i by δ ( x ( j ) ) = x ( j +1) . i i [ F ] = [ f 1 , . . . , f n ] ⊂ K { X } differential ideal associated with (Σ). Gabriela Jeronimo Index reduction for DAE systems
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