Validated simulation of DAEs Julien Alexandre dit Sandretto - PowerPoint PPT Presentation

Validated simulation of DAEs Julien Alexandre dit Sandretto Department U2IS ENSTA Paris SSC310-2020

Contents Guaranteed simulation of differential equations Differential Algebraic Equations Approach to simulate DAE Examples Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 2

Guaranteed simulation of differential equations Recall of Ordinary differential equations Given by y ′ = f ( y , t ) Initial Value Problems y ′ = f ( y , t ) , y ( 0 ) = y 0 Numerical simulation of IVPs till a time t n Compute y i ≈ y ( t i ) with t i ∈ { 0 , t 1 , . . . , t n } Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 3

Guaranteed simulation of differential equations Validated simulation of IVPs Produces a list of boxes [ y i ] and [˜ y i ] such that ◮ y ( t i ) ∈ [ y i ] with t i ∈ { 0 , t 1 , . . . , t n } ◮ y ( t ) ∈ [˜ y i ] for all t ∈ [ t i , t i + 1 ] Method of Lohner 1. Find [˜ y i ] with Picard-Lindelof operator 2. Compute [ y i ] with a validated integration scheme : Taylor (Vnode-LP) or Runge-Kutta (DynIbex) ~ [y j ] [y j ] [y j+1 ] t t j t j+1 h j Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 4

Differential Algebraic Equations Differential Algebraic Equations General form: implicit F ( t , y , y ′ , ... ) = 0, t 0 ≤ t ≤ t end y ′ = DAE 1 st order, y ′′ = DAE 2 nd , etc. (all DAEs can be rewritten in DAE of 1 st order) Hessenberg form: Semi-explicit (index: distance to ODE) ✎ ☞ ✎ ☞ � y ′ = f ( t , x , y ) � y ′ = f ( t , x , y ) index 1 : index 2 : 0 = g ( t , x , y ) 0 = g ( t , x ) ✍ ✌ ✍ ✌ ⇒ Focus on Hessenberg index-1: Simulink, Modelica-like, etc. Different from ODE + constraint � y ′ = f ( t , y ) , t 0 ≤ t ≤ t end 0 = g ( y , y ′ ) ⇒ Direct with contractor approach Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 5

Differential Algebraic Equations Differential Algebraic Equations General form: implicit F ( t , y , y ′ , ... ) = 0, t 0 ≤ t ≤ t end y ′ = DAE 1 st order, y ′′ = DAE 2 nd , etc. (all DAEs can be rewritten in DAE of 1 st order) Hessenberg form: Semi-explicit (index: distance to ODE) ✎ ☞ ✎ ☞ � y ′ = f ( t , x , y ) � y ′ = f ( t , x , y ) index 1 : index 2 : 0 = g ( t , x , y ) 0 = g ( t , x ) ✍ ✌ ✍ ✌ Some of dependent variables occur without their derivatives ! Different from ODE + constraint � y ′ = f ( t , y ) , t 0 ≤ t ≤ t end 0 = g ( y , y ′ ) ⇒ Direct with contractor approach Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 5

Differential Algebraic Equations A basic example System in Hessenberg index-1 form y ′ = y + x + 1 � y ( 0 ) = 1 . 0 and x ( 0 ) = 0 . 0 ( y + 1 ) ∗ x + 2 = 0 Simulation ⇒ stiffness (in general) Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 6

Differential Algebraic Equations Simulation of a DAE As ODE: a list of boxes [ y i ] and [˜ y i ] such that ◮ y ( t i ) ∈ [ y i ] with t i ∈ { 0 , t 1 , . . . , t n } ◮ y ( t ) ∈ [˜ y i ] for all t ∈ [ t i , t i + 1 ] But in addition: a list of boxes [ x i ] and [˜ x i ] such that ◮ x ( t i ) ∈ [ x i ] with t i ∈ { 0 , t 1 , . . . , t n } ◮ x ( t ) ∈ [˜ x i ] for all t ∈ [ t i , t i + 1 ] Both validate ◮ y ′ ( t i ) ∈ f ( t i , [ x i ] , [ y i ]) ◮ ∃ x ∈ [ x i ] , ∃ y ∈ [ y i ] : g ( t i , x , y ) = 0 ◮ y ′ ( t ) ∈ f ( t , [˜ y i ]) , ∀ t ∈ [ t i , t i + 1 ] x i ] , [˜ ◮ ∀ t ∈ [ t i , t i + 1 ] , ∃ x ∈ [˜ x i ] , ∃ y ∈ [˜ y i ] : g ( t , x , y ) = 0 Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 7

Differential Algebraic Equations Simulation of a DAE As ODE: a list of boxes [ y i ] and [˜ y i ] such that ◮ y ( t i ) ∈ [ y i ] with t i ∈ { 0 , t 1 , . . . , t n } ◮ y ( t ) ∈ [˜ y i ] for all t ∈ [ t i , t i + 1 ] But in addition: a list of boxes [ x i ] and [˜ x i ] such that ◮ x ( t i ) ∈ [ x i ] with t i ∈ { 0 , t 1 , . . . , t n } ◮ x ( t ) ∈ [˜ x i ] for all t ∈ [ t i , t i + 1 ] Both validate ◮ y ′ ( t i ) ∈ f ( t i , [ x i ] , [ y i ]) ◮ ∃ x ∈ [ x i ] , ∃ y ∈ [ y i ] : g ( t i , x , y ) = 0 ◮ y ′ ( t ) ∈ f ( t , [˜ y i ]) , ∀ t ∈ [ t i , t i + 1 ] x i ] , [˜ ◮ ∀ t ∈ [ t i , t i + 1 ] , ∃ x ∈ [˜ x i ] , ∃ y ∈ [˜ y i ] : g ( t , x , y ) = 0 Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 7

Differential Algebraic Equations Simulation of a DAE As ODE: a list of boxes [ y i ] and [˜ y i ] such that ◮ y ( t i ) ∈ [ y i ] with t i ∈ { 0 , t 1 , . . . , t n } ◮ y ( t ) ∈ [˜ y i ] for all t ∈ [ t i , t i + 1 ] But in addition: a list of boxes [ x i ] and [˜ x i ] such that ◮ x ( t i ) ∈ [ x i ] with t i ∈ { 0 , t 1 , . . . , t n } ◮ x ( t ) ∈ [˜ x i ] for all t ∈ [ t i , t i + 1 ] Both validate ◮ y ′ ( t i ) ∈ f ( t i , [ x i ] , [ y i ]) ◮ ∃ x ∈ [ x i ] , ∃ y ∈ [ y i ] : g ( t i , x , y ) = 0 ◮ y ′ ( t ) ∈ f ( t , [˜ y i ]) , ∀ t ∈ [ t i , t i + 1 ] x i ] , [˜ ◮ ∀ t ∈ [ t i , t i + 1 ] , ∃ x ∈ [˜ x i ] , ∃ y ∈ [˜ y i ] : g ( t , x , y ) = 0 Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 7

Approach to simulate DAE Based on Lohner two-step approach Step 1- A priori enclosure of state and algebraic variables How find the enclosure [˜ x ] on integration step ? Assume that ∂ g ∂ x is locally reversal we are able to find the unique x = ψ ( y ) (implicit function theorem), and then: y ′ = f ( ψ ( y ) , y ) and finally we could apply Picard-Lindelof to prove existence and uniqueness , but... Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 8

Approach to simulate DAE Based on Lohner two-step approach Step 1- A priori enclosure of state and algebraic variables How find the enclosure [˜ x ] on integration step ? Assume that ∂ g ∂ x is locally reversal we are able to find the unique x = ψ ( y ) (implicit function theorem), and then: y ′ = f ( ψ ( y ) , y ) and finally we could apply Picard-Lindelof to prove existence and uniqueness , but... ✞ ☎ ψ is unknown ! ✝ ✆ Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 8

Approach to simulate DAE Based on Lohner two-step approach Step 1- A priori enclosure of state and algebraic variables Solution If we are able to find [˜ x ] such that for each y ∈ [˜ y ] , ∃ ! x ∈ [˜ x ] : g ( x , y ) = 0, then ∃ ! h on the neighborhood of [˜ x ] , and the solution of DAE ∃ ! in [˜ y ] (Picard with [˜ x ] as a parameter) A novel operator Picard-Krawczyk PK : � P ([˜ y ] , [˜ x ]) � � [˜ y ] � If ⊂ Int then ∃ ! solution of DAE K ([˜ y ] , [˜ x ]) [˜ x ] ◮ P a Picard-Lindelof for y ′ ∈ f ([˜ x ] , y ) ◮ K a parametrized preconditioned Krawczyk operator for g ( x , y ) = 0 , ∀ y ∈ [˜ y ] Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 9

Approach to simulate DAE Parametric Krawczyk Parametric preconditioned Krawczyk operator K ([˜ y ] , [˜ x ]) = m ([˜ x ]) − Cg ( m ([˜ x ]) , m ([˜ y ])) − ( C ∂ g ∂ x ([˜ x ] , [˜ y ]) − I )([˜ x ] − m ([˜ x ])) − C ∂ g ∂ y ( m ([˜ x ]) , [˜ y ])([˜ y ] − m ([˜ y ])) (1) Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 10

Approach to simulate DAE Parametric Krawczyk Interval Newton operator N ([ x ]) : repeat [ A ] = J ([ x ]) [ b ] = F ( m ([ x ])) Solve [ A ] s = [ b ] with a linear system solver method (Gauss elimination for example) [ x ] = [ x ] ∩ s + m ([ x ]) until Fixed point If N ([ x ]) ⊂ Int ([ x ]) , then F has a unique solution and this solution is in N ([ x ]) Parametric preconditioned Krawczyk A better version of Newton, with parameter and preconditioning Julien Alexandre dit Sandretto - Validated simulation of DAEs October 22, 2020- 11