Composition and Splitting Methods Book Sections II.4 and II.5 - PowerPoint PPT Presentation

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Composition and Splitting Methods Book Sections II.4 and II.5 Claude Gittelson Seminar on Geometric Numerical Integration 21.11.2005

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Outline The Adjoint of a Method 1 Definition Properties Composition Methods 2 Definition Order Increase Splitting Methods 3 Idea Examples Connection to Composition Methods

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Preliminaries Notation autonomous differential equation y = f ( y ) , ˙ y ( t 0 ) = y 0 , its exact flow ϕ t , and numerical method Φ h , i.e. y 1 = Φ h ( y 0 ) .

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Preliminaries Notation Basic Facts autonomous differential ϕ h ( y ) = y + O ( h ) equation p : order of Φ h e := Φ h ( y ) − ϕ h ( y ) error y = f ( y ) , ˙ y ( t 0 ) = y 0 , e = C ( y ) h p + 1 + O ( h p + 2 ) its exact flow ϕ t , and numerical method Φ h , i.e. y 1 = Φ h ( y 0 ) .

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Definition of the Adjoint Method Definition The adjoint of Φ h is h := Φ − 1 Φ ∗ − h . It is defined implicitly by y 1 = Φ ∗ h ( y 0 ) iff y 0 = Φ − h ( y 1 ) . Φ h is symmetric, if Φ ∗ h = Φ h .

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Properties of the Adjoint Method Remark Note that ϕ − 1 − t = ϕ t , but in h = Φ − 1 general Φ ∗ − h � = Φ h . The adjoint method h ) ∗ = Φ h and satisfies (Φ ∗ (Φ h ◦ Ψ h ) ∗ = Ψ ∗ h ◦ Φ ∗ h .

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Properties of the Adjoint Method Remark Note that ϕ − 1 − t = ϕ t , but in h = Φ − 1 general Φ ∗ − h � = Φ h . The adjoint method h ) ∗ = Φ h and satisfies (Φ ∗ (Φ h ◦ Ψ h ) ∗ = Ψ ∗ h ◦ Φ ∗ h . Example ( explicit Euler ) ∗ = implicit Euler ( implicit midpoint ) ∗ = implicit midpoint

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Order of the Adjoint Method Theorem If Φ h has order p and satisfies 1 Φ h ( y 0 ) − ϕ h ( y 0 ) = C ( y 0 ) h p + 1 + O ( h p + 2 ) , then Φ ∗ h also has order p and satisfies h ( y 0 ) − ϕ h ( y 0 ) = ( − 1 ) p C ( y 0 ) h p + 1 + O ( h p + 2 ) . Φ ∗ In particular, if Φ h is symmetric, its order is even. 2

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Definition of Composition Methods Definition Let Φ 1 h , . . . , Φ s h be one step methods. Composition Ψ h := Φ s γ s h ◦ . . . ◦ Φ 1 γ 1 h , where γ 1 , . . . , γ s ∈ R . Example Φ 1 h = . . . = Φ s h =: Φ h 1 Φ 2 k h = Φ h and Φ 2 k − 1 = Φ ∗ 2 h h

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Order Increase of General Composition Methods Theorem Let Ψ h := Φ s γ s h ◦ . . . ◦ Φ 1 γ 1 h with Φ k h of order p and h ( y ) − ϕ h ( y ) = C k ( y ) h p + 1 + O ( h p + 2 ) . Φ k If γ 1 + . . . + γ s = 1 , then Ψ h has order p + 1 if and only if γ p + 1 C 1 ( y ) + . . . + γ p + 1 C s ( y ) = 0 . s 1

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Order Increase of Compositions of a Single Method Corollary If Ψ h = Φ γ s h ◦ . . . ◦ Φ γ 1 h , then the conditions are γ 1 + . . . + γ s = 1 γ p + 1 + . . . + γ p + 1 = 0 . s 1 Remark A solution only exists if p is even.

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Order Increase of Compositions of a Single Method Corollary Example If Ψ h = Φ γ s h ◦ . . . ◦ Φ γ 1 h , then s = 3 , Φ h symmetric, order p = 2 , γ 1 = γ 3 . the conditions are Then Ψ h = Φ γ 3 h ◦ Φ γ 2 h ◦ Φ γ 1 h is γ 1 + . . . + γ s = 1 also symmetric, order ≥ 3 . γ p + 1 + . . . + γ p + 1 Symmetric ⇒ order even ⇒ = 0 . s 1 order 4 . So repeated application is possible. Remark A solution only exists if p is even.

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Order Increase of Compositions with the Adjoint Method Corollary If Ψ h = Φ α s h ◦ Φ ∗ β s h ◦ . . . ◦ Φ ∗ β 2 h ◦ Φ α 1 h ◦ Φ ∗ β 1 h , then the conditions are β 1 + α 1 + . . . + β s + α s = 1 ( − 1 ) p β p + 1 + α p + 1 + . . . + ( − 1 ) p β p + 1 + α p + 1 = 0 . 1 1 s s Example Ψ h := Φ h 2 ◦ Φ ∗ 2 is symmetric, order p + 1 . h Φ h explicit Euler ⇒ Ψ h implicit midpoint Φ h implicit Euler ⇒ Ψ h trapezoidal rule

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Idea: Split the Vector Field Idea Split the vector field f into y = f ( y ) = f [ 1 ] ( y ) + f [ 2 ] ( y ) + . . . + f [ N ] ( y ) ˙ Calculate exact flow ϕ [ i ] y = f [ i ] explicitly t of ˙ Use “composition” of ϕ [ i ] h to solve ˙ y = f ( y ) , e.g. Ψ h = ϕ [ 1 ] a s h ◦ ϕ [ 2 ] b s h ◦ . . . ◦ ϕ [ 1 ] a 1 h ◦ ϕ [ 2 ] b 1 h

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Motivation Example y = ( a + b ) y , then ϕ a t ( y 0 ) = e at y 0 and ϕ b t ( y 0 ) = e bt y 0 , so ˙ ( ϕ a t ◦ ϕ b t )( y 0 ) = e at e bt y 0 = e ( a + b ) t y 0 = ϕ t ( y 0 )

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Motivation Example y = ( a + b ) y , then ϕ a t ( y 0 ) = e at y 0 and ϕ b t ( y 0 ) = e bt y 0 , so ˙ ( ϕ a t ◦ ϕ b t )( y 0 ) = e at e bt y 0 = e ( a + b ) t y 0 = ϕ t ( y 0 ) Lie-Trotter Formula for A , B ∈ C N × N . y = ( A + B ) y ˙ ϕ A t ( y 0 ) = e At y 0 ϕ B t ( y 0 ) = e Bt y 0 and � n � e A t n e B t = e ( A + B ) t Lie Trotter formula lim n n →∞ � n � ϕ A n ◦ ϕ B so ( y 0 ) → ϕ t ( y 0 ) t t n

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Examples of Splittings Example (Lie-Trotter Splitting) ϕ [ 1 ] h ◦ ϕ [ 2 ] Φ h = h ϕ [ 2 ] h ◦ ϕ [ 1 ] Φ ∗ = h h

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Examples of Splittings Example (Lie-Trotter Splitting) Example (Strang Splitting) ϕ [ 1 ] h ◦ ϕ [ 2 ] Φ h = h Φ h = ϕ [ 1 ] 2 ◦ ϕ [ 2 ] h ◦ ϕ [ 1 ] 2 = Φ ∗ ϕ [ 2 ] h ◦ ϕ [ 1 ] Φ ∗ = h h h h h

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Application to Separable Hamiltonian Systems Example Separable Hamiltonian H ( p , q ) = T ( p ) + U ( q ) � 0 � ˙ � � − H q � � � − U q � p = = + q ˙ H p T p 0

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Application to Separable Hamiltonian Systems Example Separable Hamiltonian H ( p , q ) = T ( p ) + U ( q ) � 0 � ˙ � � − H q � � � − U q � p = = + q ˙ H p T p 0 Exact flows � p 0 � � � � p 0 � � p 0 − t U q ( q 0 ) � p 0 ϕ T ϕ U = , = t t q 0 q 0 + t T p ( p 0 ) q 0 q 0

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Application to Separable Hamiltonian Systems Example Separable Hamiltonian H ( p , q ) = T ( p ) + U ( q ) � 0 � ˙ � � − H q � � � − U q � p = = + q ˙ H p T p 0 Exact flows � p 0 � � � � p 0 � � p 0 − t U q ( q 0 ) � p 0 ϕ T ϕ U = , = t t q 0 q 0 + t T p ( p 0 ) q 0 q 0 Lie-Trotter splitting Φ h = ϕ T h ◦ ϕ U h = p n − h · U q ( q n ) p n + 1 = q n + h · T p ( p n + 1 ) q n + 1

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Application to Separable Hamiltonian Systems Example Separable Hamiltonian H ( p , q ) = T ( p ) + U ( q ) � 0 � ˙ � � − H q � � � − U q � p = = + q ˙ H p T p 0 Exact flows � p 0 � � � � p 0 � � p 0 − t U q ( q 0 ) � p 0 ϕ T ϕ U = , = t t q 0 q 0 + t T p ( p 0 ) q 0 q 0 Lie-Trotter splitting Φ h = ϕ T h ◦ ϕ U h � symplectic Euler = p n − h · U q ( p n + 1 , q n ) p n + 1 = q n + h · T p ( p n + 1 , q n ) q n + 1

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Construction as a Composition Method Lemma Φ [ i ] y = f [ i ] ( y ) . h consistent method for ˙ Φ h := Φ [ 1 ] h ◦ Φ [ 2 ] h ◦ . . . ◦ Φ [ N ] h , y = f ( y ) = f [ 1 ] ( y ) + f [ 2 ] ( y ) + . . . + f [ N ] ( y ) . then Φ h has order 1 for ˙

Preliminaries The Adjoint of a Method Composition Methods Splitting Methods Summary Construction as a Composition Method Lemma Φ [ i ] y = f [ i ] ( y ) . h consistent method for ˙ Φ h := Φ [ 1 ] h ◦ Φ [ 2 ] h ◦ . . . ◦ Φ [ N ] h , y = f ( y ) = f [ 1 ] ( y ) + f [ 2 ] ( y ) + . . . + f [ N ] ( y ) . then Φ h has order 1 for ˙ Idea Compose Φ h , Φ ∗ h to construct method Ψ h of higher order. In the case N = 2 : Φ h = Φ [ 1 ] h ◦ Φ [ 2 ] h = Φ [ 2 ] ∗ ◦ Φ [ 1 ] ∗ h , Φ ∗ and h h Ψ h = Φ α s h ◦ Φ ∗ β s h ◦ . . . ◦ Φ ∗ β 2 h ◦ Φ α 1 h ◦ Φ ∗ β 1 h Φ [ 1 ] α s h ◦ Φ [ 2 ] α s h ◦ Φ [ 2 ] ∗ β s h ◦ Φ [ 1 ] ∗ β s h ◦ . . . ◦ Φ [ 2 ] α 1 h ◦ Φ [ 2 ] ∗ β 1 h ◦ Φ [ 1 ] ∗ = β 1 h