Why Algorithmic and Rigorous Polynomial Approximations? Rigorous - PowerPoint PPT Presentation

Why Algorithmic and Rigorous Polynomial Approximations? ▸ Rigorous Polynomial Approximation = Polynomial + error bound Rigorous methods Algorithmic methods Efficient and accurate 1/16 Multinormval

Why Algorithmic and Rigorous Polynomial Approximations? ▸ Rigorous Polynomial Approximation = ▸ Solutions of coupled systems of linear Polynomial + error bound ordinary differential equations. ▸ with componentwise error bounds. Rigorous methods Algorithmic methods Efficient and accurate 1/16 Multinormval

Why Algorithmic and Rigorous Polynomial Approximations? ▸ Rigorous Polynomial Approximation = ▸ Solutions of coupled systems of linear Polynomial + error bound ordinary differential equations. ▸ with componentwise error bounds. ▸ Various fields of applications: 1 . 2 1 . 1 d ( h ) y P ( h ) h 1 Rigorous methods 0 . 9 0 . 9 1 1 . 1 1 . 2 1 . 3 x Algorithmic methods Safety-critical Computer-aided engineering mathematics Efficient and accurate 1/16 Multinormval

Outline 1 Introduction 2 A New Framework for Componentwise Validation 3 Componentwise Validation of Coupled Systems of Linear ODEs in Cheby- shev Basis A Sparse Linear Algebra Problem A Newton-like A Posteriori Fixed-Point Operator for Validation 4 Conclusion and Future Work Multinormval

Outline 1 Introduction 2 A New Framework for Componentwise Validation 3 Componentwise Validation of Coupled Systems of Linear ODEs in Cheby- shev Basis A Sparse Linear Algebra Problem A Newton-like A Posteriori Fixed-Point Operator for Validation 4 Conclusion and Future Work Multinormval

Banach Fixed-Point Theorem for A Posteriori Validation ▸ Fixed-point equation T ⋅ x = x with T contracting, ▸ Approximation x ○ to exact solution x ⋆ , General scheme ▸ Compute a posteriori error bounds with Banach theorem. 2/16 Multinormval

Banach Fixed-Point Theorem for A Posteriori Validation ▸ Fixed-point equation T ⋅ x = x with T contracting, ▸ Approximation x ○ to exact solution x ⋆ , General scheme ▸ Compute a posteriori error bounds with Banach theorem. Banach Fixed-Point Theorem If ( X , d ) is complete and T contracting of ratio µ < 1, ▸ T admits a unique fixed-point x ⋆ , and ▸ For all x ○ ∈ X , d ( x ○ , T ⋅ x ○ ) ⩽ d ( x ○ , x ⋆ ) ⩽ d ( x ○ , T ⋅ x ○ ) . 1 + µ 1 − µ 2/16 Multinormval

Banach Fixed-Point Theorem for A Posteriori Validation ▸ Fixed-point equation T ⋅ x = x with T contracting, ▸ Approximation x ○ to exact solution x ⋆ , General scheme ▸ Compute a posteriori error bounds with Banach theorem. Quasi-Newton Method for F ⋅ x = 0 Banach Fixed-Point Theorem Compute A ≈ ( D F ) − 1 x ○ in order to define: If ( X , d ) is complete and T contracting of ratio µ < 1, T ⋅ x = x − A ⋅ F ⋅ x . ▸ T admits a unique fixed-point x ⋆ , and ▸ For all x ○ ∈ X , Banach fixed-point theorem applies if for some r > 0: d ( x ○ , T ⋅ x ○ ) ⩽ d ( x ○ , x ⋆ ) ⩽ d ( x ○ , T ⋅ x ○ ) . µ = sup x ∈ B ( x ○ , r ) ∥ 1 − A ⋅ D F x ∥ < 1, 1 + µ 1 − µ ∥ x ○ − T ⋅ x ○ ∥ + µ r < r . 2/16 Multinormval

Banach Fixed-Point Theorem for A Posteriori Validation ▸ Fixed-point equation T ⋅ x = x with T contracting, ▸ Approximation x ○ to exact solution x ⋆ , General scheme ▸ Compute a posteriori error bounds with Banach theorem. Quasi-Newton Method for F ⋅ x = 0 Banach Fixed-Point Theorem Compute A ≈ ( D F ) − 1 x ○ in order to define: If ( X , d ) is complete and T contracting of ratio µ < 1, T ⋅ x = x − A ⋅ F ⋅ x . ▸ T admits a unique fixed-point x ⋆ , and ▸ For all x ○ ∈ X , Banach fixed-point theorem applies if for some r > 0: d ( x ○ , T ⋅ x ○ ) ⩽ d ( x ○ , x ⋆ ) ⩽ d ( x ○ , T ⋅ x ○ ) . µ = sup x ∈ B ( x ○ , r ) ∥ 1 − A ⋅ D F x ∥ < 1, 1 + µ 1 − µ ∥ x ○ − T ⋅ x ○ ∥ + µ r < r . Applications to function space problems: Early works by Kaucher, Miranker, Yamamoto et al ( ∼ 80’s, ∼ 90’s). Lessard et al (2007 - today). Benoit, Joldes, Mezzarobba (2011); Bréhard, Brisebarre, Joldes (2017). 2/16 Multinormval

Example: Polynomial Equation in the Plane 1 sin 3 x + cos 3 x = 0 ⇔ s 0 � s 3 + 4 c 3 − 3 c = 0 c 2 + s 2 − 1 = 0 − 1 c − 1 0 1 3/16 Multinormval

Example: Polynomial Equation in the Plane 1 x ◦ x ⋆ s 0 � s 3 + 4 c 3 − 3 c = 0 c 2 + s 2 − 1 = 0 − 1 c − 1 0 1 3/16 Multinormval

Example: Polynomial Equation in the Plane 0 . 56 x ◦ 0 . 55 s x ⋆ c 0 . 54 0 . 83 0 . 84 0 . 85 3/16 Multinormval

Example: Polynomial Equation in the Plane 0 . 56 x ◦ 0 . 55 s T · x ◦ x ⋆ � � 0 . 25 − 0 . 20 A = − 0 . 37 1 . 2 c 0 . 54 0 . 83 0 . 84 0 . 85 3/16 Multinormval

Example: Polynomial Equation in the Plane 0 . 56 r ( · 10 − 3 ) µ = � 1 − A · D F � 2 Stability 3 5.98e-2 No 4.5 7.55e-2 No 5 8.09e-2 Yes 100 1.21e+0 Yes x ◦ 0 . 55 s T · x ◦ x ⋆ � � 0 . 25 − 0 . 20 A = − 0 . 37 1 . 2 c 0 . 54 0 . 83 0 . 84 0 . 85 3/16 Multinormval

Example: Polynomial Equation in the Plane 0 . 56 ≤ � x ◦ − x ⋆ � 2 ≤ � x ◦ − T · x ◦ � 2 � x ◦ − T · x ◦ � 2 1+ µ 1 − µ x ◦ 0 . 55 s T · x ◦ x ⋆ c 0 . 54 0 . 83 0 . 84 0 . 85 3/16 Multinormval

Example: Polynomial Equation in the Plane 0 . 56 3 . 97 · 10 − 3 ≤ � x ◦ − x ⋆ � 2 ≤ 4 . 68 · 10 − 3 x ◦ 0 . 55 s T · x ◦ x ⋆ c 0 . 54 0 . 83 0 . 84 0 . 85 3/16 Multinormval

Vector-valued Metric and Perov Theorem Vector-Valued Metric ( X i , d i ) 1 ⩽ i ⩽ p complete metric spaces. d ( x , y ) = ( d 1 ( x 1 , y 1 ) , . . . , d p ( x p , y p )) ∈ R p + vector-valued metric. F ∶ X → X is Λ -Lipschitz for Λ ∈ R p × p : + d ( F ⋅ x , F ⋅ y ) ⩽ Λ ⋅ d ( x , y ) ∀ x , y ∈ X 4/16 Multinormval

Vector-valued Metric and Perov Theorem Vector-Valued Metric ( X i , d i ) 1 ⩽ i ⩽ p complete metric spaces. d ( x , y ) = ( d 1 ( x 1 , y 1 ) , . . . , d p ( x p , y p )) ∈ R p + vector-valued metric. F ∶ X → X is Λ -Lipschitz for Λ ∈ R p × p : + d ( F ⋅ x , F ⋅ y ) ⩽ Λ ⋅ d ( x , y ) ∀ x , y ∈ X Convergent to Zero Matrices Λ ∈ R p × p is convergent to zero if: Λ k → 0 as k → ∞ , ⇔ ρ ( Λ ) < 1. Generalized Contractions F ∶ X → X is a generalized contraction if it is Λ -Lipschitz for Λ convergent to zero. 4/16 Multinormval

Vector-valued Metric and Perov Theorem Vector-Valued Metric d ( x ○ , x ⋆ ) ⩽ d ( x ○ , T ⋅ x ○ ) + d ( T ⋅ x ○ , x ⋆ ) ( X i , d i ) 1 ⩽ i ⩽ p complete metric spaces. d ( x , y ) = ( d 1 ( x 1 , y 1 ) , . . . , d p ( x p , y p )) ∈ R p + vector-valued metric. F ∶ X → X is Λ -Lipschitz for Λ ∈ R p × p : + d ( F ⋅ x , F ⋅ y ) ⩽ Λ ⋅ d ( x , y ) ∀ x , y ∈ X Convergent to Zero Matrices Λ ∈ R p × p is convergent to zero if: Λ k → 0 as k → ∞ , ⇔ ρ ( Λ ) < 1. Generalized Contractions F ∶ X → X is a generalized contraction if it is Λ -Lipschitz for Λ convergent to zero. 4/16 Multinormval

Vector-valued Metric and Perov Theorem Vector-Valued Metric d ( x ○ , x ⋆ ) ⩽ d ( x ○ , T ⋅ x ○ ) + Λ ⋅ d ( x ○ , x ⋆ ) ( X i , d i ) 1 ⩽ i ⩽ p complete metric spaces. d ( x , y ) = ( d 1 ( x 1 , y 1 ) , . . . , d p ( x p , y p )) ∈ R p + vector-valued metric. F ∶ X → X is Λ -Lipschitz for Λ ∈ R p × p : + d ( F ⋅ x , F ⋅ y ) ⩽ Λ ⋅ d ( x , y ) ∀ x , y ∈ X Convergent to Zero Matrices Λ ∈ R p × p is convergent to zero if: Λ k → 0 as k → ∞ , ⇔ ρ ( Λ ) < 1. Generalized Contractions F ∶ X → X is a generalized contraction if it is Λ -Lipschitz for Λ convergent to zero. 4/16 Multinormval

Vector-valued Metric and Perov Theorem Vector-Valued Metric ( 1 − Λ ) ⋅ d ( x ○ , x ⋆ ) ⩽ d ( x ○ , T ⋅ x ○ ) ( X i , d i ) 1 ⩽ i ⩽ p complete metric spaces. d ( x , y ) = ( d 1 ( x 1 , y 1 ) , . . . , d p ( x p , y p )) ∈ R p + vector-valued metric. F ∶ X → X is Λ -Lipschitz for Λ ∈ R p × p : + d ( F ⋅ x , F ⋅ y ) ⩽ Λ ⋅ d ( x , y ) ∀ x , y ∈ X Convergent to Zero Matrices Λ ∈ R p × p is convergent to zero if: Λ k → 0 as k → ∞ , ⇔ ρ ( Λ ) < 1. Generalized Contractions F ∶ X → X is a generalized contraction if it is Λ -Lipschitz for Λ convergent to zero. 4/16 Multinormval

Vector-valued Metric and Perov Theorem Vector-Valued Metric ( 1 − Λ ) ⋅ d ( x ○ , x ⋆ ) ⩽ d ( x ○ , T ⋅ x ○ ) ( X i , d i ) 1 ⩽ i ⩽ p complete metric spaces. ( 1 − Λ ) − 1 = 1 + Λ + Λ 2 +⋅ ⋅ ⋅+ Λ k + . . . ⩾ 0 . d ( x , y ) = ( d 1 ( x 1 , y 1 ) , . . . , d p ( x p , y p )) ∈ R p ⇒ d ( x ○ , x ⋆ ) ⩽ ( 1 − Λ ) − 1 ⋅ d ( x ○ , T ⋅ x ○ ) . + vector-valued metric. F ∶ X → X is Λ -Lipschitz for Λ ∈ R p × p : + d ( F ⋅ x , F ⋅ y ) ⩽ Λ ⋅ d ( x , y ) ∀ x , y ∈ X Convergent to Zero Matrices Λ ∈ R p × p is convergent to zero if: Λ k → 0 as k → ∞ , ⇔ ρ ( Λ ) < 1. Generalized Contractions F ∶ X → X is a generalized contraction if it is Λ -Lipschitz for Λ convergent to zero. 4/16 Multinormval