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Introduction to Interval and Taylor Model Methods Markus Neher KARLSRUHE INSTITUTE OF TECHNOLOGY (KIT) 0 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016 www.kit.edu KIT – The Research University in the Helmholtz Association

Outline 1 Enclosure methods 2 Interval analysis 3 Taylor models 4 Interval and Taylor model methods for ODEs 1 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Enclosure Methods 2 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Enclosure Methods Also called: guaranteed methods, rigorous methods, validated methods, verified methods, . . . Aim: Compute guaranteed bounds for the solution of a problem, including Discretization errors (ODEs, PDEs, optimization), Truncation errors (Newton’s method, summation), Roundoff errors. Used for Modelling of uncertain data, Sensitivity analysis, Bounding of roundoff errors. 3 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Arithmetic 4 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Arithmetic Compact real intervals (Warmus 1956, Sunaga 1958, Moore&Yang 1959, Moore 1962, 1966): IR = { x = [ x , x ] | x ≤ x } ( x , x ∈ R ) . Width: w ( x ) = x − x . Hausdorff distance: � [ x , x ] , [ y , y ] � = max � � � � � x − y � , | x − y | q ( x , y ) = q . Interval vectors, interval matrices: Analogously. n x = ( x i ) ∈ IR n : w ( x ) = max i = 1 w ( x i ) . 5 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Arithmetic Basic arithmetic operations: x ◦ y : = { x ◦ y | x ∈ x , y ∈ y } , ◦ ∈ { + , − , ∗ , / } ( 0 �∈ y for / ) . x + y = [ x + y , x + y ] , x − y = [ x − y , x − y ] , x ∗ y = [ min { xy , xy , xy , xy , } , max { xy , xy , xy , xy , } ] , x / y = x ∗ [ 1 / y , 1 / y ] . No inverses: � 1 � [ 0 , 1 ] − [ 0 , 1 ] = [ − 1 , 1 ] , [ 1 , 2 ] / [ 1 , 2 ] = 2 , 2 . Floating point IA: � y = y ] , x x y , x etc. 6 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Ranges and Inclusion Functions Range of f : D → E : Rg ( f , D ) : = { f ( x ) | x ∈ D } . Inclusion function F : IR → IR of f : D ⊆ R → R : F ( x ) ⊇ Rg ( f , x ) for all x ⊆ D . Examples: Rational functions: For f : x �→ f ( x ) , replace each occurrence of x by x , evaluate using interval arithmetic: � = � [ 0 , 1 ] x 1 + [ 0 , 1 ] = [ 0 , 1 ] [ 0 , 1 ] f ( x ) = 1 + x ⇒ F [ 1 , 2 ] = [ 0 , 1 ] . Elementary functions: For f ∈ { exp , ln , cos , arcsin , tanh , . . . } let F ( x ) : = Rg ( f , x ) : � [ − 1 , 1 ] � = � � � [ 1 , 2 ] � = � � e − 1 , exp e sin sin 1 , 1 , . 7 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

IA: Dependency Multiple instances of the same variable are considered independent. x 1 f ( x ) = 1 + x = 1 − 1 + x , x = [ 0 , 1 ] : 1 + x = [ 0 , 1 ] x [ 1 , 2 ] = [ 0 , 1 ] , � = � = Rg ( f , x ) . � 1 � 1 1 0 , 1 1 − 1 + x = 1 − [ 1 , 2 ] = 1 − 2 , 1 2 Theorem Let f : D ⊂ R m → R be Lipschitz-continuous and let x ⊆ x 0 ⊆ D. Then � ≤ γ w ( x ) , � q F ( x ) , Rg ( f , x ) γ ≥ 0 . 8 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Ranges and Inclusion Functions Centered form: f ( x ) = f ( c ) + g ( x , c )( x − c ) ⇒ Rg ( f , x ) ⊆ F ( c ) + G ( x , c )( x − c ) . Rg ( f , x ) ⊆ f ( c ) + F ′ ( x )( x − c ) , c ∈ x . Mean value form: x c = 1 f ( x ) = x = [ 0 , 1 ] , 2 : 1 + x , � 1 � � � � � � � [ 0 , 1 ] − 1 − 1 6 , 5 0 , 1 + F ′ ([ 0 , 1 ]) = ⊃ = Rg ( f , x ) . f 2 2 6 2 Theorem Let f : D ⊂ R m → R be a sufficiently smooth function and be represented in the centered form. Furthermore, let x ⊆ x 0 ⊆ D. Then � � ≤ γ � � 2 , q f ( x ) , Rg ( f , x ) w ( x ) γ ≥ 0 . 9 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

IA: Wrapping Effect Enclosing non-intervals by intervals causes overestimation. √ 2 f : ( x , y ) �→ 2 ( x + y , y − x ) Example: (Rotation) � [ − 1 , 1 ] , [ − 1 , 1 ] � Interval evaluation of f on : y y 2 2 1 1 x x − 2 − 1 − 2 − 1 1 2 1 2 − 1 − 1 − 2 − 2 10 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Taylor Models 11 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Taylor Models Taylor model T n , f = ( p n , f , i n , f ) of order n of f : x → R (Berz and Hoffstätter 1994, Makino and Berz 1996): ∀ x ∈ x : f ( x ) ∈ p n , f ( x − x 0 ) + i n , f � = � ⊂ R m . � ( p , i ) � p ( x ) + ξ | x ∈ x , ξ ∈ i Rg x = [ − 1 , 1 ] , x 0 = 0: Example: e x = 1 + x + 1 2 x 2 + 1 6 x 3 + 1 24 x 4 e ξ , x , ξ ∈ x , 2 x 2 + 1 6 x 3 + [ 0 . 015 , 0 . 114 ] , T 3 , e x = 1 + x + 1 x ∈ x . 12 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Taylor Models Minimized dependency: p n , f is processed symbolically to order n . Higher order terms are enclosed into the interval of the result. x 2 Minimized wrapping effect: TMs can represent non-convex sets. 2 � � x 1 Example: T 2 , f = , x i ∈ [ − 1 , 1 ] . 2 + x 2 1 + x 2 x 1 − 1 1 13 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval and Taylor Model Methods for IVPs 14 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Initial Value Problem IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ R m , t end > t 0 15 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Initial Value Problem IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ R m , t end > t 0 15 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Initial Value Problem IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ R m , t end > t 0 15 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Initial Value Problem IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ R m , t end > t 0 15 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Verified Initial Value Problem IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ R m , t end > t 0 16 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Method for IVP with Uncertainty Interval IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 ∈ u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ IR m , t end > t 0 17 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Method for IVP with Uncertainty Interval IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 ∈ u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ IR m , t end > t 0 17 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Method for IVP with Uncertainty Interval IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 ∈ u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ IR m , t end > t 0 17 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Method for IVP with Uncertainty Interval IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 ∈ u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ IR m , t end > t 0 17 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Method for IVP with Uncertainty Interval IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 ∈ u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ IR m , t end > t 0 17 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Method for IVP with Uncertainty Interval IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 ∈ u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ IR m , t end > t 0 17 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Method for IVP with Uncertainty Interval IVP: u ′ = f ( t , u ) , u ( t 0 ) = u 0 ∈ u 0 , t ∈ t = [ t 0 , t end ] f : R × R m → R m sufficiently smooth, u 0 ∈ IR m , t end > t 0 17 Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016