Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples On the Taylor models approach for solving ODEs Dr. Florian Bünger florian.buenger@tuhh.de Institute for Reliable Computing Hamburg University of Technology Schwarzenberg-Campus 3 D-21073 Hamburg Germany November 30th, 2017 Florian Bünger Taylor models 1 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples Outline Introduction 1 Definition and arithmetic of Taylor models 2 Algorithm for solving ODEs 3 Accompanying methods 4 Numerical examples 5 Florian Bünger Taylor models 2 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples Taylor models have been used successfully to calculate verified inclusions of the solutions of initial value problems for ODEs. This was done by - Makino (PhD 1998) and Berz and their group implementation: [Fortran] COSY INFINITY - Eble (PhD 2007, Univ. Karlsruhe, Alefeld, Lohner, Neher) implementation: [C++] Riot - Dzetkuliˇ c (2012) implementation: [C++] ODEintegrator - Xin Chen et al. (PhD 2015, RWTH Aachen) implementation: [C++] FLOW Florian Bünger Taylor models 4 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples Especially Berz, Makino, and their group focused on Taylor models and invented several accompanying methods like shrink wrapping blunting preconditioning error parametrization dynamic domain decomposition We give a short description of definition and arithmetic of Taylor models main algorithm steps for solving ODEs shrink wrapping and preconditioning numerical results Florian Bünger Taylor models 5 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples Definition of a Taylor model dimension: := { 1 , 2 , . . . } k ∈ N order: m ∈ N 0 := { 0 , 1 , 2 , . . . } D ∈ IR k := { [ a , b ] := k domain: [ a i , b i ] | a , b ∈ R k , a ≤ b } × i = 1 error interval: I ∈ IR polynomial part: p = � p α x α ∈ R [ x 1 , . . . , x k ] α ∈ N k 0 | α |≤ m x α := � k | α | := � k i = 1 x α i where and i = 1 α i i evaluation point: x 0 ∈ D Taylor model (of order m ): { f ∈ C 0 ( D , R ) | ∀ x ∈ D : f ( x ) ∈ p ( x − x 0 ) + I } p + I := where p ( x − x 0 ) + I := { p ( x − x 0 ) + y | y ∈ I } Florian Bünger Taylor models 7 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples underlying idea: p is the Taylor polynomial of order m with evaluation point x 0 of some f ∈ C m + 1 ( D , R ) such that R ( x ) := f ( x ) − p ( x − x 0 ) ∈ I for all x ∈ D is an m -th order Taylor model for f ∈ C m + 1 ( D , R ) . in short: p + I This explains the name “Taylor model”. BUT: The formal definition does not (!) necessitate any Taylor expansion of some unknown function f in the background. Florian Bünger Taylor models 8 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples Taylor model vectors are analogously denoted by p + I : p = ( p 1 , . . . , p n ) ∈ ( R [ x 1 , . . . , x k ]) n I = I 1 × · · · × I n ∈ IR n is an interval vector is a Taylor model as defined before, i = 1 , . . . , n p i + I i The range of p + I is p ( D − x 0 ) + I = { p ( x − x 0 ) + y | x ∈ D , y ∈ I } ⊆ R n . D = [ − 1 , 1 ] 2 , I := [ − 0 . 05 , 0 . 05 ] 2 Example: k := 2 =: n , m := 3, x 0 := ( 0 , 0 ) , p 1 ( x 1 , x 2 ) := − 0 . 2 x 3 p 2 ( x 1 , x 2 ) := 0 . 3 x 2 2 + 0 . 9 x 1 , 1 x 2 + x 2 + 0 . 1 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 Florian Bünger Taylor models 9 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples Now, arithmetic operations + , − , × , ÷ standard functions exp, log, sin, cos, ... integration w.r.t. one of the n variables can be defined on m -th order Taylor models with fixed D and x 0 . Examples: a) Summation / Subtraction: ( p + I ) ± ( q + J ) := ( p ± q ) + ( I ± J ) b) Multiplication: ( p + I ) ∗ ( q + J ) := r + K c) Exponential function: exp ( p + I ) := r + K � d) Antiderivative: p + I dx i := r + K Florian Bünger Taylor models 10 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples b) Multiplication: ( p + I ) ∗ ( q + J ) := r + K pq = r + s , deg ( r ) ≤ m , s : terms of order > m ( p + I )( q + J ) = pq + pJ + I ( q + J ) = r + ( s + pJ + I ( q + J )) � �� � to be contained in K s ( D − x 0 ) + p ( D − x 0 ) J + I ( q ( D − x 0 ) + J ) ⊆ K 1 ∈ IR By symmetry: s ( D − x 0 ) + q ( D − x 0 ) I + J ( p ( D − x 0 ) + I ) ⊆ K 2 ∈ IR K := K 1 ∩ K 2 Florian Bünger Taylor models 11 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples c) Exponential function: exp ( p + I ) := r + K The image of p is thought to be centered around c := p ( 0 ) . Thus, only a Taylor expansion of exp around c makes sense: m + ( x − c ) ( m + 1 ) ( x − c ) i � exp ( x ) = exp ( c ) exp ( ξ ) ξ ∈ c +[ 0 , 1 ]( x − c ) . , ( m + 1 )! i ! i = 0 � �� � � �� � R ( x ):= q ( x ):= q ( p + I ) = r + J , deg ( r ) ≤ m , J ∈ IR exp ( p + I ) = q ( p + I ) + R ( p + I ) = r + ( J + R ( p + I )) � �� � to be contained in K p ( D − x 0 ) + I ⊆ K 1 ∈ IR R ( K 1 ) = ( K 1 − c ) ( m + 1 ) exp ( c ) exp ([ 0 , 1 ]( K 1 − c )) ⊆ K 2 ∈ IR ( m + 1 )! K := J + K 2 Florian Bünger Taylor models 12 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples x i � d) Antiderivative: p + I dx i := r + K , x i ∈ [ x i , x i ] x 0 i Integrate p with respect to x i . The resulting polynomial ˆ p has degree at most m + 1. Split ˆ p = r + s such that s contains all terms of degree m + 1. x i � p + I dx i ⊆ r + s ( D − x 0 ) + I ([ x i , x i ] − x 0 i ) x 0 i � �� � to be contained in K ∈ IR Florian Bünger Taylor models 13 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples IIVP: y ′ f ( y , t ) ∈ R n , (1) = t ∈ ( t 0 , t e ) Y 0 ∈ IR n y ( t 0 ) ∈ Solve ODE stepwise on appropriate subintervals [ t i , t i + 1 ] , t 0 < t 1 < · · · < t N = t e , N ∈ N . A Taylor model n -vector p ( i ) ( x , t ) + I ( i ) is supplied such that for all y 0 ∈ Y 0 the solution ˜ y of (1) with ˜ y ( t 0 ) = y 0 fulfills y ( t ) ∈ p ( i ) ([ − 1 , 1 ] n , t − t i ) + I ( i ) ∀ t ∈ [ t i , t i + 1 ] , i = 0 , . . . , N − 1 . ˜ Florian Bünger Taylor models 15 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples Lotka-Volterra equations: Example: y ′ 1 = 2 y 1 ( 1 − y 2 ) y 1 ( 0 ) ∈ [ 0 . 95 , 1 . 05 ] y ′ 2 = y 2 ( y 1 − 1 ) y 2 ( 0 ) ∈ [ 2 . 95 , 3 . 05 ] Remark: The Taylor model method supplies enclosures at all points t ∈ [ t i , t i + 1 ] Florian Bünger Taylor models 16 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples Standard form: dimension: k := n + 1 dim of ODE + 1 D := [ − 1 , 1 ] n × [ t i , t i + 1 ] , domain: B := [ − 1 , 1 ] n evaluation point: x 0 := ( 0 , . . . , 0 , t i ) p ( i ) ( B , t i + 1 − t i ) + I ( i ) = p ( i + 1 ) ( B , 0 ) + I ( i + 1 ) The range (only implicitly given) is taken as set of initial conditions for the next integration step from t i + 1 to t i + 2 . I ( 0 ) , I ( 1 ) , . . . , I ( N ) , The error interval vectors which cover all numerical and rounding errors, may strongly grow due to the wrapping effect of interval arithmetic. Florian Bünger Taylor models 17 / 45
Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples Main steps of integration algorithm IIVP: y ′ f ( y , t ) ∈ R n , = t ∈ ( t 0 , t e ) Y 0 = y 0 + [ − r , r ] ∈ IR n , r ∈ R n y ( t 0 ) ∈ > 0 q ( x , t ) = q ( x ) := y 0 + diag ( r ) x = ( y 0 i + r i x i ) 1 ≤ i ≤ n 1 J := [ 0 , 0 ] ∈ IR n 2 Range of q + J q ( B ) = Y 0 3 t � Integral operator K ( p + I ) := q + J + f ( p + I , τ ) d τ 4 t 0 5 K ( p + I ) ⊆ p + I ⇒ y ( t ) ∈ p ( B , t − t 0 ) + I ∀ t ∈ [ t 0 , t 1 ] Schauder Set q ( x ) := p ( x , t 1 − t 0 ) , J := I for next integration on [ t 1 , t 2 ] . 6 Florian Bünger Taylor models 18 / 45
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