Taylor Models and Their Applications Martin Berz and Kyoko Makino Department of Physics and Astronomy Michigan State University Contributions: Alex Wittig (MSU) R. Armellin, P. Di Lizia (Politecnico di Milano)
Outline • Motivation • Introduction to Taylor models Basics. Arithmetic. Comparisons with interval computations. • Applications – Demonstrate with examples — Range bounding Moore’s simple 1D function — Global optimizations Moore’s simple 1D function. Normal form defect functions. — ODE integrations The Volterra equations. Near earth asteroid Apophis. The Lorenz equations.[Long time integrations, Poincare pro- jections, covering huge size initial conditions, manifolds] • Work in progress
Introduction Taylor model (TM) methods were originally developed for a practical problem from nonlinear dynamics, range bounding of normal form defect functions. • Functions consist of code lists of 10 4 to 10 5 terms • Have about the worst imaginable cancellation problem • Are obtained via validated integration of large initial condition boxes. Originally nearly universally considered intractable by the community. But ... a small challenge goes a long way towards generating new ideas! Idea: represent all functional dependencies as a pair of a polynomial P and a remainder bound I introduce arithmetic, and a new ODE solver. Obtain the following properties: • The ability to provide enclosures of any function given by a fi nite com- puter code list by a Taylor polynomial and a remainder bound with a sharpness that scales with order ( n + 1) of the width of the domain. • The ability to alleviate the dependency problem in the calculation. • The ability to scale favorable to higher dimensional problems.
Find guaranteed bound of such functions ===> The original motivation of Taylor models
Motion in the Tevatron Speed of Light: 3x10 8 m/sec • Circumference: 6.28x10 3 m • 4x10 4 revs/sec. Need to store about 10 hours, or 4x10 5 sec • 2x10 10 revolutions total. • 10,000 magnets in ring 2x10 14 contacts with fields! • Extremely challenging computationally •Need for several State-Of-The-Art Methods: •Phase Space Maps •Perturbation Theory •Lyapunov- and other Stability Theories •High-Performance Verified Methods
Bounds Estimates RDA Interval Method Taylor Model Method f U f R I f �� �� �� �� P f f L a a b b R Bounds: [f ] ,f Bounds: P +I L U f f : Polynomial P f R : Remainder Bounds I f
Interval Arithmetic A method to perform guaranteed calculations on computer by pre- senting all numbers by intervals. [ a ] + [ ] = [ a + + ] [ a ] − [ ] = [ a − − ] [ a ] · [ ] = [min( a a ) max( a a )] [ a ] / [ ] = [min( a/ a/ / / ) max( a/ a/ / / )] Not a group because [ a ] − [ ] 6 = [0 0] unless a = = . In particular, [ a ] − [ a ] = [ a − − a ] [ a ] / [ a ] = [min(1 a/ /a ) max(1 a/ /a )] Thus, operations lead to over estimation, which can become large with time to blow up.
Moore’s Simple 1D Function f ( x ) = 1 + x 5 − x 4 . Study on [0 , 1] . Trivial-looking, but dependency and high order. Assumes shallow min at 0 . 8 . y y 1 0.98 0.96 0.94 0.92 0 0.25 0.5 0.75 1 x x
Moore 1D function f(x)=x^5-x^4+1 in [0,1] 1.02 1 0.98 0.96 0.94 0.92 0.9 0 0.2 0.4 0.6 0.8 1
Moore 1D function f(x)=x^5-x^4+1 in [0,1]. Bounding by 16 intervals 1.3 1.2 1.1 1 0.9 0.8 0.7 0 0.2 0.4 0.6 0.8 1
Moore 1D function f(x)=x^5-x^4+1 in [0,1]. Bounding by 128 intervals 1.3 1.2 1.1 1 0.9 0.8 0.7 0 0.2 0.4 0.6 0.8 1
Moore 1D function f(x)=x^5-x^4+1 in [0,1]. Bounding by 128 intervals 1.02 1 0.98 0.96 0.94 0.92 0.9 0 0.2 0.4 0.6 0.8 1
Moore 1D function f(x)=x^5-x^4+1 in [0,1]. Bounding by 256 intervals 1.02 1 0.98 0.96 0.94 0.92 0.9 0 0.2 0.4 0.6 0.8 1
Moore 1D function f(x)=x^5-x^4+1 in [0,1]. Bounding by 512 intervals 1.02 1 0.98 0.96 0.94 0.92 0.9 0 0.2 0.4 0.6 0.8 1
Moore 1D function f(x)=x^5-x^4+1 in [0,1]. Bounding by 1024 intervals 1.02 1 0.98 0.96 0.94 0.92 0.9 0 0.2 0.4 0.6 0.8 1
De fi nitions - Taylor Models and Operations We begin with a review of the de fi nitions of the basic operations. De fi nition (Taylor Model) Let f : D ⊂ R v → R be a function that is ( n +1) times continuously partially di ff erentiable on an open set containing the domain v -dimensional domain D. Let x 0 be a point in D and P the n -th order Taylor polynomial of f around x 0 . Let I be an interval such that f ( x ) ∈ P ( x − x 0 ) + I for all x ∈ D. Then we call the pair ( P, I ) an n -th order Taylor model of f around x 0 on D. De fi nition (Addition and Multiplication) Let T 1 , 2 = ( P 1 , 2 , I 1 , 2 ) be n -th order Taylor models around x 0 over the domain D . We de fi ne T 1 + T 2 = ( P 1 + P 2 , I 1 + I 2 ) T 1 · T 2 = ( P 1 · 2 , I 1 · 2 ) where P 1 · 2 is the part of the polynomial P 1 · P 2 up to order n and I 1 · 2 = B ( P e ) + B ( P 1 ) · I 2 + B ( P 2 ) · I 1 + I 1 · I 2 where P e is the part of the polynomial P 1 · P 2 of orders ( n + 1) to 2 n , and B ( P ) denotes a bound of P on the domain D. We demand that B ( P ) is at least as sharp as direct interval evaluation of P ( x − x 0 ) on D.
De fi nitions - Taylor Model Intrinsics De fi nition (Intrinsic Functions of Taylor Models) Let T = ( P, I ) be a Taylor model of order n over the v -dimensional domain D = [ a, b ] around the point x 0 . We de fi ne intrinsic functions for the Taylor models by performing various manipulations that will allow the computation of Taylor models for the intrinsics from those of the arguments. In the following, let f ( x ) ∈ P ( x − x 0 ) + I be any function in the Taylor model, and let c f = f ( x 0 ) , and ¯ f be de fi ned by ¯ f ( x ) = f ( x ) − c f . Likewise we de fi ne ¯ P by P ( x − x 0 ) = P ( x − x 0 ) − c f , so that ( ¯ ¯ P, I ) is a Taylor model for ¯ f. For the various intrinsics, we proceed as follows. Exponential. We fi rst write ¡ ¢ ¡ ¯ ¢ c f + ¯ exp( f ( x )) = exp f ( x ) = exp( c f ) · exp f ( x ) ½ f ( x ) + 1 f ( x )) 2 + · · · + 1 1 + ¯ 2!( ¯ k !( ¯ f ( x )) k = exp( c f ) · ¢¾ ¡ 1 f ( x )) k +1 exp ( k + 1)!( ¯ θ · ¯ + f ( x ) , where 0 < θ < 1 .
De fi nitions - Taylor Model Exponential, cont. Taking k ≥ n , the part ½ ¾ f ( x ) + 1 f ( x )) 2 + · · · + 1 1 + ¯ 2!( ¯ n !( ¯ f ( x )) n exp( c f ) · is merely a polynomial of ¯ f , of which we can obtain the Taylor model via Taylor model addition and multiplication. The remainder part of exp( f ( x )) , the expression ½ 1 ( n + 1)!( ¯ f ( x )) n +1 exp( c f ) · ¢¾ ¡ 1 f ( x )) k +1 exp ( k + 1)!( ¯ θ · ¯ + · · · + f ( x ) , will be bounded by an interval. First observe that since the Taylor polyno- mial of ¯ f does not have a constant part, the ( n + 1) -st through ( k + 1) -st P, I ) of ¯ powers of the Taylor model ( ¯ f will have vanishing polynomial part, and thus so does the entire remainder part. The remainder bound interval for the Lagrange remainder term
De fi nitions - Taylor Model Exponential, cont. ¡ ¢ 1 f ( x )) k +1 exp ( k + 1)!( ¯ θ · ¯ exp( c f ) f ( x ) can be estimated because, for any x ∈ D , ¯ P ( x − x 0 ) ∈ B ( ¯ P ) , and 0 < θ < 1 , and so ¡ ¢ ¡ ¢ k +1 f ( x )) k +1 exp ( ¯ θ · ¯ B ( ¯ f ( x ) ∈ P ) + I ¡ ¢ [0 , 1] · ( B ( ¯ × exp P ) + I ) . The evaluation of the “ exp ” term is mere standard interval arithmetic. In the actual implementation, one may choose k = n for simplicity, but it is not a priori clear which value of k would yield the sharpest enclosures.
De fi nitions - Taylor Model Arc Sine Arcsine. Under the condition ∀ x ∈ D, B ( P ( x − x 0 ) + I ) ⊂ ( − 1 , 1) , using an addition formula for the arcsine, we re-write q ³ ´ p 1 − c 2 1 − ( f ( x )) 2 arcsin( f ( x )) = arcsin( c f ) + arcsin f ( x ) · f − c f · . Utilizing that q p 1 − c 2 1 − ( f ( x )) 2 g ( x ) ≡ f ( x ) · f − c f · does not have a constant part, we have 5!( g ( x )) 5 + 3 2 · 5 2 3!( g ( x )) 3 + 3 2 arcsin( g ( x )) = g ( x ) + 1 ( g ( x )) 7 7! 1 ( k + 1)!( g ( x )) k +1 · arcsin ( k +1) ( θ · g ( x )) , + · · · + where p arcsin 0 ( a ) = 1 / arcsin 00 ( a ) = a/ (1 − a 2 ) 3 / 2 , 1 − a 2 , arcsin (3) ( a ) = (1 + 2 a 2 ) / (1 − a 2 ) 5 / 2 , ...
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