16 an interval method for linear IVPs for ODEs nedialkok

NED NEDIALKOV Department of Computing and Software McMaster University, Canada nedialk@mcmaster.ca Joint work with Qiang Song McMaster University Improved version of the talk given a

NED NEDIALKOV Department of Computing and Software

McMaster University, Canada

nedialk@mcmaster.ca

Joint work with Qiang Song McMaster University

Improved version of the talk given at the

Workshop on Taylor Models 17-20 December 2003, Miami, Florida

Idea (Lohner, Nickel)

e Perform (n+ 1) integrations of points specifying a parallelepiped

at t; and enclose each point solution at £;+1

We have (n + 1) boxes

e Find (n+ 1) points that determine a parallelepiped, which

encloses all the parallelepipeds with vertices in these boxes

e Repeat.

solution

Figure 2: The same computation as in the previous figure, except that

the width of each component of the enclosures is 2 x 10719 The boxes are denoted by +.

solution

Figure 4: The same computation as in the previous figure, except that

the width of each component of the enclosures is 2 x 10719 The boxes are denoted by +.

Advantages

e We enclose point solutions:

Taylor series + remainder term

e The method does not impose restrictions on the size of the initial box

e An automatic differentiation package for computing Taylor

coefficients for the solution to Y’ = A(t)Y, Y(0) = J is not

needed

These coefficients are computed in AWA and VNODE

Difficulties

e How to compute (n+ 1) points on each step such that the

parallelepiped specified by them encloses the solution set

e How to achieve small overestimations and reduce the wrapping effect.

y(t; to, Yo) € [yo] for all t € 0, Al

Att=h,

?—Ì U(h: to, 1o) € o + 3 h' filyo) +h? fp((ol)-

„—=]l

Assume that at a point ¢;, for all o € [0o],

N — ros : : : i = oy 3 : t wa Pes LIÊN ¬s § TẤT N Ỷ

since for a € |0, 1J”, œø; € [w,], and e; = w,; —c; € [e;| (j =90,

Ee {co +Ca+t > [ej] + (n— Deo] | a € (0, " }

= {co + Cat [e] |a€ [0,1]"}

(Note that each [e,] is symmetric )

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We want to find gg and G such that

feg+Cat+e|aeé |0,1]", e € fe] } C {go + Ga | a € [0,1]” }

Let H € R”*” be nonsingular

Denote

Ir] =(H~*C)[0, 1)" + H-*[e] and D = diag (w((r]))

Then

{co + Cat+e|ae [0,1]", e € [e]}

={co+ H((H`*C)œ+ H*e) |ae€ |0,1]”, e € [e|}

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Now, for all ¿ € {bọ + Ba | a € |0,1]” },

We integrate go, (Go + 91), -, (Go + Gn) )

Transformation Matrix

Parallelepiped method

H=ZC,

ir] = (H~ˆ*Ø)|0,1]” + H~'[e] = [0,1]” + Ơ~ˆ|{e]

This method breaks down when C its close to singular

QR-factorization method

C=QRk, H=Q,

ir] = (A~*C) 0,1)" + H~*[e] = RO, 1)” + Q* [e]

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On some problems, with a large initial box, the QR method can

produce large overestimations

step 1

0.8F 0.6F

Can we combine them, or switch between them at run time?

Two ad-hoc solutions: Approach | and II.

Approach I

We can (roughly) measure the overestimations in the parallelepiped

and QR methods by |[w(C[rp])|| and |[w(Q[ra]) ||, respectively

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0.8 0.6

0.5 1 1.5 step 6, Par

0.8 0.6 0.4 0.2

Approach II

Let Gmax be the largest angle among the angles between every two

columns of C’

Let Gmin be the smallest such angle

Let 6,0 < 6 <7, bea constant

0.8 0.6 0.4 0.2

0.6 0.4 0.2 -02 0 02 04 06 0.8

0.6 0.4}

e To reduce the wrapping effect when propagating larger sets, a

combination of the parallelepiped and QR-factorization methods

may be necessary

e When to switch from one method to the other?

e An eigenvalue, or stability type analysis of a combined approach

may be necessary

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