Validating an A Priori Enclosure Using High-Order Taylor Series George F. Corliss and Robert Rihm Abstract We use Taylor series plus an enclosure of the remainder term to validate the existence of a unique solution for initial value problems in ordinary dierential equations and to compute a coarse enclosure of that solution. The signicance of this result is its application to Lohner's AWA algorithm for validated solutions, not to the theory of ordinary dierential equations. By using high-order Taylor series in Lohner's Algorithm I, we are able to validate the solution over much longer time steps than is done in the current AWA code. For Lohner's enclosure by polynomials, the enclosures are expensive to compute, but it is easy to check for enclosure. For our enclosure by Taylor series, the enclosures are free because they are already being computed, but checking for enclosure requires 2 n polynomial rootndings. Work is continuing on an implementation that will allow direct computational comparisons of the eectiveness of the two methods. Keywords: ordinary dierential equations, Lohner's algorithm, validated computation, Taylor series, enclosure methods. Introduction The AWA (Anfangswertaufgabe) program by Rudolf Lohner [4, 5, 6, 7, 8] computes an enclosure of the solution of an initial value problem (IVP) in ordinary dierential equations (ODE) u = f(u); u(t) = u; (1) where we only know an interval enclosure [u] of the vector u in general. Without loss of generality, Lohner assumes the system is autonomous only to simplify the proofs. We assume that f is at least p? 1 times continuously dierentiable in a domain D with [u] D IR n, p 2. Then there is a unique at least p times continuously dierentiable solution u(t) in a neighborhood of t. AWA is a single-step method. At each integration time step, AWA applies two algorithms: Algorithm I: (Existence and enclosure) Find a step size h and a coarse enclosure interval [u ] D such that for t 2 [t] := [t; t + h], the solution u(t) exists and satises u(t) 2 [u ].
2 George F. Corliss and Robert Rihm Algorithm II: (Tightening) Compute a tight enclosure for u(t) at t = t1 := t + h. Throughout this paper, we use the term \validation" to mean \prove existence of u(t) for some t and nd a coarse enclosure [u ]". The eciency of AWA has been limited by the Euler step size enforced by Algorithm I as used in the current program. We show that Taylor series plus an enclosure of the remainder term can be used not only in Algorithm II to tighten the enclosure, but also in Algorithm I for the validation step. The consequence of this result for AWA is that Algorithm I can validate the existence of the unique solution over time steps appropriate for high-order methods employed for tightening during Algorithm II. The basic idea of AWA is to enclose the Taylor coecients and the remainder term of the solution. It goes back to Moore who presented his method in 1965 (see [1, 11]). The Taylor coecients are computed using recurrence relations derived from the ODE: f () (u) := f(u); (u) := u; f (i) @f (i?1) (u) := @u f (u) ; i = 1; 2; : : :; p? 1 ; (u) i := 1 i f(i?1) (u); i = 1; 2; : : :; p : (2) The recurrences given by Equation (2) are not as intimidating as they might appear; this is just an expression of the computation of the Taylor coecients for the solution. They can be evaluated by using automatic dierentiation (see e.g. [12]). If we begin the recurrences given by Equation (2) with the initial value u and compute in exact arithmetic, we get the Taylor coecients (u) i for the solution expanded at t = t. If we begin the recurrences given by Equation (2) with an interval vector [u] and compute in rounded interval arithmetic, then we get enclosures ([u]) i of the Taylor coecients for all solutions starting from u 2 [u]. We can also begin with a coarse interval [u ] and compute the interval vectors ([u ]) i. Then (b t? t ) p ([u ]) p contains the remainder of the p? 1st degree Taylor polynomial of u(b t), if u(t) 2 [u ](t) for all t 2 [t ;b t]. We emphasize that the computation of these values does not pre-suppose the existence of a solution. We are just computing sequences of numbers. If we cannot compute any of these values, then we are not able to proceed with validation. However, if we know an a priori enclosure [u ] of the solution on [t], then p?1 u(t) 2 (t? t) i (u) i + (t? t) p ([u ]) p =: T p (t; u; [u ]) or p?1 u(t) 2 (t? t) i ([u]) i + (t? t) p ([u ]) p = T p (t; [u]; [u ]), resp. (3)
Validating an A Priori Enclosure 3 for t 2 [t], if only the interval polynomial T p stays in [u ]. Moore as well as Lohner use (modications of) this formula to tighten an a priori enclosure [u ] (see e.g. [13]). 1 Current AWA Approach Validation and the computation of a tight enclosure are separate, though related, issues. This paper addresses only the issue of validation, the role of Lohner's Algorithm I. The techniques of his Algorithm II for tightening the enclosure are discussed in [4, 5, 6, 7, 8]. Algorithm I uses the following theorem for validation. Theorem 1 ([6]) Let [u ] D be an interval vector satisfying [u 1 ] := u + [; h] f? [u ] [u ] : Then u(t) 2 [u 1 ] (and hence u(t) 2 [u ]) for t 2 [t] = [t; t + h]. It follows directly from Theorem 2 Let [u](t) and [v](t) D be interval vector valued functions, and let each p times continuously dierentiable function v(t) 2 [v](t) satisfy Z t (v(t)) := u + for t 2 [t]. Then the solution u of Equation (1) satises for t 2 [t]. t f(v()) d 2 [u](t) [v](t) (4) u(t) 2 [u](t) Proof: If (4) holds and we start a Picard-Lindelof iteration Z t u (k) (t) = (u (k?1) (t)) = u + t f(u (k?1) ()) d ; k = 1; 2; : : : with a p times dierentiable function u () (t) 2 [v](t), then every successor is again p times dierentiable and lies in [u](t). However, the sequence of Picard iterates should leave this interval function if the solution did so.
4 George F. Corliss and Robert Rihm It seems strange to let v(t) be p times dierentiable. However, this assumption is used in the following section. AWA nds a step size h and an interval [u ] such that [u ] [u 1 ] := [u] + [; h]f? [u ]? [u ] ; (5) for all t 2 [t] := [t; t+h], and hence validates that the solution of Equation (1) exists on [t] and is contained in [u ]. Figure 1 shows the a priori bound [u ] = [:2; :5] and ([u ]) for the logistic equation u = u(1? u), u() = :3, on the interval t 2 [; :6]. In this gure, ([u ]) [u ] in the interval t 2 [; :5]. If we tighten [u ] = [:29; :5], then ([u ]) [u ] in the larger interval t 2 [; :563]..6.5.4 u.3.2.1.1.2.3.4.5.6 t Figure 1. A priori bounds with [u ] = [:2; :5]. 2 Lohner's Enclosure of u (p) The allowable step size h for which validation can be done using Theorem 1 as described in Section 1 is limited to a step appropriate for Euler's method, no matter how high the order of the method used during Algorithm II to tighten the enclosure. Stetter showed in [14] that for the problem u = Au, A 2 IR nn, the step size h obtained by applying Theorem 1 may be multiplied by the degree p of the Taylor
Validating an A Priori Enclosure 5 polynomial used in Algorithm II. Algorithm II yields a validated tight enclosure for t = t + ph because p!([u ]) p = A p [u ] contains the p-th derivative u (p) (t) for all t 2 [t; t + ph], even though [u ] does not enclose the solution on this enlarged interval, in general. Lohner [9] suggests using coarse enclosures for higher derivatives also in the general case. Actually, he uses Theorem 2 instead of Theorem 1 and a special kind of interval polynomials for [v](t) instead of a constant coarse enclosure as in the current AWA program. Lohner guesses an enclosure [v ] of u (p) (t)=p! satisfying p?1 [v](t) := (t? t) i (u) i + (t? t) p [v ] D: The Picard-Lindelof operator reproduces the Taylor coecients of the exact solution. There is an interval vector [v 1 ] such that each p times dierentiable function v(t) 2 [v](t) satises p?1 (v(t)) 2 [u](t) := (t? t) i (u) i + (t? t) p [v 1 ]: If [v 1 ] [v ] holds, then we also have [u](t) 2 [v](t), and according to Theorem 2, u(t) 2 [u](t). To obtain [v 1 ], one has to compute an enclosure of the remainder coecient of the p? 2nd degree Taylor polynomial of f(v(t)) for all p times dierentiable functions v(t) 2 [v](t). It also contains the remainder coecient of the p? 1st degree Taylor polynomial of (v(t)). For computing or enclosing these coecients on a computer, one has to apply polynomial interval machine arithmetic given by Eiermann [3], which provides the enclosure of a xed number of Taylor coecients of standard operations and standard functions on interval polynomials as well as the enclosure of the respective remainders. The advantage of this idea of Lohner over the validation strategy discussed in Section 1 is that validation is possible for step sizes appropriate for a method of local order p. Example 1 The solution of the initial value problem u =?u; u() = 1 satises u(t) = e?t 2 [; 1] for t. However, the application of Theorem 1 for [u ] = [; 1] requires a step size 1 since u + [; h]f([u ]) [u ], 1 + [; h](?[; 1]) [; 1], h 1:
6 George F. Corliss and Robert Rihm However, Lohner's approach yields validation for much longer steps. p = 6 and [v ] = [; 1], we get If we choose [v](t) = 1? t + t2 2? t3 6 + t4 24? t5 12 + t6 [; 1]; 72 f([v](t)) =?[v](t)?1 + t? t2 2 + t3 6? t4 24 + t5 12 [1? h ; 1]; and 6 [u](t) = ([v](t)) = 1? t + t2 2? t3 6 + t4 24? t5 12 + t6 72 [1? h 6 ; 1] for t 2 [t; t + h]. We have [1? h 6 ; 1] [v ] = [; 1], h 6 : Of course, we could enclose the sixth Taylor coecient of f instead of the fth one to obtain h = 7. However, in general we only assume f to be p? 1 times dierentiable. In this example, the coecients of f(v(t)) can easily be calculated. However, there is no implementation of the method for the general case yet, although Lohner has described [9] such an implementation using the Eiermann operators for interval polynomial arithmetic. 3 Validated Enclosure Using Taylor Series The use of Taylor series for validation, as well as for tight enclosure, is suggested by pictures such as those in Figure 2, which shows Taylor series enclosures for the solution of the Lorenz system. 2 18 16 14 12 1 8 6 4 2.1.2.3.4.5 t
Validating an A Priori Enclosure 7 3 2 1.1.2.3.4.5 t -1 4 3 2 1.1.2.3.4.5 t Figure 2. Taylor Series Enclosures for the Solution of the Lorenz System. As already mentioned, AWA uses the recurrence relations (2) and the formula (3) for computing a tight enclosure in Algorithm II after having validated a coarse enclosure [u ] for a step size h in Algorithm I. It would cause no additional costs if the Taylor expansion (3) could also be used in Algorithm I. The following theorem shows that this can actually be done. Theorem 3 Let u 2 int([u ]) D. Let for t 2 [t]. Then p?1 [u](t) := (t? t) i (u) i + (t? t) p ([u ]) p [u ] ; (6) u(t) 2 [u](t) for t 2 [t] : Proof: It follows from our assumptions that Equation (1) has a solution u in some neighborhood of the point (t; u), for every u 2 [u]. We must prove that that neighborhood includes all of [t]. If (6) holds, then u(t) 2 [u](t) as long as u(t) 2 [u ].
8 George F. Corliss and Robert Rihm Assume that u(t) leaves [u](t) =: [u(t); u(t)], and hence [u ] at b t 2 int([t ]). Without loss of generality, we assume u j (b t) = uj (b t) = sup[u ]j, for some j 2 f1; : : :; ng. From u (p) (t) 2 p!([u ]) p, it follows that p?2 u (t) 2 (t? t) i (u) i + (t? t) p?1 p([u ]) p ; and hence u (t) u (t) on [t;b t]. Therefore, uj (b t) = uj (b t) implies uj (t) = u j (t) for all t 2 [t;b t]. If uj is constant, then we have u = u j (b t) = sup[u ]j, which contradicts the assumption u 2 int([u ]). Otherwise, u j (b t) is an isolated maximum of the p-th degree polynomial u j. The same holds for u j (t), since this function is at least p times continuously dierentiable in [t]. Hence, u(t) cannot leave [u](t) at b t. This approach avoids complicated calculations and nevertheless takes the order p into account. With p = 6, we can validate the coarse enclosure [; 1] in Example 1 for h = 2:1, compared with h = 1 for a constant bound enclosure and h = 6 using Lohner's polynomial enclosure. Example 2 We want to validate the solution u(t) = 1=t of the initial value problem u =?u 2 ; u(1) = 1 in the interval [u ] = [; 2]. For the constant bound enclosure, we have [u 1 ] := 1 + [; h](?[; 2] 2 ) = [1? 4h; 1] [u ], h 1=4 as the maximal step size we can reach by applying Theorem 1. However, the approach of this section for p = 2 yields a longer step: [u](t) = 1? (t? 1) + [; 8](t? 1) 2 [; 2], h 1 + p 33 16 4 Implications for AWA Using the results of the previous section, AWA Algorithm I is :42 : Task 1: Approximate solution. Compute Taylor coecients of an approximate solution p?1 bu(t) := (t? t) i (u) i from recurrence relation (2). In practice, we must compute (u) i using rounded interval arithmetic to capture any round-o errors. The work of Task 1 is already being done in AWA Algorithm II.
Validating an A Priori Enclosure 9 Task 2: Guess a step h. Use heuristics based on the radius of convergence [1] of the truncated series for bu or on previous steps sizes. Task 3: Guess a constant enclosure [u ]. Bound the range of bu([t; t+h]), and inate it a little. Task 4: Compute the remainder terms ([u ]) i, for i = ; : : : ; p from the recurrence relation (2). The work of Task 4 is already being done in AWA Algorithm II. Task 5: Compute stepsize for validation. The stepsize is the point nearest to t (and in [t]) at which p?1 [u](t) := (t? t) i (u) i + (t? t) p ([u ]) p leaves [u ]. This requires 2 n polynomial rootndings. For each polynomial rootnding problem, it is sucient to compute only relatively coarse lower bounds for the smallest real root in the interval (t; t + h], so a few iterations of a simplied interval Newton method should suce. Remark. Variable order. We must compute ([u ]), : : :, ([u ]) p?1 in order to compute ([u ]) p. Hence, it is relatively inexpensive to compute the step size in Task 5 for each order p. The order used for validation in Algorithm I can be dierent from the order used for tightening in Algorithm II. The information is readily available to make both Algorithms I and II fully variable order. Remark. Iteration. If Task 5 nds that [u](t) [u ] on [t], then we can repeat Tasks 4 and 5 with a larger h or else with [u ] = [u]([t]). Remark. Optimization. Given guesses for h from Task 2 and for [u ] from Task 3, we compute in Task 5 the largest step for which we can validate existence and containment. Numerical experiments have shown [2] that a carefully chosen [u ] allows step sizes 1 times as long as step sizes corresponding to [u ] given by reasonable heuristics. That suggests viewing Algorithm I as an optimization problem: Maximize step size h by varying guessed h and [u ] p?1 subject to [u](t) := (t? t) i (u) i + (t? t) p ([u ]) p [u ] Remark. Eectiveness. Either Lohner's enclosure by polynomials or our enclosure by Taylor series is signicantly more expensive than Picard-Lindelof iteration of constant bounds currently used by AWA. However, AWA Algorithm II involves the solution of an ODE system of dimension n n, so almost any eort we expend in Algorithm I that allows AWA to take longer steps improves the overall eciency of AWA.
1 George F. Corliss and Robert Rihm Remark. Which is better? For Lohner's enclosure by polynomials, the enclosures are expensive to compute, but it is easy to check for enclosure. For our enclosure by Taylor series, the enclosures are free because they are already being computed, but checking for enclosure requires 2n polynomial rootndings. Work is continuing on an implementation that will allow direct computational comparisons of the eectiveness of the two methods. Acknowledgments We thank the anonymous referee for helpful suggestions. The work of Corliss was supported in part by National Science Foundation Grant No. DMS{9413525 and in part by SUN Academic Equipment Grant No. EDUD{US{9428. References [1] Y. F. Chang and G. Corliss, Solving Ordinary Dierential Equations Using Taylor Series, ACM Trans. Math. Software, 8(1982), 114{144. [2] G. Corliss, Validating an A Priori Enclosure Using High-Order Taylor Series, presented at SCAN '95, Wuppertal, 1995. [3] M. Ch. Eiermann, Adaptive Berechnung von Integraltransformationen mit Fehlerschranken, PhD thesis, Universitat Freiburg, 1989. [4] R. J. Lohner, Anfangswertaufgaben im IR n mit kompakten Mengen fur Anfangswerte und Parameter, Diplomarbeit, Inst. f. Angew. Math., Universitat Karlsruhe, 1978. [5] R. J. Lohner, Enclosing the Solutions of Ordinary Initial and Boundary Value Problems, in Computer Arithmetic: Scientic Computation and Programming Languages, E. W. Kaucher, U. Kulisch, and C. Ullrich, eds., Wiley-Teubner Series in Computer Science, Stuttgart, 1987, pp. 255{286. [6] R. J. Lohner, Einschlieung der Losung gewohnlicher Anfangs{ und Randwertaufgaben und Anwendungen, PhD thesis, Universitat Karlsruhe, 1988. [7] R. J. Lohner, Interval Arithmetic in Staggered Correction Format, in Scientic Computing with Automatic Result Verication, E. Adams and U. Kulisch, eds., Academic Press, San Diego, 1993. [8] R. J. Lohner, On Step Size and Order Control in the Veried Solution of Ordinary Initial Value Problems, presented at SCAN '93, September 1993, Vienna.
Validating an A Priori Enclosure 11 [9] R. J. Lohner, Step Size and Order Control in the Veried Solution of IVP with ODE's, presented at the SciCADE '95 International Conference on Scientic Computation and Dierential Equations, Stanford, Calif., March 28 { April 1, 1995. [1] R. E. Moore, The Automatic Analysis and Control of Error in Digital Computation Based on the Use of Interval Numbers, in Error in Digital Computation, Vol. I, L. B. Rall, ed., John Wiley and Sons, New York, 1965, pp. 61{13. [11] R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Clis, N.J., 1966. [12] L. B. Rall, Automatic Dierentiation: Techniques and Applications, vol. 12 of Lecture Notes in Computer Science, Springer Verlag, Berlin, 1981. [13] R. Rihm, Interval Methods for Initial Value Problems in ODEs, in Topics in Validated Computations, J. Herzberger, ed., Elsevier Science B.V., Amsterdam, 1994, pp. 173{27. [14] H. J. Stetter, Validated Solution of Initial Value Problems for ODE, in Computer Arithmetic and Self-Validating Numerical Methods Ch. Ullrich, ed., Academic Press, San Diego, 199, pp. 171{186. Addresses: G.F. Corliss, Marquette University, Department of Mathematics, Statistics, and Computer Science, P.O. Box 1881 Milwaukee, Wisconsin 5321-1881, USA. georgec@mscs.mu.edu R. Rihm, Universitat Karlsruhe, Institut fur Angewandte Mathematik, 76128 Karlsruhe, Germany. robert.rihm@math.uni-karlsruhe.de