ing Euler's method, based on the domain of interval valued functions of a real variable. In contrast to other interval based techniques, the actual computation.
A Domain Theoretic Account of Euler’s Method for Solving Initial Value Problems Abbas Edalat, Dirk Pattinson Department of Computing, Imperial College, London, UK
Abstract. This paper presents a method of solving initial value problems using Euler’s method, based on the domain of interval valued functions of a real variable. In contrast to other interval based techniques, the actual computation of enclosures to the solution is not based on the code list (term representation) of the vector field that defines the equation, but assumes instead that the vector field is approximated to an arbitrary degree of accuracy. By using approximations defined over rational or dyadic numbers, we obtain proper data types for approximating both the vector field and the solution. As a consequence, we can guarantee the speed of convergence also for an implementation of the method. Furthermore, we give estimates on the algebraic complexity for computing approximate solutions.
1 Introduction We consider the initial value problem given by the system of differential equations y˙ i = vi (y), yi (0) = 0 (i = 1, . . . , n) (1) where the vector field v : [−K, K]n → [−M, M ]n is continuous in a rectangle containing the origin. Our goal is to compute a a differentiable function y = (y1 , . . . , yn ) : [0, a] → Rn which satisfies (1), up to any given degree of accuracy. Standard numerical packages usually compute approximations to a solution with good precision, but there is no guarantee on the correctness of the computed values; indeed it is easy to find examples where they output inaccurate results [9]. Interval Analysis [13,12] provides a method for computing guaranteed upper and lower enclosures of the solution of initial value problems, see [3,15] and the references therein for a survey of interval techniques. In the approach of interval analysis, real numbers are represented as intervals which are guaranteed to contain the number, and outward rounding is applied if the result of an arithmetic operation is not machine representable. For many practical applications, these methods produce good enclosures, but one has no control over the rounding, which can produce unduly large intervals. As
a consequence, implementations of interval methods are not guaranteed to produce approximations which actually converge to a solution of the problem, or satisfy an a priori estimate on the actual convergence speed. These questions are addressed in the present paper using the framework of domain theory [1,8]. Based on the domain of interval valued functions of a real variable, we construct enclosures of the solutions of an IVP with an a priori guaranteed width. Moreover, our construction gives rise to proper data types, which can be directly implemented on a digital computer. This allows us to guarantee the speed of convergence also for an implementation. The main difference to standard interval techniques is that our approach is not based on the code list (term representation) of the vector field, but instead on a sequence of successively finer approximations to the vector field. Using these approximations, we obtain enclosures of the solution of the problem, which are then shown to converge to solution of the problem. As the approximations of the vector field can be defined using rational (or dyadic) numbers, no loss of precision is incurred, and we can therefore guarantee the convergence speed also for an implementation of our techniques. In this paper, we use an interval Euler technique to construct approximating enclosures to the solution of an IVP. The representation of the vector field that defines the IVP as a limit of approximations requires us to verify that our construction satisfies a continuity property w.r.t. the vector field, where we use the Scott topology (formally introduced in the next section) to topologise the space of interval vector fields. The main contributions of the paper are (i) to show that we can compute arbitrary tight enclosures of the solution using approximations of the vector field, and (ii) to show that these computations can be carried out on data types, defined over the rational or dyadic numbers. Furthermore, we give an estimate on the speed of convergence to the solution and an estimate of the algebraic complexity of computing approximations for two different realisations of Euler’s technique. Of course, this paper is only a first investigation into using domain theoretic techniques for differential equations, and much remains to be done. Most pressingly, we have to incorporate information on the derivatives of the vector field and a good stepsize control. Plan of the paper: We recall basic notions from domain theory in Section 2, and introduce two realisations of Euler’s technique in Section 3, which are shown to produce approximations to the solution of the problem in the subsequent section. We then add approximations of the vector field that defines the IVP(Section 5), and give an estimate on the speed of convergence of our method. Section 7 shows how our techniques can actually be implemented on a digital 2
computer and gives the promised estimates on the algebraic complexity. Finally, the last section puts our results into perspective with related research.
2 Preliminaries and Notation First note that the continuity assumption on v entails that v attains its maximum, and we can therefore restrict the range of v to [−M, M ]n without loss of generality. For the expression (1) to be well defined, we make the standard assumption aM ≤ K. Geometrically, M is a bound on the derivative of yi for each i = 1, . . . , n and our assumption therefore guarantees that each yi (x) ∈ [−K, K] as long as 0 ≤ x ≤ a. In this paper, we give an account of Euler’s method for solving initial value problems in the framework of domain theory [8,1]. Our investigations are based on the interval domain (IR, ⊑) where IR = {[a, a] | a, a ∈ R} ∪ {R} ordered by reverse inclusion, i.e. α ⊑ β if β ⊆ α. For a compact interval R ⊆ R, the sub-domain of compact intervals [a, a] ⊆ R is denoted by IR with inherited order relation. Note that both IR and IR, for R ⊆ R a compact interval, are directed F complete: For a directed set D T ⊆ IR of intervals, the least upper bound D always exists and is given by D. In interval terms, suprema of a directed subset of IR is equivalent to Moore’s principle of nested convergence of [12, Chapter 4]. The order on an arbitrary directed complete partial order (dcpo, for short) (D, ⊑) induces a topology on D, the so called Scott topology: We call a set o ⊆ D open, if (i) it is upward closed, i.e. d ∈ o and d ⊑ e implies e ∈ o F (ii) it is inaccessible by directed suprema, i.e. if A ⊆ D is directed and A ∈ o, then a ∈ o for some a ∈ A. In the case of the interval domain IR, it can easily be seen that convergence in the Scott topology implies convergence in the metric topology, used by Moore [13], but not vice versa. In the sequel of the paper, we always consider a dcpo, or a space of intervals, as equipped with the Scott topology. Given an arbitrary set X, every function f : X → IR can be represented by a pair (f , f ) representing the upper and the lower interval boundary of f , that is, f (x) = [f (x), f (x)] for all x ∈ X. We write this as f = [f , f ]. We often make use of the following crucial fact, which directly follows from the definition of the topology:
3
Fact 1. Suppose f = [f , f ] : R → IR is continuous w.r.t the Euclidean topology on R. Then f is Scott continuous iff f is lower and f is upper semi continuous. If the domain of a function is also a dcpo (topologised with the Scott topology), we have the following alternative characterisation of continuity: Fact 2. Suppose (D, ⊑) F and (E, ⊑) are dcpos. Then a monotone function f : F D → E is continuous iff a∈A f (a) = f ( A) for all directed A ⊆ D.
We also note that the space X ⇒ D of continuous functions of type X → D, for a topological space X and a dcpo D, is again a dcpo in the pointwise order: given f, g : X → D, we put f ⊑ g if f (x) ⊑ g(x) for all x ∈ X. Hence we can view the space of continuous functions X → D as a topological space w.r.t. the Scott topology on X ⇒ D. In case X = {1, . . . , n} with the discrete topology, we write D n for X ⇒ D and obtain the n-fold cartesian product of the dcpo D with itself. In the special case D ⊆ IR is a sub-dcpo, D n is canonically isomorphic to the dcpo of n-dimensional compact rectangles, and we will use this isomorphism without further mention. For the purpose of solving initial value problems, the following spaces of functions are of particular interest: (i) The space S = [0, a] ⇒ I[−K, K]n (with the Euclidean topology on [0, a]) for constructing solutions of (1) (ii) The space V = I[−K, K]n ⇒ I[−M, M ]n of interval vector fields.
We use the standard notion of width to measure the quality of an approximation. Given α = ([a0 , a0 ], . . . [an , an ]) ∈ IRn , we put w(α) = max{ai − ai | 1 ≤ i ≤ n}, and for a function f : X → IRn we let w(f ) = sup{w(f (x)) | x ∈ X}. The relation between vector fields in the classical sense and interval vector fields is given by the notion of extension: we say that u ∈ V extends v : [−K, K]n → [−M, M ]n if u({x}) = {v(x)} for all x ∈ [−K, K]n . In the sequel, we assume that u ∈ V is an extension of the classical vector field v. Note that every continuous v = (v1 , . . . , vn ) : [−K, K]n → [−M, M ] has an extension, the canonical extension canv whose i-th component is given by I[−K, K]n ∋ α 7→ {vi (x) | x ∈ α}. We emphasise that our development does not require us to work with the canonical extension of the classical vector field v. Finally, we introduce integrals of interval valued functions, which we use in the construction of solutions of the IVP. Suppose p ≤ q Rand f : [p,Rq] → IR. Rq q q Then the integral of f = [f , f ] is defined as p f (t)dt = [ p f (t)dt, p f (t)dt]. 4
The existence of the integrals follows from lower (resp. upper) semi R q continuity of f (resp. f ). If f = (f1 , . . . , fn ) : [p, q] → IR, we let p f (t)dt = Rq Rq ( p f1 (t)dt, . . . , p fn (t)dt). The following property follows easily from the monotone convergence theorem: Rq Fact 3. The integration operator p : ([p, q] ⇒ IR) → ([p, q] ⇒ IR), defined Rx by f 7→ λx. p f (t)dt, is monotone and continuous.
3 Euler’s Operator in Domain Theory
The formalisation of Euler’s method for solving initial value problems relies on the notion of partitions of the interval [0, a]: Definition 1 (Partitions). (i) A partition of [0, a] is a finite sequence (q0 , . . . , qk ) of real numbers 0 = q0 < · · · < qk = a; the set of partitions of [0, a] is denoted by P. (ii) The norm |Q| of a partition Q = (q0 , . . . , qk ) is given by |Q| = max1≤i≤k qi − qi−1 and its minimal width is m(Q) = min1≤i≤k qi − qi−1 . We let rQ = |Q|/m(Q) denote the ratio of minimal and maximal distance between two partition points. (iii) A partition Q = (q0 , . . . , qk ) refines a partition P = (p0 , . . . , pl ) if {p0 , . . . , pl } ⊆ {q0 , . . . , qk }; this is denoted by P ⊑ Q. We can now define and study Euler’s Operator for approximating the solutions of initial value problems for an arbitrary interval valued vector field. In fact, we introduce two such operators, the first having better convergence properties whereas computing with the second will be more efficient, as we shall demonstrate later. For the remainder of the section, we fix an extension u : I[−K, K]n → I[−M, M ]n of the classical vector field v. Notationally, if α = ([a1 , a1 ], . . . , [an , an ]) ∈ IRn and r ∈ R, we write α ⊕ r = ([a1 − r, a1 + r], . . . , [an − r, an + r]) for the symmetric expansion of the interval vector α with the real constant r. Definition 2. Suppose Q = (q0 , . . . , qn ) ∈ P. Then the Euler operator with linear expansion E l : P × V → [0, a] ⇒ I[−K, K]n is defined by ( (0, . . . , 0) x=0 l Rx Eu (Q)(x) = l l Eu (Q)(qi ) + qi u(Eu (Q)(qi ) ⊕ (x − qi )M )dt qi ≤ x ≤ qi+1
for x ∈ [0, a]. The Euler operator with constant expansion is given similarly by ( (0, . . . , 0) x=0 c R Eu (Q)(x) = x Euc (Q)(qi ) + qi u(Euc (Q)(qi ) ⊕ ∆qi M )dt qi ≤ x ≤ qi+1 5
where ∆qi = qi+1 − qi . In the sequel, E stands for either E l or E c . The operator with constant expansion represents is an interval version of Euler’s method for constructing solutions of differential equations, as described by Moore [13]. An equivalent definition could also be given without the use of integration. However, the definition we gave allows us to treat both operators in the same framework, and therefore enables us to use the same proof techniques for both. In the remainder of the paper, we show that if u is an extension of a classical vector field, which satisfies a Lipschitz property, then Eu (Q) approximates the solution of the IVP (1) to an arbitrary degree of accuracy as the width of the partition Q tends to zero. We then add approximations to the vector field u, and show that this preserves convergence to the solution. As the approximations can be faithfully represented using rational or dyadic numbers, we obtain proper data types for computing solutions, and can actually guarantee the speed of convergence for an implementation. We continue by collecting some simple properties of Eu . Lemma 4. If f = [f , f ] = Eu (Q)i is the i-th component of Eu (Q), then both f and f satisfy a Lipschitz condition with Lipschitz constant M . In particular, Eu (Q)(x) ∈ I[−K, K]n for all x ∈ [0, a]. The is immediate by boundedness of u, and entails that both Eul (Q) and Euc (Q) are Scott continuous. Corollary 5. Eu (Q) is Scott continuous. Proof. Follows from Lemma 4 and Fact 1. Note that the last two statements in particular entail that both Euler operators are actually well defined. The remainder of this section is devoted to the proof of monotonicity of Eu , i.e. Eu (P ) ⊑ Eu (Q) if P ⊑ Q. Intuitively, this assures that the information on the solution of an IVP increases as we move to a finer partition. We need one technical lemma for the proof of monotonicity. Lemma 6. Suppose f = [f , f ] : [c, d] → IR and both f , f satisfy a Lipschitz condition with Lipschitz constant M . Then f (x) ⊕ |y − x|M ⊑ f (y) for all x, y ∈ [c, d]. In particular, Eu (Q)(x) ⊕ |y − x|M ⊑ Eu (Q)(y). Proof. From Lipschitz continuity of f , we obtain that f (x) + |y − x|M ≤ f (y) and similarly f (x) − |y − x|M ≥ f (y), hence f (x) ⊕ |y − x|M ⊑ f (y). The claim on Eu (P ) now follows from Lemma 4. 6
We can now continue and prove the monotonicity of Eu . Proposition 7. Suppose P ⊑ Q ∈ P. Then Eu (P ) ⊑ Eu (Q). Proof. We only consider the case of E l ; the case of E c is exactly analogous. Suppose P = (p0 , . . . , pk ) and Q = (q0 , . . . , ql ). We show that Eul (P ) ↾ [0, qi ] ⊑ Eul (Q)↾ [0, qi ] for all i = 0, . . . , l by induction on i. The case i = 0 is trivial. To show the statement for i + 1, let j = max{0 ≤ m < k | pm ≤ qj }. For x ∈ [qi , qi+1 ], we obtain Z x l l Eu (P )(x) = Eu (P )(qi ) + u(Eul (P )(pj ) ⊕ (t − pi )M )dt qi
by additivity of integrals. Using monotonicity of u, it therefore follows from the definition of Eul (Q) that it is sufficient to show that Eul (P )(pj ) ⊕ (x − pj )M ⊑ Eul (Q)(qi ) ⊕ (t − qi )M for all t ∈ (qi , qi+1 ]. Using Lemma 6, this follows from calculating Eul (P )(pj ) ⊕ (x − pj )M = Eul (P )(pj ) ⊕ (qi − pj )M ⊕ (x − qi )M ⊑ Eul (P )(qi ) ⊕ (x − qi )M
(by Lemma 6)
⊑ Eul (Q)(qi ) ⊕ (x − qi )M
(by Ind’n Hyp)
which concludes the proof. Informally, the last proposition states that a finer partition provides us with a better approximation to the solution. In the following section, we show that – in presence of a Lipschitz condition on the vector field – the approximations actually converge to the unique solution of the IVP.
4 Convergence to the Solution of the Classical Problem In this section, we show that the approximations of the solutions computed by applying the Euler operator to increasingly fine partitions, we obtain a solution for the classical problem. The proof of convergence to the solution is split into two parts: We first show that the approximations are enclosures of all possible solutions and then – in presence of a Lipschitz condition – establish that the approximations converge to a real valued function. We begin by showing soundness of our method, i.e. our approximations are enclosures of all possible solutions. If z : [0, a] → Rn is a real valued function, we write Iz : [0, a] → IRn for its trivial extension Iz(x) = {z(x)}. We follow the convention of the previous section and use Eu to stand for both Euc or Eul . 7
Proposition 8. Suppose P is a partition and z : [0, a] → [−K, K]n is a solution of the IVP (1). Then Eu (Q) ⊑ Iz. Proof. Suppose Q = (q0 , . . . , qk ) ∈ P, we show that that Eu (Q)↾ [0, qi ] ⊑ Iz↾ [0, qi ] for all i ≤ n by induction on i. For i = 0, this is trivial. To obtain the claim for i + 1, we calculate Z x (t − qi ) M )dt u(Eu (Q)(qi ) ⊕ Eu (Q)(x) = Eu (Q)(qi ) + ∆qi qi Z x ⊑ Iz(qi ) + u(Iz(t))dt qi
= Iz(x)
where we have used monotonicity of u and the fact that z is a solution of (1) that satisfies a Lipschitz condition with Lipschitz constant M . As immediate corollary, we obtain that – in case the approximations converge to a real valued function – that this function is a solution of the problem. F Corollary 9. Suppose P is a directed set of partitions and y = [y, y] = Q∈P Eu (Q) and y − = y + . Then y − = y + is a solution of (1). Proof. By the previous proposition, as Peano’s theorem [2] guarantees that there is at least one solution. We therefore have to investigate conditions, which ensure that the approximations converge to a real valued function. As in the classical theory, this is achieved by imposing a Lipschitz condition on the vector field. In an interval setting, this translates to the following property of interval extensions: Definition 3 (Interval Lipschitz Condition). The function u : I[−K, K]n → I[−M, M ]n satisfies an interval Lipschitz condition with Lipschitz constant L if w(u(α)) ≤ L · w(α) for all α ∈ I[−K, K]n . Note that if u is interval Lipschitz, then it maps real numbers to real numbers, hence it is an extension of a classical function. The following proposition (due to Moore [13]) shows that this is in accordance with assuming a Lipschitz condition on the classical vector field v. Proposition 10. The following are equivalent: (i) v satisfies a Lipschitz condition. (ii) the canonical extension of v satisfies an interval Lipschitz condition 8
(iii) v has an extension that satisfies an interval Lipschitz condition. Proof. We only show (iii) =⇒ (i) as all other claims are trivial. So assume that u = (u1 , . . . , un ) is an extension of v = (v1 , . . . , vn ) with interval Lipschitz constant L and suppose x, y ∈ [−K, K]n . Let R = x ⊓ y denote the greatest lower bound of x and y, i.e. the smallest rectangle containing both x and y. Since u extends v, we have both v(x), v(y) ∈ u(x ⊓ y) by monotonicity of u, hence ||v(x)−v(y)|| ≤ w(v(x⊓y)) = L||x−y||, where ||·|| is the maximum norm. For the rest of the paper, we assume that u is an interval Lipschitz extension of a classical vector field v : [−K, K]n → [−M, M ]n with interval Lipschitz constant L. We begin with an auxiliary lemma which helps to show that in this case, the approximations converge to a real valued function. In particular, this lemma also shows that E l has better convergence properties than E c . Lemma 11. Suppose Q = (q0 , . . . , qn ) is a partition. Then w(Eu∗ (Q)(x)) ≤ w(Eu∗ (Q)(qi ))(1 + |Q| · L) + C|Q|2 LM for x ∈ [qi , qi+1 ], where C = 1 for ∗ = l and C = 2 for ∗ = c. Proof. We only consider Eul ; the estimate for Euc is similar. We abbreviate y = Eul (Q). Suppose x ∈ [qi , qi+1 ]. We calculate Z x u(y(qi ) ⊕ (t − qi )M )dt w(y(x)) = w(y(qi ) + qi Z x = w(y(qi )) + w(u(y(qi ) ⊕ (t − qi )M )dt qi Z x L · w(y(qi ) ⊕ (t − qi )M )dt ≤ w(y(qi )) + qi
≤ w(y(qi )) + ∆qi · L · (w(y(qi )) + ∆qi M ) = w(y(qi ))(1 + ∆qi L) + (∆qi )2 LM ≤ w(y(qi ))(1 + |Q|L) + |Q|2 LM
where ∆qi = qi+1 − qi . Based on the previous lemma, we can now give an estimate on the speed of convergence; recall from Definition 1 that rQ is the ratio of the largest and smallest distance between two partition points. 9
Proposition 12. Suppose P is a partition of [0, a]. Then w(Eu∗ (P )) ≤ C · |Q|M (eaLrQ − 1) where C = 1 for ∗ = l and C = 2 for ∗ = c. Proof. Again, we only treat the case ∗ = l; the case ∗ = c is completely analogous. Assume Q = (q0 , . . . , qk ) and put c = |Q|2 LM and d = 1 + |Q|L. Using the previous lemma, we claim that w(Eul (Q)(x) ≤ c(1 + · · · + di ) for x ∈ [qi , qi+1 ]: For i = 0 this is the statement of Lemma 11. For i > 0, it is a matter of calculating w(Eul (Q)(x) ≤ w(Eul (Q)(qi ) · d + c = c(1 + · · · + di−1 ) · d + c
(Lemma 11) (Ind’n Hyp.)
i
= c(1 + · · · + d ). We now obtain w(Eul (Q)) ≤ c(1 + · · · + dk−1 ) dn − 1 =c d−1 c = ((1 + |Q|L)n − 1) |Q|L = |Q|M ((1 + LrQ m(Q))n − 1) arQ L n ≤ |Q|M ((1 + ) − 1) n ≤ |Q|M (earQ L − 1) as required. The estimates for both arbitrary and equally spaced partitions are now straightforward corollaries. The first statement treats the case of equidistant partitions, that is partitions Q = (q0 , . . . , qk ) where qi − qi−1 = qj = qj−1 for all 1 ≤ i, j ≤ k. Corollary 13 (Speed of Convergence). (i) If Q is equidistant, then w(Eu∗ (Q)) ≤ C · |Q|M (eaL − 1). (ii) If Q is arbitrary, then w(Eu∗ (Q)) ≤ 2C|Q|M · (e4aL − 1). where, in both cases, C = 1 for ∗ = l and C = 2 for ∗ = c. 10
Proof. Again, we only treat Eul . The first claim is immediate. For the second claim, suppose Q is a partition of [0, a]. By deleting appropriate partition points of Q, we can find a partition P ⊑ Q s.t. |P | ≤ 2|Q| and m(P ) ≥ |Q|/2. Using Lemma 7, we obtain w(Eul (Q)) ≤ w(Eul (P )) ≤ M |Q|(e4aL − 1). We can now collect our observations about approximating the solution of the IVP (1) as follows: Theorem 14. Suppose (Qn )F n∈N is an increasing sequence of partitions with limn→∞ |Qn | = 0 and y = n∈N Eu (Qn ). Then w(y) = 0 and y − = y + is a solution of (1).
5 Approximation of the Vector Field In the previous section, we have seen that we can compute the solution of an initial value problem in domain theory if we are given an interval extension u : I[−K, K]n → I[−M, M ] of the vector field v : [−K, K]n → [−M, M ]n . For computational purposes, we cannot expect to have a an exact interval extension of the vector field: it is rather the case that one can approximate the values of the vector field up to an arbitrary degree of accuracy. In this section, we show that Euler’s operator Eu is continuous in u, which will allow us to use approximations of the vector field for computing the solution of the IVP up to an arbitrary degree of accuracy. These results are then used in the following section to present and analyse a framework for implementing the domain theoretic framework. This section presents the main difference to the traditional interval based approach to solving ODEs: instead of assuming that the vector field is given as a term involving certain basic functions like addition and sine, we assume that the vector field is given as a supremum of simple functions, taking only finitely many values. As we will be made precise in the next section, the use of simple functions allows us to compute without loss of accuracy, and we can therefore guarantee the convergence also for an implementation of the method. We follow the convention of the precious section and use Eu to stand for both Eul or Euc and begin with showing that Eu is monotone in u. Lemma 15. Suppose u1 , u2 : I[−K, K]n → I[−M, M ]n with u1 ⊑ u2 . Then Eu1 ⊑ Eu2 , i.e. Eu1 (Q) ⊑ Eu2 (Q) for all partitions Q. Proof. We fix an arbitrary partition Q = (q0 , . . . , qk ) and show that Eu1 (Q)↾ [0, qi ] ⊑ Eu2 (Q)↾ [0, qi ] for all i ≤ k by induction on i. For i = 0 this is obvious from the definition of Eu . For i > 0 we have Eu1 (Q)(qi ) ⊑ Eu2 (Q)(qi ) by 11
induction hypothesis, hence Eu1 (Q)(qi ) ⊕ tM ⊑ Eu2 (Q)(qi ) ⊕ tM for all t ∈ R. The claim now follows from monotonicity of u and monotonicity of integration (Fact 3). This means that if both u1 , u2 ⊑ u and u2 contains more information than u1 , i.e. u1 ⊑ u2 , the operator Eu2 associated with u2 produces better approximations of the solution than the operator associated with u1 . We now continue to show that Eu is actually continuous in u, that is, we can replace u by a sequence of approximations for the purpose of actual computations. Proposition 16. Suppose (uj )j∈J is F a directed collection of F vector fields uj : I[−K, K]n → I[−M, M ]n with u = j∈J uj . Then Eu = j∈J Euj .
l Proof. Again, we only consider fix a partition Q = (q0 , . . . , qk ) and F E . We l show that Eu (Q) ↾ [0, qi ] = j∈J Eul j (Q) ↾ [0, qi ] by induction on i. Again for i = 0 thisFis trivial, so assume 0 < i. By induction hypothesis F we have Eul (Q)(qi ) = j∈J Eul j (Q)(qi ), hence Eul (Q)(qi )⊕(x−qi )M = j∈J Euj (Q)(qi )⊕ (x − qi )M . Using the interchange-of-suprema law [1, Proposition 2.1.12], we now calculate for x ∈ [qi , qi+1 ] that G Eul j (Q)(x) i
=
Eul j (Q)(qi ) +
Z
G
Eul j (Q)(qi ) +
GZ
j∈J
=
x
G
i
j∈J
=Eul (Q)(qi ) + =Eul (Q)(qi ) + =Eul (Q)(qi ) +
qi
Z
x
G
(
qi j∈J x
qi
x qi
uj (Eul j (Q)(qi ) ⊕ (t − qi )M dt
(uj (Eul j (Q)(qi ) ⊕ (t − qi )M )dt
qi j∈J Z x G
Z
ui (Eul j (Q)(qi ) ⊕ (t − qi )M dt
uj )(
G
(Continuity of Addition) (Continuity of Integration)
Eul j (Q)(qi ) ⊕ (t − qi )M )dt (Interchange of Suprema)
j∈J
u(Eul (Q)(qi ) ⊕ (t − qi )M dt
(as seen above)
=Eul (Q)(qi )(x)
which shows the claim. Continuity of Eu in u allows us to approximate u by a sequence of functions (un ) while still obtaining a sequence of functions approximating the solution of the IVP. We note this as corollary before establishing an estimate for the speed of convergence which incorporates the approximations for u. 12
F Corollary 17. Suppose (un )n∈N is a sequence in V with u = F n∈N un and (Qn )n∈N is a sequence of partitions with limn→∞ |Qn | = 0. Then n∈N Eun (Qn ) is real valued an satisfies (1). Proof. By the interchange-of-suprema law [1, Prop. 2.1.12] from the last Proposition. In presence of approximations un of u, the speed of convergence will clearly depend on the speed of convergence of the un to u. The next definition introduces the measure which we use to express the convergence rate of un to u. Definition 4. If α = (α1 , . . . , αn ) and β = (β1 , . . . , βn ) ∈ IRn , we let d(α, β) = max{|ai − bi |, |ai − bi |, i = 1, . . . , n} where αi = [ai , ai ] and βi = [bi , bi ]. For u, u′ ∈ V, we put d(u, u′ ) = supα∈I[−K,K]n d(u(α), u′ (α)). Note that d(α, β) is the interval distance already used by Moore [13]. The following lemma is the key for obtaining a result on the speed of convergence in presence of approximations of the vector field. Lemma 18. Suppose u′ ⊑ u and Q ∈ P. Then w(Eu∗′ (Q)(x)) ≤ w(Eu∗′ (Q)(pi ))(1 + |Q| · L) + C|Q|2 LM + |Q|d(u, u′ ) for x ∈ [qi , qi+1 ], where C = 1 for ∗ = l and C = 2 for ∗ = c. Proof. We use Lemma 11 and obtain the following estimate for the case of Eul (Q) and x ∈ [qi , qi+1 ]: Z x l l w(Eu′ (Q)) = w(Eu′ (Q)(qi ) + w( u′ (Eul ′ (Q)(qi ) ⊕ (t − qi )M )dt) q Z xi ≤ w(Euc ′ (Q)(qi )) + u(Eul ′ (Q)(qi ) ⊕ (t − qi )M ) + d(u, u′ )dt qi
≤ w(Eu′ (Q)(qi ))(1 + |Q|L) + |Q|2 LM + |Q|d(u, u′ ).
The estimate for Euc is completely analogous. As in the previous section, this lemma allows to give an estimate for the speed of convergence in presence of approximations of the vector field. Proposition 19. Suppose Q is a partition of [0, a] and u′ ⊑ u. Then w(Eu∗′ (Q)) ≤ C · (|Q|M +
d(u, u′ ) aLrQ )(e − 1) L
where C = 1 for ∗ = l and C = 2 for ∗ = c. 13
Proof. As in the proof of Proposition 12, we have that w(Euc ′ (x) ≤ c(1 + · · · + di ) for x ∈ [qi , qi+1 ], where c = |Q|2 LM + |Q|d(u, u′ ) and d = 1 + |Q|L, and the remaining steps are similar to the proof of Proposition 12. In summary, adding approximations of the vector field when computing solutions of the IVP does not destroy the rate of convergence, as long as the approximations to the vector field converge at the same rate as the norm of the partitions tend to zero. This is the content of the following theorem, which summarises the findings of this section. Theorem 20. Suppose (Qn )n∈N is F a monotone sequence of partitions of [0, a] with limn→∞ |Qn | = 0 and u = n∈N un with d(u, un ) ≤ C0 · |Qn | for some constant C0 ≥ 0. Putting yn = Eun (Qn ), we have that 1. F w(yn ) ≤ C1 · |Qn | for some constant C1 ≥ 0 2. n∈N yn is real valued and solves (1).
In the next section, we show how we can use information on the derivative of the vector field to obtain solutions of the initial value problem, before we describe how our methods can be implemented in Section 7.
6 A Second Order Method We now assume that the vector field v is not only continuous, but continuously differentiable, and show, how we can use the derivative information in order to improve the width of the approximations. In contrast to the first order methods, described in Section 3, we replace the (global) Lipschitz constant, used to give a conservative estimate on the solution between two partition points, locally with the derivative of the vector field, giving rise to tighter approximations. In order to account for the derivative information, we need to replace the domain V = I[−K, K]n ⇒ I[−M, M ]n , which we have used to approximate continuous vector fields, with the domain V 1 of differentiable vector fields, which we now briefly describe. Suppose that f : I[−K, K]n → I[−M, M ] is Scott-continuous. We recall from [6] (see also [5] for the case of one variable) that f has an interval Lipschitz constant b ∈ IRn in a ∈ I[−K, K]n (denoted f ∈ δ(a, b)), if f (x) − f (y) ⊑ b(x − y) for all x, y ⊆ ao contained in the interior of a. This allows us to define the domain theoretic derivative of f by putting G df = aցb dx f ∈δ(a,b)
14
where a ց b(x) = a if x
