VALIDATED CONTINUATION FOR EQUILIBRIA OF PDES
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- Osborn Rodgers
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1 VALIDATED CONTINUATION FOR EQUILIBRIA OF PDES SARAH DAY, JEAN-PHILIPPE LESSARD, AND KONSTANTIN MISCHAIKOW Abstract. One of the most efficient methods for determining the equiibria of a continuous parameterized famiy of differentia equations is to use predictor-corrector continuation techniques. In the case of partia differentia equations this procedure must be appied to some finite dimensiona approximation which of course raises the question of the vaidity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numericay produced equiibrium for the finite dimensiona system can be used to expicity define a set which contains a unique equiibrium for the infinite dimensiona partia differentia equation. Using the Cahn-Hiiard and Swift-Hohenberg equations as modes we demonstrate that the cost of this new vaidated continuation is ess than twice the cost of the standard continuation method aone. 1. Introduction. The first step in understanding the dynamics of a noninear system of differentia equations u t = f(u, ν) (1.1) on a Hibert space is to identify the set of equiibria E := {(u, ν) f(u, ν) = 0}. For many appications this can ony be done using numerica methods. In particuar, continuation provides an efficient technique for determining eements on branches of E. Reca, that this method invoves a predictor and corrector step: given, within a prescribed toerance, an equiibrium u 0 at parameter vaue ν 0, the predictor step produces an approximate equiibrium ũ 1 at nearby parameter vaue ν 1, and the corrector step, often based on a Newton-ie operator, taes ũ 1 as its input and produces, once again within the prescribed toerance, an equiibrium u 1 at ν 1. With any numerica method there is the question of vaidity of the output as compared with the cost of computation. The goa of this paper is to argue that for a arge and important cass of partia differentia equations the cost of vaidating the existence and uniqueness of equiibria is sma when compared to the cost of identifying potentia equiibria by means of a continuation method. Our interest in this question was motivated by the increasing deveopment of computer-assisted proofs in the dynamics of infinite dimensiona systems (see [3], [10] and references therein). As mathematicians we are wiing to argue forcefuy for the importance of rigorous verification and thus marginaize the cost. However, in reaity for many appications, researchers are often interested in investigating a variety of mode partia differentia equations at a mutitude of parameter vaues to gain scientific insight rather than an answer to a particuar question. This paces a premium on minimizing computationa cost, often eading to acceptance of the vaidity of numerica resuts simpy based upon the reproducibiity of the resut at different eves of refinement. As we sha argue, the resuts of this paper suggest that this dichotomy need not exist and we provide exampes wherein it is demonstrated that by judicious use of the computations invoved in the continuation method it is cheaper to vaidate the resuts The Coege of Wiiam and Mary, Department of Mathematics, P.O.Box 8795, Wiiamsburg, VA, (sday@math.wm.edu). Schoo of Mathematics, Georgia Institute of Technoogy, Atanta, Georgia USA (essard@math.gatech.edu). Department of Mathematics, Hi Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Freinghusen Rd, Piscataway, NJ , USA and Schoo of Mathematics, Georgia Institute of Technoogy, Atanta, Georgia USA (mischai@math.rutgers.edu). 1
2 than to re-perform the continuation computation. We refer to the method we propose as vaidated continuation. As is made cear towards the end of the introduction, vaidated continuation is sighty weaer and computationay cheaper than rigorous continuation. To the best of our nowedge this is the first attempt to integrate the techniques of rigorous computations with a continuation method, thus we focus on a cear presentation of the ideas as opposed to presenting the resuts in the most genera possibe setting. We mae use of spectra methods as they provide us with considerabe contro on truncation errors. To be more precise, assume that (1.1) taes the form u t = L(u, ν) + c p (ν)u p (1.2) where L(, ν) is a inear operator at parameter vaue ν and d is the degree of the poynomia noninearity. Typicay, c 1 (ν) = 0 since inear terms are grouped under L(, ν). Expanding (1.2) using an orthogona basis chosen appropriatey in terms of the eigenfunctions of the inear operator L(, ν), the particuar domain and the boundary conditions, resuts in a countabe system of differentia equations on the coefficients of the expanded soution. To simpify the exposition, et us assume the expansion taes the form u = f (u, ν) := µ u + P p=0 ni= p=0 (c p ) n0 u n1 u np = 0, 1, 2,... (1.3) where µ = µ (ν) are the parameter dependent eigenvaues of L(, ν) and {u n } and {(c p ) n } are the coefficients of the corresponding expansions of the functions u and c p (ν) respectivey with u n = u n and (c p ) n = (c p ) n for a n. In order to simpify the notation, for a fixed parameter ν, we use f(u) to denote f(u, ν). The continuation method is appied to the m-dimensiona system of ODEs of the form u = µ u + P p=0 ni = n i <m (c p ) n0 u n1 u np = 0, 1,..., m 1. (1.4) obtained by performing a Gaerin projection on (1.3). It is this truncation that introduces the most substantia concern for the vaidity of the resuts of the continuation method. In Section 3 we present estimates that provide us with bounds on the errors. We obtain these bounds under the assumption of power decay rates in the coefficients {u n }. Of course, such decay rates are directy reated to the spatia smoothness of the equiibria which in turn is governed, at east in part, by the inear operator L(, ν). The theoretica justification for our proof of existence and uniqueness of equiibria is based on a component-wise version of the Banach fixed point theorem (see Theorem 2.1) which itsef represents a minor modification of a resut of Yamamoto [9, Theorem 2.1]. A simiar formuation can aso be found in [4]. Reca that to appy the Banach fixed point theorem one must have a contraction mapping T : X X. With this in mind, we can state that it is appropriate to view our approach as a method by which the Newton-ie iteration of the corrector step in the continuation process is used to construct a set X and the above estimates are used to verify that an appropriate generaization of the Newton-ie operator is in fact a contraction. More 2
3 precisey, et ū be a numerica zero obtained from (1.4). In the orthogona basis used to obtain (1.3) consider the set X = ū + W (r) of ū where W (r) is of the form W (r) = [ r, r] m 1 =0 =m [ A s s, A ] s s. Observe that s indicates the decay rate of the coefficients and r is referred to as the vaidation radius. Our strategy which is described in detai in Section 3 is to produce a set of radii poynomias, {P (r)} =0,1,..., whose coefficients are given expicity in terms of the constants A s, s, and (1.3). Theorem 3.4 guarantees that if there exists a vaidation radius r > 0 such that P (r) < 0 for a, then there exists a unique equiibrium soution to (1.2) in the set X = ū + W (r) buit around the numerica equiibrium ū produced by the continuation procedure. It is important to remar that the conditions of Theorem 3.4 can be checed with a finite number of cacuations. As is indicated above the focus of this paper is on the computationa efficacy of vaidated continuation, and hence, the foowing organization of the materia. Section 2 contains the statement and proof of the aforementioned component-wise version of the Banach fixed point theorem, Theorem 2.1, without any indication of how this resut can be used in practice. Section 3 provides the opposite extreme, an expicit set of formuas and steps and the assertion that their successfu impementation eads, via Theorem 3.4, to the existence of a unique equiibrium in a specified set. The justification of this assertion and the reationship between Theorems 2.1 and 3.4 is presented in Section 6. However, presenting the formuae in this fashion has two advantages. First, they contain a the necessary information shoud the reader wish to independenty code and test the techniques suggested in this paper. Second, it aows for the presentation in Section 4 of the comparison of the computationa costs between traditiona and vaidated continuation. It shoud be emphasized that how one shoud best compare the costs between the two methods of continuation is not competey cear. In the standard approach m, the dimension of the system on which continuation is performed, is fixed. Thus traditionay, a particuar Gaerin projection dimension is chosen and continuation is performed. The resuts are checed by choosing a higher dimensiona projection, re-performing the continuation and then deciding if the two cacuations agree within a certain eve of numerica toerance. In vaidated continuation, m becomes a variabe. In particuar, if vaidation fais then one has the option of choosing a higher dimensiona Gaerin projection. Equay important, faiure of vaidation may be an indication that a higher dimensiona projection is necessary. In summary, vaidated continuation provides an interna chec of consistency on the dimension of truncation from the infinite to finite dimensiona probem a feature which is not present in the traditiona appication of continuation methods. With this in mind we have chosen to compare the computationa costs as foows. First we restrict our attention to cubic noninearities. As is made cear by the formuae of Section 3 in this case the cost of evauating the noninearities and performing Newton s method are both of order m 3. Thus, we can obtain a rough bound on the ratio of the cost of traditiona versus vaidated continuation by counting the number of m 3 operations which need to be performed. These cacuations suggest that for fixed m the cost of vaidated continuation is ess than twice the cost of traditiona continuation, that is it appears that it is cheaper to perform vaidated continuation, than to perform traditiona continuation and then chec it against continuation performed on a higher dimensiona projection. In Section 5 this estimate is tested against 3
4 actua computations for the Swift-Hohenberg equation and the Cahn-Hiiard equation. To ensure that these comparisons are fair, we empoy standard foating point computations in both cases. This ast point raises an important distinction: vaidated continuation versus rigorous continuation. Using foating point cacuations at a steps of the vaidated continuation, does not aow one to contro for roundoff errors and hence one cannot rigorousy concuded the existence of an equiibrium. Because the current computer technoogy treats foating point and interva arithmetic differenty we chose not to mae and present timed comparisons between the two for this paper. However, if specific steps in the vaidation argument are performed using interva arithmetic, then one obtains rigorous resuts on the existence of equiibria. Resuts of this type are presented in Section 5 for a branch of equiibria of the Swift-Hohenberg equation. We see the resuts of this paper as a first step in the direction of combining continuation methods with rigorous computations. With this in mind we concude the paper in Section 7 with a discussion of open questions and on going wor. In particuar, we return to the issue of the necessity of interva arithmetic computations. 2. Computationa Proofs for Equiibria. Assume that foowing the expansion of a PDE into an appropriate orthogona basis, we have a system of the form (1.3). Our goa is to prove that there is a unique equiibrium for (1.3) which ies in a sma set containing a computed numerica equiibrium. Suppose ū F is a numerica equiibrium computed using an m-dimensiona continuation procedure (as described in Section 3) and ū := (ū F, 0,...) is the corresponding point in the infinite dimensiona space. We wi consider a set of the form ū + W where W = Π w, { [ r, r] 0 < m w = [ ] A s (2.1), As s m s for some constants r, A s > 0 and s 2. A particuary nice norm to use for this set (simiar to the one used by Yamamoto in [9]) is the normaized sup norm u W := sup { } u w where w := max { x x w }. In this norm, W = B(0, 1) is the unit ba around 0, and ū + W = B(ū, 1) is the unit ba around ū. We wi now reformuate our probem of studying equiibria for (1.3) by estabishing an equivaent fixed point probem on ū + W. Suppose J is an invertibe operator. Then u is an equiibrium soution of (1.3) if and ony if u is a fixed point of T (u) = u Jf(u) (2.2) where f is given by (1.3). In practice, T is constructed to be a contraction (Newtonie) operator with J (Df(ū)) 1 so that we may use Banach s fixed point theorem. We now frame this fixed point theorem in a more computationa setting. In the process of showing that T is a contraction, we first consider the foowing Lipschitz condition on ū + W : T (x) T (y) W K x y W for x, y ū + W. (2.3) The question now becomes whether we can compute a contraction constant K < 1 satisfying (2.3). We begin by computing Lipschitz constants, K n, for the component 4
5 functions T n on ū + W satisfying the foowing T n (x) T n (y) K n x y W for x, y ū + W. (2.4) If T is C 1, we may tae K n to be a bound on the derivative of T n over ū + W. More expicity, K n sup DT n (ū + W ) W := sup DT n (ū + b) c. b,c W A constant K n computed in this manner satisfies (2.4) by the foowing argument. For x, y ū + W, et g n (s) := T n [sx + (1 s)y]. Appying the mean vaue theorem to g n, we get the existence of s n [0, 1] such that g n (1) g n (0) = g (s n ). Since the set ū + W is convex, we get the existence of z n := s n x + (1 s n )y ū + W such that T n (x) T n (y) = DT n (z n )(x y) = DT x y n(z n ) x y W x y W. x y K By construction of W, x y W W. Now if K := sup n n w n <, then, as the foowing argument shows, it satisfies (2.3) T (x) T (y) W = sup n = sup n T n (x) T n (y) w n x y DT n (z n ) x y W x y W w n K n sup n w n x y W = K x y W. Theorem 2.1. (Existence and Uniqueness) If for a n there exist bounds Y n T n (ū) ū n and K n satisfying (2.4) such that and Y n + K n w n < 0 (2.5) K := sup n K n w n < 1 (2.6) then there exists a unique fixed point of T in ū + W. Proof. The first inequaity ensures that T (ū + W ) ū + W. This is true if and ony if for every u ū + W, T (u) ū W 1, or equivaenty, Tn(u) ūn w n < 1 for a n. Let u ū + W. Then u ū W 1 and for each n, T n (u) ū n = T n (u) T n (ū) + T n (ū) ū n T n (u) T n (ū) + T n (ū) ū n K n u ū W + Y n Y n + K n < w n 5
6 by assumption (2.5). Therefore, T (ū + W ) ū + W. The second inequaity guarantees that T is aso a contraction. Thus, the resut foows from Banach s fixed point theorem. Let us mae the comment here that sufficient reguarity of the equiibrium soutions wi effectivey reduce the infinite set of conditions isted in Theorem 2.1 to a finite ist. In essence, the strong decay in the higher modes may be used to verify (2.5) simutaneousy for a n > N for some N. (In our case N is determined by the dimension used for continuation and the degree of the noninearity.) Furthermore, reguarity of the equiibria may aso be used to show that K n w n 1 becomes a decreasing sequence. Therefore, (2.6) foows automaticay from (2.5). Perhaps an even more important point to mae for our intended agorithmic approach in this paper is that Y n + K n w n wi be given as a poynomia in the vaidation radius r, the width of the set W in the ow modes. Therefore, vaidating the existence of a unique equiibrium near ū wi amount to showing that it is possibe to simutaneousy sove a (finite) ist of poynomia inequaities in r. 3. Vaidated Continuation. The ideas outined in Section 2 for proving the existence of unique equiibria fit naturay with traditiona continuation techniques for foowing branches of numerica equiibria. In particuar, an approximation of a projection of the Newton operator given in (2.2) onto the appropriate m-dimensiona subspace is an intrinsic eement of the continuation agorithm. In this section, we discuss expoiting this reationship to automaticay produce a vaidation of the existence of unique equiibria at each step of the continuation procedure. Reca that foowing the expansion of the system in the appropriate basis, we have where for = 0, 1, 2,..., µ = µ (ν), (c p ) n = (c p (ν)) n and u = f (u) = µ u + u = f(u, ν) (3.1) P p=0 ni= (c p ) n0 u n1 u np (3.2) A first step for impementing a continuation agorithm for studying a PDE is to perform a Gaerin projection. Let m be a fixed projection dimension and consider the foowing truncated version of our origina expansion of the PDE given in (3.2). For u F := (u 0,..., u m 1 ) R m, define f (m) : R m R m by f (m) (u F ) = (f (m) 0 (u F ),..., f (m) m 1 (u F )) where for = 0,..., m 1, f (m) (u F ) = µ u + P p=0 ni = n i <m (c p ) n0 u n1 u np The corresponding Gaerin projection of the origina system (3.1) is then u F = f (m) (u F, ν) (3.3) This is the m-dimensiona system to be studied numericay. Intuitivey, we expect that if m is sufficienty arge, (3.3) wi capture the essentia dynamics for the origina system (3.1). In particuar, given an equiibrium ū F for (3.3) we expect that there is a sma set around ū := (ū F, 0,...) which contains a unique equiibrium soution for (3.1). Our approach is to study this reationship via the toos outined in Section 2. 6
7 3.1. Continuation for ODEs and Newton-ie Operator. A traditiona continuation procedure invoves iteration of predictor and corrector steps to trace out branches of equiibria. Under the assumption that at some parameter ν = ν 0 we have an equiibrium soution for (3.3), we want to continue the equiibrium as we vary ν. 1) Euer Predictor: Given an approximate equiibrium x 0 at ν 0, the predictor at ν 1 = ν 0 + ν is x (0) 1 = x 0 + ẋ 0 ν, where ẋ 0 = f x (m) (x 0, ν 0 ) 1 f ν (m) (x 0, ν 0 ). (3.4) 2) Quasi-Newton Corrector: We now use the foowing quasi-newton iterative scheme to improve our approximation at ν 1 x (n+1) 1 = x (n) 1 f x (m) (x 1 (0), ν 1 ) 1 f (m) (x (n) 1, ν 1) (3.5) If is the tota number of iterations of (3.5), then ū F := x () 1 and f (m) (ū F, ν 1 ) 0. As before, define the corresponding point ū = (ū F, 0,...) in the infinite dimensiona space. We now use the information required for the next predictor step, the numerica inverse of f x (m) (ū F, ν 1 ), to construct a Newton-ie operator near ū at the parameter vaue ν 1. Let J F F be the numerica inverse of f x (m) (ū F, ν 1 ) and define the Newton-ie operator T by T (u) = u Jf(u) (3.6) where J := J F F 0 0 µ 1 m µ 1 m+1... is the boc diagona matrix which we expect to be cose to (Df(ū, ν 1 )) 1. Note that T, J, and f a depend on the parameter ν. As in Section 2, we wi attempt to show that T is a contraction on a set of the form ū + W where W has the form (2.1). We now emphasize the dependence of this set W = W (r) on the vaidation radius r since this approach reies on finding an appropriate r > 0 to satisfy a set of conditions. The constants A s and s may be determined by reguarity arguments or otherwise set prior to the computations. As seen in the definition of W (r), these constants determine the size of the region in which we are attempting to show the unique existence of an equiibrium soution Radii Poynomias. We now present the formuae for radii poynomias. In order to focus on the appicabiity of vaidated continuation the justification that these poynomias do, in fact, encode the required bounds Y n and K n in (2.5) for the Newton-ie operator constructed in (3.6) is deayed to Section 6. Since the formuae for the poynomias are rather ungainy, et us begin by expicity stating the information that is used to construct the coefficients. d is the degree of the noninearity of (1.2). m is the number of modes used in the Gaerin projection. 7
8 where and M m is a computationa parameter that aows for the use of expicit vaues for coefficients of M m additiona modes to decrease truncation error bounds. m + m is a computationa parameter that aows for the use of additiona structure in the mode to get tighter truncation error bounds. ū F R m is the numerica zero produced by the predictor-corrector step. J F F is the numerica inverse obtained from the predictor-corrector step. (c p ) n, n < m are the coefficients from the expansion (1.3). µ, 0 are the eigenvaues for the inear operator L as expressed in (1.3) and µ := im inf n m + µ n. Note that if µ n is monotonicay increasing for n m +, then µ = µ m+. s and A s are positive constants that are reated to the reguarity of the equiibria. Observe that given this information we can evauate the vectors We can aso set f F (ū) := f 0 (ū). f m 1 (ū) f n (ū) = µ n ū n + f n(ū) = µ n + p p=1 and f (ū) := p=0 n 0 + +np=n n 1,..., n p <m n 0 + +np=n n 1,..., n p 1 <m f 0(ū) f 1(ū). (c p ) n0 ū n1 ū np (c p ) n0 ū n1 ū np 1. Y { P d p=2 (cpūp ) µ if m d(m 1) 0 if > d(m 1) (3.7) where (c p ū p ) = P ni= (c p ) n0 ū n1 ū np. The foowing constants are a reated to asymptotic bounds on the expansions of the numerica equiibrium ū, and the set ū + W. As such they are reated to the 8
9 reguarity of the equiibrium and the coefficients of (1.2). Define α := s s 1 C p := max { (c p) 0, (c p ) s } Ā := max { ū 0, ū s } 1 <m A = A(r) := max{a s, r(m 1) s } ( ) p C(Ā, A) := α p C p Ā p A(r) =1 p=max{2,} max n m+ { 1 Jf (ū) n n s }, Ā inear := max { 0 if f n (ū) = 0 for a n m + d p=2 pαp C p Ā p 1 µ 1 otherwise { d d C + (Ā, A) := =1 p=max{2,} ( p ) α p C p Ā p A if Y = 0 for a m + d p=0 αp C p Ā p + d d =1 p=max{2,} ( p ) α p C p Ā p A otherwise. Observe that this impies that ū Ā [ 1, 1] and w s A [ 1, 1] for a. Aso, s decreasing µ by increasing m + decreases the asymptotic bound constant Āinear. The vaidation procedure aso requires bounds on the errors due to truncating modes m. These bounds come in the foowing form: ( ) p ɛ n := ɛ n (p,, M) (3.8) where and =1 p= ɛ n (p,, M) := (3.9) { pα p 1 C p Ā p A [ ] 1 min (M 1) s 1 (s 1) (M n) s + 1 (M + n) s, αp C p Ā p A } n s, C n (p, j,, M) := (3.10) n <(p )(m 1)+M P ni = n n 0 <M n 1,..., n p <m (c p ) n0 ū n1 ū np P ni + n=n m n 1,..., n j <M } A j s n 1 s n j s. For notationa purposes, we aso define m-vectors containing these bounds for modes n = 0,..., m 1 as foows. ɛ 0 C 0 (p, j,, M) ɛ F :=., and C F (p, j,, M) :=.. C m 1 (p, j,, M) ɛ m 1 Note that these bounds are computabe in that they require ony a finite number of computations. In addition, increasing the computationa parameter M has the effect of increasing the computationa wor in order to decrease the bounds. 9
10 We now use bounds (3.9) and (3.10) to define radii poynomias, P n (r). These poynomias are designed to encode the bounds required by Theorem 2.1. More specificay, as is demonstrated in Section 6, the poynomias are constructed so that P n (r) < 0 impies that Y n + K n w n < 0 on the set ū + W (r) = ū + ( m 1 [ r, r] =0 =m [ A s s, A ] ) s s. (3.11) Definition 3.1. To simpify notation, the finite radii poynomias, P 0,..., P m 1, are given as an m-vector P F (r) = (P 0 (r),..., P m 1 (r)) t. Define P F (r) := C F (n)r n (3.12) n=0 where the coefficients are CF Y + CK F (0) n = 0 C F (n) := CF K (1) 1 n = 1 CF K (n) n = 2,..., d. The right hand terms are defined as foows. The individua terms of the vectors CF K (i) are chosen to satisfy ( )( ) C K (i) p J F F C F (p, i,, M) i =max{2,i} p= J F F ɛ F + αāineara s i = 0 + 2Āinear s i = 1. (3.13) 0 otherwise and simiary C Y F J F F f F (ū). (3.14) where and the bounds are computed component-wise. Observe, again, that determining these bounds require ony a finite number of computations. Definition 3.2. For m, the tai radii poynomia is P (r) = P d p=0 (cpūp ) µ + C(Ā,A(r)) µ As s s m < m + C +(Ā,A) µ s As s m + where, again, (c p ū p ) = P ni= (c p ) n0 ū n1 ū np. Definition 3.3. Consider the radii poynomias consisting of the finite radii poynomias P, = 0,..., m 1, and the tai radii poynomias, P, m. A positive rea number r is a vaidation radius if P (r) < 0 for a 0. 10
11 The proof of the foowing theorem is presented in Section 6 Theorem 3.4. If there exists a vaidation radius r > 0 and the eigenvaues µ satisfy µ, then there exists a unique equiibrium soution of (3.1) in ū+w (r). We now present a procedure for computing a vaidation radius that satisfies the hypotheses of Theorem 3.4. In particuar, this procedure describes a natura order for defining the decay constants A s, s, and A. The constants A s and s refect reguarity properties of the equation and shoud be chosen either from numerica simuations or anaysis. In this approach, we choose to treat A = A(r) as a constant. The rationae for this choice is that from a computationa perspective, we woud ie to find r > 0 soving simpe constructions of the finite radii inequaities P 0 (r) < 0,, P m 1 (r) < 0 without having to simutaneousy contro the more compicated effects from A on the coefficients of these poynomias as we as on the tai poynomias P, m. A practica way to achieve this goa is to set A = A s at the beginning of the procedure and then chec in the end that a soution r > 0 to P 0 (r) < 0,, P m 1 (r) < 0 aso satisfies r(m 1) s A s. Here, for the sae of simpicity, we set M = m. If the truncation error bounds prove too arge for the computations, then M shoud be increased as described in Remar 6.4 in Section 6. Finay, we add a condition which reduces the chec of the tai poynomias P (r) < 0, > m to a finite number of computations. The foowing procedure outines this approach. Procedure 3.5. Suppose that the eigenvaues µ are such that µ. Suppose further that we may choose m, m +, m N, m m + m, and µ > 0 such that 1. m is the Gaerin projection dimension used for numerica continuation, 2. m + is the parameter used in the computation of C + (Ā, A), and 3. m measures eigenvaue monotonicity as foows: for a m, µ µ. Set M = m. Remar: m shoud be chosen to give the expected nonzero modes aong the bifurcation branch under study and m = m + = (2d + 1)(m 1) + 1 if (c p ) n = 0 for a n 0 and the eigenvaues, µ are monotonicay increasing in magnitude after = (2d + 1)(m 1). Fix the decay constants s 2 and A s > 0. (3.15) Remar: In practice, A s and s shoud be determined by reguarity properties of the equation. Set A := A s. Using the finite radii poynomias given in Definition 3.1, for = 0,, m 1, numericay compute I := {r > 0 P (r) < 0} and Chec that I =. I := m 1 =0 I. (3.16) Remar: If I =, begin the procedure again either by choosing m arger or by choosing s arger and/or A s smaer in (3.15). Chec that there exists r I such that r A s (m 1) s. (3.17) 11
12 Remar: If such an r exists, then A = A s = max{a s, r(m 1) s }. This in turn impies that component-wise P F ( r) < 0. If r does not exist, then begin the procedure again either by choosing m arger or by choosing s arger and/or A s smaer in (3.15). Chec the inequaities P m ( r) < 0,, P m 1 ( r) < 0 and C(Ā, A) µ A s < 0. Remar: If any of these inequaities fais, begin the procedure again either by choosing m arger or by choosing s arger and/or A s smaer in (3.15). Observe that if Procedure 3.5 is successfu, the hypotheses of Theorem 3.4 are satisfied with vaidation radius r. 4. Computationa Cost. We now provide a rough comparison of the cost of continuation with the cost of vaidated continuation for PDEs of the form u t = L(u, ν) u 3. (4.1) Since the degree of the poynomia noninearity in (4.1) is cubic and we use a Newtonie operator in the continuation procedure, the most expensive terms of the computation invove m 3 operations, where m is the number of modes used in the Gaerin projection f (m) (u F, ν) = µ (ν)u n 1 +n 2 +n 3 = n i <m u n1 u n2 u n3, = 0,..., m 1. (4.2) With this in mind we count the number of m 3 operations for both approaches to obtain an estimate for the asymptotic costs and concude with statistics obtained from cacuations for the Swift-Hohenberg and Cahn-Hiiard equations Cost of Continuation. We decompose the anaysis of the cost of continuation into four steps, assuming that we begin with an approximate zero x 0 at ν 0. Step 1.. In order to get the Euer predictor (3.4), we need to evauate the vector (x 0, ν 0 ) 1 f ν (m) where for 0 i, j < m, f (m) x [ f (m) x (x (0) 0, ν 0) (x 0, ν 0 ). This requires computing the m by m matrix f (m) x (x (0) 0, ν 0), ] i+1,j+1 ( = δ i,j µ i 3 + n 1 +n 2 j=i n i <m n 1 +n 2 +j=i n i <m [x (0) 0 ] n 1 [x(0) 0 ] n 2 [x (0) 0 ] n 1 [x(0) 0 ] n 2 This invoves the evauation of 2m 2 sums demanding 2m 1 mutipications and 2m 2 additions each. Therefore, determining f x (m) (x (0) 0, ν 0) requires 8m 3 operations. Next, we compute the LU decomposition of f x (m) (x (0) 0, ν 0) in order to compute the action of its inverse on f ν (m) (x 0, ν 0 ). This invoves 2 3 m3 operations. In our case, f ν (m) (x 0, ν 0 ) = x 0, requiring no additiona cost. The predictor is then { x (0) 1 = x 0 νf x (m) (x 0, ν 0 ) 1 x 0 ν 1 = ν 0 + ν. 12 ).
13 Step 2.. We now start the corrector. To construct the quasi-newton operator (3.5), we need the action of the inverse of f x (m) at the predictor (x (0) 1, ν 1). As seen before, it costs 8m 3 to evauate f x (m) (x (0) 1, ν 1) and 2 3 m3 to compute its inverse using LU decomposition. Note that we need to compute the LU decomposition ony at the first step. Step 3.. At the j th iteration of (3.5), we need to evauate f (m) (x (j 1) 1, ν 1 ). Its i th component is [f (m) (x (j 1) 1, ν 1 )] i = µ i (ν 1 )[x (j 1) 1 ] i [x (j 1) 1 ] n1 [x(j 1) 1 ] n2 [x(j 1) 1 ] n3 n 1 +n 2 +n 3 =i n i <m which requires at east 3m 2 operations to evauate. Since f (m) has m components, we get a tota of 3m 3. If is the tota number of iterations of the corrector, then this step requires 3m 3 operations. Step 4.. The corrector ends when f (m) (x () 1, ν 1) < toerance. Let ā F := x () 1. Evauating the function at (ū F, ν 1 ) is another 3m 3. Now, note that we have to compute the action of the inverse of f x (m) (ū F, ν 1 ) to get the predictor for the next step. Reca J F F is the numerica inverse of f x (m) (ū F, ν 1 ) computed as before using an LU decomposition. Expicity computing a the coefficients in f x (m) (ū F, ν 1 ) requires an extra 2m 3 operations. We do not count the m 3 invoved to get the next predictor, since that is part of the next predictor-corrector step. Combining the costs of the four above mentioned steps suggests that the cost of one appication of the predictor-corrector agorithm is on the order of (20 + 3)m 3, where is the number of iterations in the quasi-newton corrector Cost of Vaidation. We now show that the extra cost of performing vaidation for a cubic function (d = 3) with constant function coefficients is of the order of 6m 3 operations where m is the projection dimension used for continuation. The additiona cost comes primariy from computing the coefficients of the radii poynomias. In the foowing, we construct m + = d(m 1) + 1 = 3m 2 poynomias P 0,..., P 3m 3 using Procedure 3.5 and cacuate the associated computationa cost. Both to simpify the presentation and because this is what is used to perform the computations presented in Section 5, we set m = m + = d(m 1) + 1, with µ µ m for a m, and M = m. As described in Procedure 3.5, A = A s and we consider fixed s > 2 and A s > 0. The ony noninear term of (4.1) is a monomia of degree 3. Thus, if p 3, then C (p, j,, M) = 0. In addition, we have set M = m. Hence, if j 0, then C (p, j,, M) = 0 (see Remar 6.4). Therefore, the ony nonzero terms of this form are C (3, 0,, m) = n 1 +n 2 +n 3 = n 1, n 2, n 3 <m 13 ū n1 ū n3. (4.3)
14 Hence, by (3.13) we set C K (0) ( J F F ɛ F ) + αāineara s (4.4) for 0 < m and denotes the component-wise absoute vaue. A necessary computations for C K F (0) are of order ess than m3. Using (3.13), C K (1) 2Āinear s for 0 < m. Comment 4.1. Observe that max IF 0 n<m F J F F Df (m) (ū) IF, Ā inear max d p=2 pαp C p Ā p 1 (min{µ m,..., µ m+ 1, µ}) 1 where I F := (1, 1,..., 1) t and denotes the component-wise absoute vaue. We used this upper bound in our impementation, which impies that computing CF K (1) does not require any m 3 operations. Finay, combining (3.13) and (4.3) C K F (2) 6 J F F C F (3, 0, 2, m) where C n (3, 0, 2, m) = ū n and C K F (3) 3 J F F C F (3, 0, 3, m) where C n (3, 0, 3, m) = 1. The ast coefficient to compute to get a the finite radii poynomias (3.14) is C Y F J F F f F (ū) where again denotes the component-wise absoute vaue. This comes with no extra m 3 cost since f F (ū) = f (m) (ū F, ν 1 ) was computed in Step 4 of the predictor-corrector agorithm. The next step in Procedure 3.5 is checing for the existence of a vaidation radius r > 0. This requires finding the numerica zeros of each of the cubic poynomias P 0,, P m 1, constructing I 0,, I m 1 where I are cosed intervas such that I {r > 0 P (r) < 0}, and finay checing for a non-empty intersection I = m 1 =0 I. A of these steps are of order ess than m 3. Assuming there exists a positive r I such that r(m 1) s A s, we construct and evauate the tai radii poynomias P m,, P 3m 1 at r. We compute Y using (3.7) which requires 6m 3 operations since we need to evauate f (ū) for = m,, 3m 3. Using Definition 3.2 and the assumption that A = A s we compute 3 C(Ā, A) = =1 ( ) 3 α 3 Ā 3 A = 3α 3 A s (Ā + A s) 2. 14
15 This atter step and the remaining computations for Procedure 3.5 are a of order ess than m 3. In summary, the m 3 cost of computing the coefficients of the radii poynomias is 6m 3. Thus the additiona cost of vaidation is on the order of 6m 3 operations Reative Cost. Combining the resuts of Sections 4.1 and 4.2 suggests that asymptoticay the ratio of the cost of vaidated continuation to the cost of traditiona continuation is where is the number of iterations performed in the corrector step. We tested this hypothesis again two fourth order partia differentia equations with cubic noninearities, Swift-Hohenberg and Cahn-Hiiard. The resuts are discussed in greater detai in Section 5. For the moment we are ony interested in the reative times of computation. We performed vaidated continuation for 46 predictor-corrector steps invoving a tota of 90 quasi-newton iterations for the cubic Swift-Hohenberg equation. We repeated the computations without vaidation. The ratio of eapsed time for vaidated continuation to the time used for continuation aone was Given that we had an average of 90/46 iterations per predictor-corrector step, this is cose to the rough estimate of / / given by the above arguments. Simiary, we performed vaidated continuation for 15 predictor-corrector steps invoving a tota of 37 quasi-newton iterations for Cahn-Hiiard. Again, we repeated the computations without vaidation. The ratio of eapsed time for vaidated continuation to the time used for continuation aone was Given that we had an average of 37/15 iterations per predictor-corrector step, the asymptotic ratio is / / The resuts of these computations are summarize in Figure 4.1. PDE m # iterations # steps Experimenta Ratio Estimated Ratio S-H C-H Fig Comparison of the Asymptotic Ratios 5. Sampe Resuts. To demonstrate the practica appicabiity of vaidated continuation we turn to two mode probems, Cahn-Hiiard and Swift-Hohenberg. In both cases we foow a branch of equiibria and vaidate at each parameter vaue of the continuation. In the case of Swift-Hohenberg we aso use interva arithmetic to evauate the radii poynomias, thus aowing us to rigorousy verify the existence and uniqueness of the equiibria Cahn-Hiiard. The Cahn-Hiiard equation was introduced in [1] as a mode for the process of phase separation of a binary aoy at a fixed temperature. On a one-dimensiona domain it taes the form u t = ( 1 ν u xx + u u 3 ) xx, x [0, 1] u x = u xxx = 0, at x = 0, 1. (5.1) 15
16 The assumption of an equa concentration of both aoys is formuated as 1 0 u(x, ) dx = 0 Note that when ooing for the equiibrium soutions of (5.1), it is sufficient to wor with the Aen-Cahn equation 1 ν u xx + u u 3 = 0 (5.2) u x = 0 at x = 0, 1. Re-writing (5.2) in the form of (1.2), the inear operator is L(, ν) = 1 2 ν x + 1 and 2 the poynomia noninearity is of degree d = 3 with coefficient functions { 1 p = 3 and n = 0 (c p ) n = 0 otherwise. Appying Procedure 3.5 with M = m = 60, s = 3, and A s = 0.01, resuts in the branch of equiibria indicated in Figure 5.1 where each point represents the center of the infinite dimensiona vaidation set of the form ū + W ( r), containing a unique equiibrium of (5.1). These are the points used to obtain the cost estimates presented in Figure 4.1. To avoid drowning the reader in arge ists of numbers, we ony provide the detaied numerica output at one parameter vaue. Vaidated Resut 5.1. Let ν = Then, r = is a vaidation radius for the numerica zero ū F given in Figure 5.2. Thus, there exists a unique equiibrium for (5.1) in the vaidation set (ū F, 0) + 59 =0 [ r, r] =60 [ , 0.01 ] Swift-Hohenberg. The Swift-Hohenberg equation { ) 2 } ( u t = f(u, ν) = ν (1 + 2 x 2 u u 3, u(, t) L 2 0, 2π ), L 0 ( u(x, t) = u x + 2π ), t, u( x, t) = u(x, t), ν > 0, (5.3) L 0 was originay introduced to describe the onset of Rayeigh-Bénard heat convection [8], where L 0 is a fundamenta wave number for the system size 2π/L 0. The parameter ν corresponds to the Rayeigh number and its increase is associated with the appearance of mutipe soutions that exhibit compicated patterns. For the computations presented here we fixed L 0 = Re-writing (5.3) in the form of (1.2), the inear operator is L(, ν) = ν (1+ 2 x ) 2 2 and the poynomia noninearity is of degree d = 3 with coefficient functions { 1 p = 3 and n = 0 (c p ) n = 0 otherwise. 16
17 Fig Vaidated continuation in ν for the Cahn-Hiiard equation on [0, 1]. Appying Procedure 3.5 with M = m = 27, s = 4, and A s = 0.002, resuts in the branch of equiibria indicated in Figure 5.3 where each point represents the center of the infinite dimensiona vaidation set of the form ū + W ( r), containing a unique equiibrium of (5.3). Again, these are the points used to obtain the cost estimates presented in Figure 4.1. As in the case of the Cahn-Hiiard equation, we ony incude the output at one point on the branch of the Figure 5.3. Vaidated Resut 5.2. Let ν = Then r = is a vaidation radius for the numerica zero ū F whose coefficient vaues are indicated in Figure 5.4. Thus, there exists a unique equiibrium soution for (5.3) in the vaidation set 26 [ (ū F, 0) + [ r, r] , ] 4. =0 =27 Observe that in a the above mentioned cacuations foating point round-off errors have not been controed, thus at this point one cannot caim that the Vaidation Resuts presented above are rigorous. However, with additiona computationa effort a computer-assisted proof can be obtain. To be more precise, our technique reies on the existence of a vaidation radius r maing a radii poynomias stricty negative. Hence, rigorous vaidation foows if the inequaities are satisfied when one incudes bounds to contro the possibe of foating point errors. The first step in checing these inequaities on this eve is to obtain foating point outer bounds for the coefficients of the poynomias. This can be done by defining each entry of ū F, f (m) (ū F, ν), J F F, f x (m) (ū F, ν), µ (ν), A s, and s to be an interva and then computing (3.13), (3.14) and the quantities in Definition 3.2 using interva arithmetic. The resuting radii poynomias, which we denote by P, have interva coefficients. Let r be the smaest representabe number such that using 17
18 ū Fig The numerica zero ū F obtained by continuation for the Cahn-Hiiard equation at ν = Note that a even coefficients are 0. interva arithmetic, the corresponding finite radii poynomias may be shown to be stricty contained in (, 0). Assume such an r exists. If, again using interva arithmetic, r(m 1) A s (, 0) and the intervas obtained from evauating tai radii poynomias at r are stricty contained in (, 0), i.e. P ( r) (, 0) for a m, then the hypotheses of Theorem 3.4 are satisfied and we obtain a proof. The above mentioned computations were performed using the interva arithmetic pacage in Matab. Thus, we can state the foowing theorem. Theorem 5.3. Each point in Figure 5.3 represents the center of an infinite dimensiona set of the form ū F + 26 =0 [ r, r] containing a unique equiibrium to (5.3). =27 18 [ , ] 4
19 Fig Vaidated continuation in ν for the Swift-Hohenberg equation at L 0 = ū Fig The numerica zero ū F obtained by continuation for the Swift-Hohenberg equation at ν = and L 0 = A even coefficients are 0. The actua vaues for the various numerica zeros and vaidation radii are of imited interest and thus not presented. Of greater interest is understanding how arge are the errors induced by the foating point computations as opposed to the magnitudes of the foating point computations of P ( r), 0, where r is the vaidation radius. Let us restrict our attention to the equiibrium described by Vaidated Resut 5.2. Foowing Procedure 3.5 at this parameter vaue, beginning using radii poynomias with interva coefficients and performing the computations with interva arithmetic eads to an interva of potentia vaidation radii I = [ , ]. Hence, we choose r = There are 53 incusions that need to be satisfied, those arising from the 2m 2 = 52 tai radii poynomias with interva coefficients and the one associated with the inequaity (3.17). The fact that the 19
20 incusions are satisfied eads to the concusion of Theorem 5.3 at this parameter vaue. Again, rather than isting a 53 incusions et us focus on the two extremes, the interva cosest to 0 P 27 ( r) = ± and the interva the farthest from ± corresponding to the inequaity (3.17). Observe that in both cases, the width of the interva induced by the foating point errors is more than ten orders of magnitude smaer than the vaue of the center. Furthermore, this behavior is typica for a the vaidation computations that were performed. This suggests that it is reasonaby safe to assume that a vaidated equiibrium is a true equiibrium. 6. Justification of radii poynomias. In this section, we describe the construction of the radii poynomias that were defined in Section 3.2 and encode the bounds required for Theorem 2.1. We begin by computing the required bounds Y n and K n in (2.5) for the Newton-ie operator constructed in (3.6). Using a Tayor expansion of the Newton-ie operator T (u) = u Jf(u) around the numerica equiibrium ū = (ū F, 0, 0,...) eads to T (ū + w )w = (1 Jf (ū + w ))w ( ( = 1 J f (ū) + f (ū)(w ) + + f () (ū) = (1 Jf (ū))w J ( =2 = (1 Jf (ū)) w J f () (ū) ( 1)! (w ) 1 =2 p= = (1 Jf (ū)) w J =2 p= ( 1)! (w ) f (d) (ū) ) w p!c p ū p (w ) 1 ( 1)!(p )! w ( ) p c p ū p (w ) 1 w )) (d 1)! (w ) d 1 w where f (ū) = L (ū) + pc p ū p 1. p=2 Expanding into Fourier modes, we get T (ū + w )w = (1 Jf (ū)) w J =2 p= = (1 Jf (ū)) w J =2 p= 20 ( ) p c p ū p (w ) 1 w ( ) p (c p ū p ) (w ) 1 w (6.1)
21 where denotes discrete convoution and 1 has Fourier expansion 1 n = { 1 n = 0 0 otherwise. Thus, (1 Jf (ū)) w = n 1 J µ + p p=2 P ni= (c p ) n0 ū n1 ū p 1 w n n n n, and (c p ū p ) ((w ) 1 ) w = n P ni= n (c p ) n0 ū n1 ū np P ni=n n w n 1 w n 1 w n. n Here, [ ] n denotes the bi-infinite vector indexed by n Z and ( ) denotes the entry at index. We use this expansion to compute the bounds K max (T (ū + W )W ) max (1 Jf (ū)) w J =2 p= ( ) p (c p ū p ) w where, as in Section 2, W has the form (2.1). The boc-diagona structure of J aows us to decompose (6.1) into a finite, m- dimensiona piece and the infinite dimensiona tai terms. For the foowing, we adopt the notation [ ] F to denote the m-vector whose nth entry is computed at index vaue n 1 for 1 n m, the subscript F to denote the bi-infinite vector in which the th entries for m are set equa to 0, and the subscript Ĩ to denote the bi-infinite vector in which the th entries for < m are set equa to 0. We begin with the foowing decomposition of the inear term. (1 Jf (ū)) w = (1 Jf (ū)) w F + (1 Jf (ū)) wĩ = w Jf (ū) w (6.2) [ ] Remar 6.1. Note that (1 Jf F (ū)) w F may be written as the matrix F mutipication ( I F F J F F Df (m) (ū) ) w F where I F F is the m m identity matrix. The matrix ( I F F J F F Df (m) (ū) ) was computed during the continuation step and is expected to be cose to zero. The term (1 Jf (ū)) w F encodes bounds on ( I F F J F F Df (m) (ū) ) as we as other inear effects from the mixing of terms in the numerica zero ū F with truncated modes. It foows that [T (ū + W )W ] F [(1 Jf (ū)) w F + (1 Jf (ū)) wĩ] F ( ) p J F F [(c p ū p ) w ] F. (6.3) 21 =2 p=
22 For m, (T (ū + W )W ) J(, ) =1 p= ( p ) ((c p ū p ) w ). (6.4) We now focus on finding bounds on the terms given in (6.3) and (6.4). First consider ((c p ū p ) w ) = n (c p ) n0 ū n1 ū np w n1 w n (6.5) P ni= n P ni+ n= where p is the degree of the origina monomia term of f and {1,..., p} is the order of the derivative being taen. One upper bound for (6.5) is given in the foowing emma, which uses asymptotic bounds first isted in Section 3.2. Lemma 6.2. Let α = 2 s s, ū Ā [ 1, 1], (c s p ) Cp [ 1, 1], and s w A [ 1, 1] for a. Then s n P ni= n α p C p Ā p A ((c p ū p ) w [ 1, 1] 0 ) s α p C p Ā p A [ 1, 1] = 0. Proof. Note that (c p ) n0 ū n1 ū np P ni+ n= P ni= P ni= w n1 w n (c p ) n0 ū n1 ū np w np +1 w np C p Ā n 0 s n 1 s Ā A n p s n p +1 s A [ 1, 1] n p s The remainder of the proof is a modification of [2, Lemma 5.8]. In most cases, especiay when is sma reative to p, this bound wi be too arge to use for the ow modes. In particuar, ū may be far from zero, resuting in a arge constant Ā. By taing sufficienty arge, the contraction given by J(, ) µ 1 wi overcome the arge bound. A more practica approach for obtaining bounds for the ow modes is given by the foowing emma. For fexibiity in baancing numerica computations (requiring a finite number of operations) with anaysis (to obtain truncation bounds), we choose M m to be the dimension used to spit these sums. Lemma 6.3. For M m, ( ) ((c p ū p ) w ) C (p, j,, M)r j + ɛ (p,, M) [ 1, 1]. j j=0 22
23 Proof. This emma is a modification of [2, Lemma 5.10] combined with Lemma 6.2. In [2, Lemma 5.10], the bound is spit into finite sums and the tai term, bounded by pα p 1 C p Ā p A [ ] 1 (M 1) s 1 (s 1) (M ) s + 1 (M + ) s [ 1, 1]. We obtain a poynomia in r by rewriting the finite sums as foows: ( )( ) (c p ) n0 ū n1 ū np w n1 w n P n ni = n n i <M = n = n = n = j=0 ( P ni = n n 1,..., n p <m n 0 <M P ni = n n 1,..., n p <m n 0 <M P ni = n n 1,..., n p <m n 0 <M ( ) j r j n <(p )(m 1)+M P ni + n= n i <M (c p ) n0 ū n1 ū np )( P ni + n= n i <M (c p ) n0 ū n1 ū np j=0 (c p ) n0 ū n1 ū np j=0 [ 1, 1] P ni = n n 1,..., n p <m n 0 <M w n1 w n ) ( ) j P ni + n= m n 1,..., n j <M n j+1,..., n <m ( ) r j [ 1, 1] j (c p ) n0 ū n1 ū np w n1 w n P ni + n= m n 1,..., n j <M n j+1,..., n <m P ni + n= m n 1,..., n j <M n j+1,..., n <m A j s n 1 s n j s A j s n 1 s n j s Remar 6.4. Note that in Lemma 6.3, C (p, j,, M) captures the contribution to the ( j)th poynomia coefficient from the -th derivative of the p-th monomia term of f in the Tayor expansion. If M = m, then C (p, j,, M) = 0 for a j > 0 and C (p, 0,, m) = (c p ) n0 ū n1 ū np. n 0 + +n p = n 0,..., n p <m For M > m there is aso a (sma) contribution to the coefficients of higher degrees of r in the poynomias, whie simutaneousy decreasing the ɛ term. This offers a method for using additiona computations to decrease the bound ɛ if this bound proves to be too arge for the vaidation procedure. Now consider the inear terms ((1 Jf (ū)) w) = ((1 Jf (ū)) w F ) + ((1 Jf (ū)) wĩ). (6.6) 23
24 If (1 Jf (ū)) 0 for ony finitey many vaues of, then direct computation of tight bounds for this term is possibe. This is the case for the systems we consider in Section 5. In order to reduce the computationa cost of the vaidation procedure, we here use a sight modification of Lemma 6.2 to aso give anaytic bounds of these terms. These bounds wi aso cover the case where (1 Jf (ū)) 0 for infinitey many vaues of. We begin by estabishing asymptotic bounds for the terms (1 Jf (ū)). Let m + m and compute 1 (J F F f 1 Jf F (ū)) 0 = 0 (ū) = (J F F f F (ū)) 0 < < m J(, )f (ū) m < m +. Using Lemma 6.2, for m + > 0, (6.7) { 0 if f 1 Jf (ū) n (ū) = 0 for a n m + µ 1 d p=2 pαp C p Ā p 1 s (6.8) otherwise. Now for a, This eads to the foowing bound 1 Jf (ū) Āinear s [ 1, 1]. (6.9) ((1 Jf (ū)) w F ) (1 Jf (ū)) n w n (6.10) n m Using a sight modification of Lemma 6.2, = r 1 Jf (ū) n [ 1, 1] n m r 2Āinear [ 1, 1]. s 1 ((1 Jf (ū)) wĩ) αāineara s s [ 1, 1]. (6.11) For notationa purposes, set ɛ F and C F (p, j,, M) to be the m-vectors as defined in Section 3.2 and for 0 n < m, define ɛ F, CF (p, j,, M), and such that ɛ n := ɛ n [ 1, 1], C K, F C n (p, j,, M) := C n (p, j,, M)[ 1, 1], C K, n := C K, n [ 1, 1]. For 0 < m, we substitute the bounds from Lemma 6.3, (6.10), and (6.11) into 24
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