Computational Theory and Methods for Finding Multiple Critical Points

Transcription

1 Computational Theory and Methods for Finding Multiple Critical Points Jianxin Zhou Introduction Let H be a Hilbert space and J C (H, ). Denote δj its Frechet derivative and J its gradient. The objective of this research is to develop computational theory and methods for finding multiple critical points, i.e., solutions to the Euler-Lagrange equation J (u) =. A critical point u is nondegenerate if J (u ) is invertible. Otherwise u is degenerate. The first candidates for a critical point are the local extrema to which the classical critical point theory was devoted in calculus of variation. Most conventional numerical algorithms focus on finding such stable solutions. Critical points that are not local extrema are unstable and called saddle points. In physical systems, saddle points appear as unstable equilibria or transient excited states. (In)Stability is one of the main concerns in system design and control. However, in many applications, performance or maneuverability is more desirable. An unstable solution can be stable enough to accomplish a short term mission before being excited to decay to another desirable state [4,5,] and it may have much higher performance or maneuverability index. Due to unstable nature, those multiple solutions are very elusive to be numerically captured. Little is known in the literature. Minimax method is one of the most popular approaches in critical point theory, But most minimax theorems, such as, the mountain pass, various linking and saddle point theorems, focus on the existence issue. They require one to solve a two-level global optimization problem and therefore are not for algorithm implementation. Inspired by the numerical work of Chio-McKenna and Ding-Costa-Chen in [9,], the idea of Nehari and Ding-Ni to define a solution manifold and the Morse theory, we present a local minimax method (LMM). Consider a semilinear elliptic BVP as a model problem u(x) + f(x, u(x)) = x Ω (.) for u H = H(Ω) where Ω N is bounded and open, f(x, t) is nonlinear satisfying f(x, ) = and other standard conditions. The variational functional is the energy J(u) = { t u(x) F (x, u(x))} dx with F (x, t) = f(x, τ)dτ. (.) Ω It can be shown that (weak) solutions to (.) coincide with critical points of J in H. Texas A&M University, College Station, Texas, U.S.A. jzhou@math.tamu.edu

2 A Local Minimax Method and Its Convergence For a subspace H H, let S H = {v v H, v = }. Let L be a closed subspace in H, called a support, for v S L, denote {L, v} = {tv + w w L, t }. Definition. A set-valued mapping P : S L H is called the peak mapping of J w.r.t. L if for v S L, P (v) is the set of all local maximum points of J on {L, v}. A single-valued mapping p: S L H is a peak selection of J w.r.t. L if p(v) P (v) v S L. For v S L, we say that J has a local peak selection w.r.t. L at v, if there is a neighborhood N (v) of v and a mapping p: N (v) S L H s.t. p(u) P (u) u N (v) S L. Lemma. [8] Let v δ S L. If J has a local peak selection p w.r.t. L at v δ s.t. p is continuous at v δ and dis(p(v δ ), L) > α > for some α >, then either J (p(v δ )) = or for any δ > with J (p(v δ )) > δ, there exists s >, s.t. J(p(v(s))) J(p(v δ )) < αδ v(s) v δ, < s < s with v(s) = v δ + sd v δ + sd, d = J (p(v δ )). Theorem. [8] If J has a local peak selection p w.r.t. L at v S L s.t. (a) p is continuous at v, (b) dis(p(v ), L) > and (c) v is a local minimum point of J(p(v)) on S L, then p(v ) is a critical point of J. Define a solution set M = {p(v) : v S L }. When L = {}, M corresponds to Nehari or Ding-Ni s solution manifold. Then a local minimum point of J on M yields a saddle point, which can be approximated by, e.g., a steepest descent method. It leads to the following local minimax algorithm Step : Given ε >, λ > and n previously found critical points w, w,..., w n of J, of which w n has the highest critical value. Set L = span{w, w,..., w n }. Let v S L be an ascent direction at w n. Let t =, v L = w n and set k = ; Step : Using the initial guess w = t k v k + vl k, solve for w k p(v k ) = arg max J(u), and denote t k v k + v u [L,v k L k = w k p(v k ); ] Step : Compute the steepest descent vector d k = J (w k ); Step 4: If d k ε then output w n = w k, stop; else goto Step 5; Step 5: Set v k (s) = vk + sd k v k + sd k in [L, v k ( λ m )] where t k and vk L and use initial guess u = tk vk ( λ m ) + v k L to find p(vk ( λ m )) are found in Step and, find { λ s k = max m, m > d k, J(p(v k ( λ m ))) m J(wk ) tk dk v k ( λ } ) m vk.

3 Step 6: Set v k+ = v k (s k ) and update k = k + then goto Step. Initial guesses in Steps and 5 are used to closely and consistently trace the position of the previous point w k = t k v k + vl k. This strategy is also to avoid the algorithm from possible oscillating between different branches of the peak mapping P. In Step, the steepest descent direction d k = J (w k ) (or the negative gradient) at a known point w k, is usually computed by solving a corresponding linearized problem (ODE or PDE). Definition. At v S L, if d = J (p(v)), denote v(s) = stepsize at v by { s(v) = max s λ s v + sd v + sd and define the λ > s d, J(p(v(s))) J(p(v)) dis(l, p(v)) d v(s) v }. (.) Theorem. Let v k and w k = p(v k ), k =,,,... be the sequence generated by the algorithm s.t. p is continuous at v k and w k L, then the algorithm is strictly decreasing and thus stable. Theorem. [9] Let p be a peak selection of J w.r.t. L and J satisfy the (PS) condition. If (a) p is continuous, (b) dis(l, w k ) > α >, k =,,,... and (c) inf v SL J(p(v)) >, then () {w k } k= has a subsequence {wk i =p(v k i )} converges to a critical point; () any limit point of {w k } is a critical point of J. In particular, any convergent subsequence {w k i } of {w k } converges to a critical point. { } { } Let us define K = w H J (w) = and K c = w H J (w) =, J(w) = c. Theorem.4 [9] Assume that V and V are open sets in H with (V S L ) (V S L ). Set V = V S L and V = V S L. If a peak selection p of J w.r.t. L determined by the algorithm is continuous in V satisfying (a) p(v ) K K c, where c = inf v V J(p(v)) and (b) there is d >, s.t. inf{j(p(v)) v V, dis(v, V ) d} = a > b = sup{j(p(v)) v V } > c. Let λ < d be chosen in the stepsize rule of the algorithm and {w k } k= be generated by the algorithm started from any v V and dis(l, wk ) > α > k =,,,.., then for any ε >, there exists an N s.t. for any k > N, dis(w k, K c ) < ε. Theorem.5 Assume the peak selection p determined by the algorithm is continuous. If J(p( v)) = loc min v SL J(p(v)) and p( v) is an isolated critical point with dis(p( v), L) >, then there exists an open set V in H, v V S L, s.t. starting from any v V S L, the sequence {w k } k= generated by the algorithm converges to p( v). The above convergence results are based on functional analysis. It is assumed that at each step of the algorithm the computation is exact. Convergence results allowing error to exist have been recently obtained by us.

4 A Local Min-L- Method In the local minimax theory, we need to check if a local peak selection p is continuous or differentiable at v. This is very difficult, since p is not explicitly defined and the graph of p is, in general, not closed, an ill condition in numerical computation. On the other hand, non minimax type saddle points, such as the monkey saddles do exist. minimax principle can not cover them. In this section, we present a more general approach to solve those problems. Applying Nehari s idea to (.), Ding-Ni defined a solution manifold [5] M = {v H (Ω) v, [ v vf(v) ] } dx =, (.) Ω and proved that a global minimizer of J on M yields a solution (MI= ). Using Green s identity, we can rewrite (.) as } { } M = {v H (Ω) v, v [ v + f(v)] dx = = v H (Ω) v, J (v), v H (Ω) =. Ω it becomes an orthogonal condition. This observation inspires us to Definition. Let L be a closed subspace of H. A set-valued mapping P : S L H is an L- mapping of J if P (v) = {u {L, v} : J (u) {L, v}} v S L. A mapping p: S L H is an L- selection of J if p(v) P (v) v S L. Let v S L and N (v) be a neighborhood of v. if P (p) is locally defined in N (v) S L, then P (p) is a local L- mapping (selection) of J at v. It is clear that a local peak selection is a local L- selection. The graph of P can be very complex, it may contain multiple branches, U-turn or bifurcation points. We show that such defined L- selection has several interesting properties. Lemma. If J is C, then the graph G = {(u, v) : v S L, u P (v) } is closed. Thus the ill-condition for a local peak selection has been removed. Definition. Let v S L and p a local L- selection of J at v. Assume u = p(v ) L, we say that u is an isolated L- point of J w.r.t. p if there exist neighborhoods N (u ) of u and N (v ) of v s.t. N (u ) L p(n (v ) S L ) = {u }, i.e., for each v N (v ) S L and v v either p(v) L or p(v) = u. Theorem. Let v S L and p be a local L- selection of J at v and continuous at v. Assume either p(v ) L or p(v ) L is an isolated L- point of J w.r.t. p, then a necessary and sufficient condition that u = p(v ) is a critical point of J is that there exists a neighborhood N (v ) of v s.t. J (p(v )) p(v) p(v ), v N (v ) S L. (.) 4

5 Remark. The nature of a critical point is about orthogonality. Except for J, J is not involved in the theorem. It implies that the result is still true for non variational problem. Replacing J by an operator A : H H in the definition of a local L- selection p, Theorem. provides a necessary and sufficient condition that u = p(v ) solves A(u) =, a potentially useful result in solving multiple solutions to non variational problems. We have the flowing local characterization of a critical point. Lemma. [] Let H be a Hilbert space, L be a closed subspace of H and J C (H, ). Let v S L and p be a local L- selection of J at v s.t. (a) p is continuous at v, (b) either p(v) L or p(v) L is an isolated L- point of J, (c) v is a local minimum point of the function J(p( )) on S L, then u = p(v) is a critical point of J. Replacing the local maximization by a local orthogonalization in Steps and 5 in the local minimax algorithm, we obtain a local min- algorithm. Next we study how to check the differentiability of an L- selection p. When an L- selection p is introduced, various implicit function theorems can be used to check its continuity or smoothness. Let L = {w, w,.., w n } and v S L. Following the definition of p, u = t v + t w t n w n = p(v) is solved from (n+) orthogonal conditions F (v, t, t,..., t n ) J (t v + t w t n w n ), v =, F j (v, t, t,..., t n ) J (t v + t w t n w n ), w j =, j =,..., n for t, t,..., t n. We have F t = J (t v + t w t n w n )v, v, F t i = J (t v + t w t n w n )w i, v, i =,,..., n. F j t = J (t v + t w t n w n )v, w j, j =,,..., n. F j t i = J (t v + t w t n w n )w i, w j, i, j =,,..., n. By the implicit function theorem, if the (n + ) (n + ) matrix J (u)v, v, J (u)w, v,... J (u)w n, v Q = J (u)v, w, J (u)w, w,... J (u)w n, w (.) J (u)v, w n, J (u)w, w n,... J (u)w n, w n is invertible or Q, then p is differentiable at and near v. This condition can be easily and numerically checked. 5

6 4 Instability Analysis by the Local Minimax Method Definition 4. A vector v H is said to be a decreasing direction of J at a critical point u H if there exists T > s.t. J(u + tv) < J(u ) T > t >. In general, the set of all decreasing vectors of J at u does not form a linear vector space. The maximum dimension of a subspace of decreasing directions of J at u can be used to measure local instability of u and is called the local instability index of u. Since such an index lacks of characterization and is too difficult to compute, let us consider several alternatives. When J (u ) : H H is a self-adjoint Fredholm operator at a critical point u, the Hilbert space H has an orthogonal spectral decomposition H = H H H + where H, H and H + are respectively the maximum negative definite, the null and the maximum positive definite subspaces of J (u ) in H with dim(h ) <. The Morse index (MI) of u is MI(u ) = dim(h ). When u is nondegenerate, i.e., H = {}, MI(u ) is the local instability index of J at u [8]. But MI is very expensive to compute and it is ineffective in measuring local instability of a degenerate critical point. In our local minimax method, u = p(v ) is a local maximum point of J in the subspace {L, v }. The dimension of {L, v }, which equals dim(l) +, is known even before we numerically find the solution u and can be used to measure local instability of u even u is degenerate. We establish some relations between MI and the number dim(l) +. Theorem 4. [] Let v S L. Assume that J has a local peak selection p w.r.t. L at v s.t. (a) p is continuous at v, (b) u p(v ) L and (c) v = arg min v SL J(p(v)). If either H {L, v } = {} or p is differentiable at v, then u is a critical point with dim(l) + = MI (u ) + dim(h {L, v }). (4.) Remark 4. The result in the last theorem is multi-fold. First, it provides a method to evaluate MI of a saddle point without actually finding dim(h ), a very expensive job. Secondly, it indicates that dim(l) + is better than MI in measuring local instability for a degenerate saddle point u. One question still remains: how to determine dim(h {L, v })? Since it is usually very expensive to find H, we point out that one does not have to find H to determine dim(h {L, v }. To see this, let L = {w,..., w n } where w,..., w n are linearly independent and define the quadratic function Q(t, t,.., t n ) J (u )(t v + t w t n w n ), (t v + t w t n w n ) = (t, t,..., t n )Q (t, t,..., t n ) T (4.) where Q is the (n + ) (n + ) matrix given in (.). Let us observe that (t v + t w t n w n ) H {L, v } implies (t, t,..., t n ) ker(q ), i.e., { } (H {L, v }) t v + t w t n w n : (t, t,..., t n ) ker{q }. ker{q } = {} or Q implies dim(h {L, v }) = even H {}. 6

7 Theorem 4. [] Let v S L. Assume J has a local peak selection p w.r.t. L at v s.t. p is continuous at v, u p(v ) L and v = arg min v SL J(p(v)). If p is differentiable at v, then u is a critical point with dim(l) + MI (u ) dim(ker{q }). 5 Selected Numerical Examples { u(x) + x For a semilinear elliptic BVP q u p (x) =, x Ω, u(x) =, x Ω, we consider the Lane-Emden equation (q =, p = ) on a dumbbell-shaped domain Ω in and the Henon equation (q =, p = ) on the same dumbbell-shaped domain Ω and (q = 9, p = ) on the unit disk Ω in. In all our numerical examples here, we set ε < Figure : Lane-Emden: The domain (left) and the ground state solution (right) Figure : Lane-Emden: The nd solution MI= (left) and the rd solution MI= (right) Figure : Lane-Emden: A solution MI= (left) and a solution MI= (right). 7

8 w axis Figure 4: Henon: (q=) The ground state solution (left) and its contours (right) w axis Figure 5: Henon: (q=) The second solution MI= (left) and its contours (right) w axis Figure 6: Henon: (q=) A solution MI= (left) and its contours (right)...8 u axis u axis 6.8 Y axis X axis 6. Figure 7: Henon: q=9. A -peak solution MI= (left) and a -peak solution MI= (right). Y axis X axis 8

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