Computational Theory and Methods for Finding
Multiple Critical Points
Jianxin Zhou∗
1
Introduction
Let H be a Hilbert space and J ∈ C 1 (H, ). Denote δJ its Frechet derivative and J 0 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 0 (u) = 0. A
critical point u∗ is nondegenerate if J 00 (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 [14,15,22] 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 [9,12], 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)) = 0 x ∈ Ω
(1.1)
for u ∈ H = H01 (Ω) where Ω ⊂ N is bounded and open, f (x, t) is nonlinear satisfying
f (x, 0) = 0 and other standard conditions. The variational functional is the energy
Z
Z t
1
2
J(u) = { |∇u(x)| − F (x, u(x))} dx with F (x, t) =
f (x, τ )dτ.
(1.2)
Ω 2
0
It can be shown that (weak) solutions to (1.1) coincide with critical points of J in H.
∗
Texas A&M University, College Station, Texas, U.S.A. jzhou@math.tamu.edu
1
2
A Local Minimax Method and Its Convergence
For a subspace H 0 ⊂ H, let SH 0 = {v|v ∈ H 0 , kvk = 1}. Let L be a closed subspace in H,
called a support, for v ∈ SL⊥ , denote {L, v} = {tv + w|w ∈ L, t ∈ }.
Definition 2.1 A set-valued mapping P : SL⊥ → 2H is called the peak mapping of J w.r.t. L
if for v ∈ SL⊥ , P (v) is the set of all local maximum points of J on {L, v}. A single-valued
mapping p: SL⊥ → H is a peak selection of J w.r.t. L if p(v) ∈ P (v) ∀v ∈ SL⊥ . For v ∈ SL⊥ ,
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) ∩ SL⊥ → H s.t. p(u) ∈ P (u) ∀u ∈ N (v) ∩ SL⊥ .
Lemma 2.1 [18] Let vδ ∈ SL⊥ . If J has a local peak selection p w.r.t. L at v δ s.t. p is
continuous at vδ and dis(p(vδ ), L) > α > 0 for some α > 0, then either J 0 (p(vδ )) = 0 or for
any δ > 0 with kJ 0 (p(vδ ))k > δ, there exists s0 > 0, s.t.
J(p(v(s)))−J(p(vδ )) < −αδkv(s)−vδ k, ∀0 < s < s0 with v(s) =
vδ + sd
,
kvδ + sdk
d = −J 0 (p(vδ )).
Theorem 2.1 [18] If J has a local peak selection p w.r.t. L at v 0 ∈ SL⊥ s.t. (a) p is
continuous at v0 , (b) dis(p(v0 ), L) > 0 and (c) v0 is a local minimum point of J(p(v)) on S L⊥ ,
then p(v0 ) is a critical point of J.
Define a solution set M = {p(v) : v ∈ SL⊥ }. When L = {0}, 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 1: Given ε > 0, λ > 0 and n − 1 previously found critical points w1 , w2, . . . , wn−1 of
J, of which wn−1 has the highest critical value. Set L = span{w1 , w2 , . . . , wn−1 }. Let
v 1 ∈ SL⊥ be an ascent direction at wn−1 . Let t00 = 1, vL0 = wn−1 and set k = 0;
Step 2: Using the initial guess w = tk0 v k + vLk , solve for
w k ≡ p(v k ) = arg max J(u),
u∈[L,v k ]
and denote tk0 v k + vLk = w k ≡ p(v k );
Step 3: Compute the steepest descent vector dk = −J 0 (w k );
Step 4: If kdk k ≤ ε then output wn = w k , stop; else goto Step 5;
v k + sdk
and use initial guess u = tk0 v k ( 2λm ) + vLk to find p(v k ( 2λm ))
kv k + sdk k
in [L, v k ( 2λm )] where tk0 and vLk are found in Step 2 and, find
Step 5: Set v k (s) =
nλ o
tk0 k
k
k λ
k
m
k
k λ
s = max m m ∈ , 2 > kd k, J(p(v ( m ))) − J(w ) ≤ − kd k kv ( m ) − v k .
2
2
2
2
k
2
Step 6: Set v k+1 = v k (sk ) and update k = k + 1 then goto Step 2.
Initial guesses in Steps 2 and 5 are used to closely and consistently trace the position of the
previous point wk = tk0 v k + vLk . This strategy is also to avoid the algorithm from possible
oscillating between different branches of the peak mapping P . In Step 3, the steepest descent
direction dk = −J 0 (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 2.2 At v ∈ SL⊥ , if d = −J 0 (p(v)) 6= 0, denote v(s) =
v + sd
and define the
kv + sdk
stepsize at v by
n o
1
s(v) = max s λ > skdk, J(p(v(s))) − J(p(v)) ≤ − dis(L, p(v))kdkkv(s) − vk .
λ≥s≥0
2
(2.1)
Theorem 2.2 Let v k and w k = p(v k ), k = 0, 1, 2, ... be the sequence generated by the algorithm
s.t. p is continuous at v k and w k 6∈ L, then the algorithm is strictly decreasing and thus stable.
Theorem 2.3 [19] 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 > 0, ∀k = 0, 1, 2, ... and (c) inf v∈SL⊥ J(p(v)) > −∞,
ki
ki
then (1) {w k }∞
k=0 has a subsequence {w = p(v )} converges to a critical point; (2) any limit
k
point of {w } is a critical point of J. In particular, any convergent subsequence {w ki } of {w k }
converges to a critical point.
o
o
n
n
Let us define K = w ∈ H J 0 (w) = 0
and Kc = w ∈ H J 0 (w) = 0, J(w) = c .
Theorem 2.4 [19] Assume that V1 and V2 are open sets in H with ∅ 6= (V2 ∩SL⊥ ) ⊂ (V1 ∩SL⊥ ).
Set V10 = V1 ∩ SL⊥ and V20 = V2 ∩ SL⊥ . If a peak selection p of J w.r.t. L determined by the
algorithm is continuous in V10 satisfying (a) p(V10 ) ∩ K ⊂ Kc , where c = inf v∈V10 J(p(v)) and
(b) there is d > 0, s.t.
inf{J(p(v))|v ∈ V10 , dis(v, ∂V10 ) ≤ d} = a > b = sup{J(p(v))|v ∈ V20 } > c.
Let λ < d be chosen in the stepsize rule of the algorithm and {w k }∞
k=0 be generated by the
algorithm started from any v 0 ∈ V20 and dis(L, w k ) > α > 0 ∀k = 0, 1, 2, .., then for any
ε > 0, there exists an N s.t. for any k > N ,
dis(w k , Kc ) < ε.
Theorem 2.5 Assume the peak selection p determined by the algorithm is continuous. If
J(p(v̄)) = loc minv∈SL⊥ J(p(v)) and p(v̄) is an isolated critical point with dis(p(v̄), L) > 0,
then there exists an open set V in H, v̄ ∈ V ∩ SL⊥ , s.t. starting from any v0 ∈ V ∩ SL⊥ , the
sequence {w k }∞
k=0 generated by the algorithm converges to p(v̄).
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.
3
3
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 (1.1), Ding-Ni defined a solution manifold [25]
Z
1
2
M = v ∈ H0 (Ω)|v 6= 0,
|∇v| − vf (v) dx = 0 ,
(3.1)
Ω
and proved that a global minimizer of J on M yields a solution (MI= 1). Using Green’s
identity, we can rewrite (3.1) as
n
Z
o
1
M = v ∈ H0 (Ω)|v 6= 0, v [∆v + f (v)] dx = 0 = v ∈ H01 (Ω)|v 6= 0, hJ 0(v), viH01 (Ω) = 0 .
Ω
it becomes an orthogonal condition. This observation inspires us to
Definition 3.1 Let L be a closed subspace of H. A set-valued mapping P : S L⊥ → 2H is an L⊥ mapping of J if P (v) = {u ∈ {L, v} : J 0 (u) ⊥ {L, v}} ∀v ∈ SL⊥ . A mapping p: SL⊥ → H
is an L-⊥ selection of J if p(v) ∈ P (v) ∀v ∈ SL⊥ . Let v ∈ SL⊥ 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 3.1 If J is C 1 , then the graph G = {(u, v) : v ∈ SL⊥ , u ∈ P (v) 6= ∅} is closed.
Thus the ill-condition for a local peak selection has been removed.
Definition 3.2 Let v ∗ ∈ SL⊥ 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 ∗) ∩ SL⊥ ) = {u∗ },
i.e., for each v ∈ N (v ∗) ∩ SL⊥ and v 6= v ∗ either p(v) 6∈ L or p(v) = u∗ .
Theorem 3.1 Let v ∗ ∈ SL⊥ and p be a local L-⊥ selection of J at v ∗ and continuous at
v ∗ . Assume either p(v ∗ ) 6∈ 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 0 (p(v ∗ )) ⊥ p(v) − p(v ∗ ),
4
∀v ∈ N (v ∗) ∩ SL⊥ .
(3.2)
Remark 3.1 The nature of a critical point is about orthogonality. Except for J 0 , J is not
involved in the theorem. It implies that the result is still true for non variational problem. Replacing J 0 by an operator A : H → H in the definition of a local L-⊥ selection p, Theorem 3.1
provides a necessary and sufficient condition that u ∗ = p(v ∗ ) solves A(u) = 0, a potentially
useful result in solving multiple solutions to non variational problems. We have the flowing
local characterization of a critical point.
Lemma 3.2 [33] Let H be a Hilbert space, L be a closed subspace of H and J ∈ C 1 (H, ).
Let v ∈ SL⊥ and p be a local L-⊥ selection of J at v s.t. (a) p is continuous at v, (b) either
p(v) 6∈ 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 SL⊥ , then u = p(v) is a critical point of J.
Replacing the local maximization by a local orthogonalization in Steps 2 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 = {w1 , w2 , .., wn } and v ∈ SL⊥ . Following the definition of p, u =
t0 v + t1 w1 + ... + tn wn = p(v) is solved from (n+1) orthogonal conditions
F0 (v, t0 , t1 , ..., tn ) ≡ hJ 0 (t0 v + t1 w1 + ... + tn wn ), vi = 0,
Fj (v, t0 , t1 , ..., tn ) ≡ hJ 0 (t0 v + t1 w1 + ... + tn wn ), wj i = 0,
j = 1, ..., n
for t0 , t1 , ..., tn . We have
∂F0
∂t0
∂F0
∂ti
∂Fj
∂t0
∂Fj
∂ti
= hJ 00 (t0 v + t1 w1 + ... + tn wn )v, vi,
= hJ 00 (t0 v + t1 w1 + ... + tn wn )wi , vi,
i = 1, 2, ..., n.
= hJ 00 (t0 v + t1 w1 + ... + tn wn )v, wj i,
j = 1, 2, ..., n.
= hJ 00 (t0 v + t1 w1 + ... + tn wn )wi , wj i,
i, j = 1, 2, ..., n.
By the implicit function theorem, if the (n + 1) × (n + 1) matrix
 00
hJ (u)v, vi, hJ 00 (u)w1, vi, ... hJ 00 (u)wn , vi
 hJ 00 (u)v, w1i, hJ 00 (u)w1, w1 i, ... hJ 00 (u)wn , w1 i
Q00 = 
 ...
...
00
00
hJ (u)v, wni, hJ (u)w1, wn i, ... hJ 00 (u)wn , wn i




(3.3)
is invertible or |Q00 | 6= 0, then p is differentiable at and near v. This condition can be easily
and numerically checked.
5
4
Instability Analysis by the Local Minimax Method
Definition 4.1 A vector v ∈ H is said to be a decreasing direction of J at a critical point
u∗ ∈ H if there exists T > 0 s.t. J(u∗ + tv) < J(u∗ ) ∀ T > t > 0.
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 00 (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 0 ⊕ H + where
H − , H 0 and H + are respectively the maximum negative definite, the null and the maximum
positive definite subspaces of J 00 (u∗ ) in H with dim(H 0 ) < ∞. The Morse index (MI) of u∗ is
MI(u∗ ) = dim(H − ). When u∗ is nondegenerate, i.e., H 0 = {0}, MI(u∗ ) is the local instability
index of J at u∗ [28]. 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) + 1, 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) + 1.
Theorem 4.1 [32] Let v ∗ ∈ SL⊥ . 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 ∗ ) 6∈ L and (c) v ∗ = arg minv∈SL⊥ J(p(v)). If either
H − ∩ {L, v ∗ }⊥ = {0} or p is differentiable at v ∗ , then u∗ is a critical point with
dim(L) + 1 = MI (u∗ ) + dim(H 0 ∩ {L, v ∗ }).
(4.1)
Remark 4.1 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) + 1 is better than MI in measuring local instability for a degenerate
saddle point u∗ .
One question still remains: how to determine dim(H 0 ∩ {L, v ∗ })? Since it is usually
very expensive to find H 0 , we point out that one does not have to find H 0 to determine
dim(H 0 ∩ {L, v ∗ }. To see this, let L = {w1 , ..., wn } where w1 , ..., wn are linearly independent
and define the quadratic function
1 00 ∗
hJ (u )(t0 v ∗ + t1 w1 + ... + tn wn ), (t0 v ∗ + t1 w1 + ... + tn wn )i
2
1
=
(t0 , t1 , ..., tn )Q00 (t0 , t1 , ..., tn )T
2
Q(t0 , t1 , .., tn ) ≡
(4.2)
where Q00 is the (n + 1) × (n + 1) matrix given in (3.3). Let us observe that (t0 v ∗ + t1 w1 +
... + tn wn ) ∈ H 0 ∩ {L, v ∗ } implies (t0 , t1 , ..., tn ) ∈ ker(Q00 ), i.e.,
n
o
(H 0 ∩ {L, v ∗ }) ⊂ t0 v ∗ + t1 w1 + ... + tn wn : (t0 , t1 , ..., tn ) ∈ ker{Q00 } .
ker{Q00 } = {0} or |Q00 | 6= 0 implies dim(H 0 ∩ {L, v ∗ }) = 0 even H 0 6= {0}.
6
Theorem 4.2 [32] Let v ∗ ∈ SL⊥ . Assume J has a local peak selection p w.r.t. L at v ∗ s.t.
p is continuous at v ∗ , u∗ ≡ p(v ∗ ) 6∈ L and v ∗ = arg minv∈SL⊥ J(p(v)). If p is differentiable at
v ∗ , then u∗ is a critical point with 0 ≤ dim(L) + 1 − MI (u∗ ) ≤ dim(ker{Q00 }).
5
Selected Numerical Examples
∆u(x) + |x|q up (x) = 0, x ∈ Ω,
u(x) = 0,
x ∈ ∂Ω,
we consider the Lane-Emden equation (q = 0, p = 3) on a dumbbell-shaped domain Ω in
2
and the Henon equation (q = 3, p = 3) on the same dumbbell-shaped domain Ω and
(q = 9, p = 3) on the unit disk Ω in 2 . In all our numerical examples here, we set ε < 10.−4 .
For a semilinear elliptic BVP
15
1
10
0
−1
−1.5
−1
0
1
2
0
3
−1
0
−5
3
2
1
0
−1 −1.5
1
Figure 1: Lane-Emden: The domain (left) and the ground state solution (right).
15
15
10
10
0
0
−5
−1
−5
−1
0
1
2
3
−1
0
1
0
1
0
1
2
3
−1
Figure 2: Lane-Emden: The 2nd solution MI= 1 (left) and the 3rd solution MI= 1 (right).
15
15
10
10
0
0
−5
1
−1
0
0
−5
3
2
1
0
−1 −1.5
−1
1
−1.5
−1
0
1
2
3
Figure 3: Lane-Emden: A solution MI= 2 (left) and a solution MI= 3 (right).
7
2
1
0.256
0.392
5
50
0
1
1
x−axis
0
−1
0.597
6
0.3
92
0.25
2
3
x−axis
s
axi
0
2
1.2 1.42
1
1.14
7
0.3
24
−1
−1
−0.5
3
1.5
51.62
.35
1.218
.01
34 0.871 1.08 1
0.66
0.46 6
1
0.324
87
0.1
0.119
0.0505
0.7
−1
−1.5
0
0.11
9
0.18
y−axis
w−axis
0
1.49
39
0.9
0.802
29
0.5
0.597
56
0.392
0.2
0.0505
1
05
0.05
197
00.1.18 1 0.529
6
0.4 0.802
0.3
24
0.66
06
0.939 .734
0.871
0.0
y−
1
Figure 4: Henon: (q=3) The ground state solution (left) and its contours (right).
7
6
1
5
y−axis
7
1.2
2.122
3.83.55
45
2
4.25
0
19
0.4
3
0.8
w−axis
4
3.4 4.68
2.98
1.7
0.41
9
1
−1
−1.5
0
−1
0
1
2
3
x−axis
−1
−1
−1
0
0
y−ax
1
is
2
1
3
is
x−ax
Figure 5: Henon: (q=3) The second solution MI= 1 (left) and its contours (right).
7
6
5
1
0.238
w−axis
4
0.4
1
1.0
5
6
1.2
0.7
1.01
0.494
38
1
1.261.0
4.34
3.82
0.4
0.2
3894
5
0.7
y−axis
5
1.52
4.85
2.2 3.31
29.54.08
4
1.77
0.238
0.238
2
3.57
2.852
2.03
1.
0
1
0.7
5
6
3.0
5.6
2
94
0.7
0.2
38
0.2
0.494
3
0
−1
−1.5
−1
−1
−1
0
1
2
3
x−axis
is
ax
y−
0
1
3
−1.5
−1
0
1
2
x−axis
Figure 6: Henon: (q=3) A solution MI= 2 (left) and its contours (right).
32.18
30
33.11
30
20
u−axis
u−axis
20
10
10
0
0
−6.13
−1
1
−6.3081
Y−axis
0
−1
−1
Y−a
is
−ax
0
xis
X
−1
0
0
1
1
X−axis
Figure 7: Henon: q=9. A 1-peak solution MI= 1 (left) and a 2-peak solution MI= 2 (right).
8
27.97
30
20
u−axis
u−axis
20
10
10
0
0
−5.327
−1
−5.848
−1
Y−a
xis
Y−
−1
0
0
1
1
ax
X−axis
is
0
−1
0
1
1
X−axis
Figure 8: Henon: q=9. A 3-peak solution MI= 3 (left) and a 4-peak solution MI= 4 (right).
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