Download slides - Optimal Point Configurations and Orthogonal Polynomials

OPCOP 2017, Peter Dragnev, IPFW
Logarithmic and Riesz Equilibrium for
Multiple Sources on the Sphere — the
Exceptional Case
P. D. Dragnev∗
Indiana University-Purdue University Fort Wayne (IPFW)
∗
with J. Brauchart - TU Graz, E. Saff - Vanderbilt, R. Womersley - UNSW
OPCOP 2017, Peter Dragnev, IPFW
Outline
•
OPC, OP and external field problems in C
•
OPC on Sd - why minimize energy?
•
OPC separation - external field on Sd
•
Multiple sources Log external fields on S2
•
Multiple sources (d − 2)-external fields on Sd
OPCOP 2017, Peter Dragnev, IPFW
OPC, OP and external field problems in C
Classical energy problems
• Electrostatics - capacity, equilibrium measures;
• Geometry - transfinite diameter (OPC);
• Polynomials - Chebyshev constant (OP)
• Classical theorem in potential theory
External field problems
• Characterization theorem of weighted equilibrium
• Examples
• Applications to orthogonal polynomials on the real line
Constrained energy problems
• Characterization theorem of constrained equilibrium
• Examples
• Applications to discrete orthogonal polynomials
OPCOP 2017, Peter Dragnev, IPFW
Classical energy problem and equilibrium measure
Electrostatics - capacity of a conductor cap(E)
E - compact set in C, µ ∈ M(E) - probability measure on E;
Equilibrium occurs when potential (logarithmic) energy I(µ) is
minimized.
Z Z
VE := inf{I(µ) := −
log |x−y | dµ(x)dµ(y )}, cap(E) := exp(−VE )
Remark: For Riesz energy we use Riesz kernel |x − y |−s instead.
Equilibrium measure µE
If cap(E) > 0, there exists unique
R µE : I(µE ) = VE .
Potential satisfies U µE (x) = − log |x − y | dµ(y ) = C on E.
Examples
• E = T, dµE = dθ/(2π)
√
• E = [−1, 1], dµE = dx/π 1 − x 2
OPCOP 2017, Peter Dragnev, IPFW
Classical theorem in potential theory
Geometry - transfinite diameter of a set δ(E)
E ⊂ C - compact, Zn = {z1 , z2 , . . . , zn } ⊂ of E maximize Vandermond
Fekete points (OPC)
2/(n(n−1))

δn (E) := max 
Zn ⊂E
Y
|zi − zj |
, δ(E) := lim δn (E)
1≤i<j≤n
Approximation Theory - Chebyshev constant τ (E)
E - compact set in C, Tn (x) - monic polynomial of minimal uniform
norm (OP for L2 -norm);
tn (E) := min{kx n − pn−1 (x)k : pn−1 ∈ Pn−1 }, τ (E) = lim tn 1/n (E)
Classical theorem (Fekete, Szegö)
cap(E) = δ(E) = τ (E)
OPCOP 2017, Peter Dragnev, IPFW
External field problem - characterization theorem
Electrostatics - add external field
E - closed set in C, Q - lower semi-continuous on E (growth cond.);
Z
VQ := inf{IQ (µ) := I(µ) + 2 Q(x) dµ(x)
Theorem - Weighted equilibrium µQ
There exists unique µQ : IQ (µQ ) = VQ .
Potential satisfies: U µQ (x) + Q(x) ≥ C q.e. on E
U µQ (x) + Q(x) ≤ C on supp(µQ ).
Applications
• Orthogonal polynomials on real line
• Approximation of functions by weighted polynomials
• Integrable systems, Random matrices
OPCOP 2017, Peter Dragnev, IPFW
Constrained energy problem
Electrostatics - add external field and upper constraint
Add constraint measure σ: σ(E) > 1
VQσ := inf{IQ (µ) := I(µ) + 2
Z
Q(x) dµ(x) : µ ≤ σ
Applications: Discrete orthogonal polynomials, random walks,
numerical linear algebra methods, etc.
Theorem (R ’96, Saff-D. ’97) - Constrained equilibrium λσQ
There exists unique λσQ : IQ (λσQ ) = VQσ .
σ
Potential satisfies: U λQ (x) + Q(x) ≥ C on supp(σ − λσQ )
σ
U λQ (x) + Q(x) ≤ C on supp(µ).
Theorem (Saff-D. ’97) - Constrained vs. weighted equilibrium
If Q ≡ 0, then σ − λσ = (kσk − 1)µQ for Q(x) = −U σ (x)/(kσk − 1)
OPCOP 2017, Peter Dragnev, IPFW
OPC on S2 - why minimize energy? Electrostatics:
Thomson Problem (1904) (“plum pudding” model of an atom)
Find the (most) stable (ground state) energy
configuration (code) of N classical electrons
(Coulomb law) constrained to move on the
sphere S2 .
Generalized Thomson Problem (1/r s potentials and log(1/r ))
A code C := {x1 , . . . , xN } ⊂ Sn−1 that minimizes Riesz s-energy
Es (C) :=
X
j6=k
1
s,
|xj − xk |
s > 0,
is called an optimal s-energy code.
Elog (ωN ) :=
X
j6=k
log
1
|xj − xk |
OPCOP 2017, Peter Dragnev, IPFW
OPC on S2 - why minimize energy? Coding:
Tammes Problem (1930)
A Dutch botanist that studied modeling of the
distribution of the orifices in pollen grain
asked the following.
Tammes Problem (Best-Packing, s = ∞)
Place N points on the unit sphere so as to
maximize the minimum distance between
any pair of points.
Definition
Codes that maximize the minimum distance are called optimal
(maximal) codes. Hence our choice of terms.
OPCOP 2017, Peter Dragnev, IPFW
OPC on S2 - why minimize energy? Nanotechnology:
Fullerenes (1985) - (Buckyballs)
Vaporizing graphite, Curl, Kroto, Smalley,
Heath, and O’Brian discovered C60
(Chemistry 1996 Nobel prize)
Duality structure: 32 electrons and C60 .
OPCOP 2017, Peter Dragnev, IPFW
Other "Fullerenes"
Under the lion paw
Montreal biosphere
OPCOP 2017, Peter Dragnev, IPFW
Computational "Fulerene" - Rob Womersley
OPCOP 2017, Peter Dragnev, IPFW
Known OPC on S2
Recall: Riesz Oprimal Configurations
A configuration ωN := {x1 , . . . , xN } ⊂ Sd that minimizes Riesz
s-energy
Es (ωN ) :=
X
j6=k
1
s,
|xj − xk |
s > 0,
E0 (ωN ) :=
X
log
j6=k
1
|xj − xk |
is called an optimal s-energy configuration.
OPC on S2
• s = 0, Smale’s problem, logarithmic points (known for
N = 1 − 6, 12);
• s = 1, Thomson Problem (known for N = 1 − 6, 12)
• s = −1, Fejes-Toth Problem (known for N = 1 − 6, 12)
• s → ∞, Tammes Problem (known for N = 1 − 12, 13, 14, 24)
OPCOP 2017, Peter Dragnev, IPFW
OPC separation on Sd and external fields
Separation Distance
δ(ωN ) := min |xj − xk | ,
j6=k
(s)
ωN = {x1 , . . . , xN }
(s)
Expect: δ(ωN ) N −1/d as N → ∞, where ωN optimal for Sd
Definition
d
A sequence of N-point configurations {ωN }∞
N=2 ⊂ S is
well-separated if there exists some c > 0 not depending on N s.t.
δ(ωN ) ≥ c N −1/d for all N.
OPCOP 2017, Peter Dragnev, IPFW
OPC separation on Sd and external fields
Separation Results for Optimal Point Configurations on Sd
d = 2, s = 0
0 < s< d −2
s =d −1
d −1≤s <d
d −2 ≤ s< d
s= d
s>d
s=∞
(0)
δ(ωN ) ≥ O(N −1/2 )
(s)
δ(ωN ) ≥?
(d−1)
δ(ωN
) ≥ O(N −1/d )
(s)
δ(ωN ) ≥ O(N −1/d )
(s)
δ(ωN ) ≥ βs,d N −1/d
(d)
−1/d
δ(ωN ) ≥ O((N log N)
)
(s)
δ(ωN ) ≥ O(N −1/d )
(∞)
δ(ωN ) ≥ O(N −1/d )
Asymptotic Results (H-vdW (1951), Bo-H-S (2007))
R-S-Z (1995)
Dahlberg (1978)
K-S-S (2007)
D-S (2007)
K-S (1998)
K-S (1998)
Conway-Sloane
OPCOP 2017, Peter Dragnev, IPFW
Logarithmic Points on S2 and external field
Separation Results for Logarithmic Configurations on S2
√
(0)
δ(ωN ) ≥ (3/5)/ N
√
(0)
δ(ωN ) ≥ (7/4)/ N
√
(0)
δ(ωN ) ≥ 2/ N − 1
R-S-Z (1995)
Dubickas (1997)
D. (2002)
OPCOP 2017, Peter Dragnev, IPFW
Logarithmic Points on S2 and external field
Separation Results for Logarithmic Configurations on S2
√
(0)
δ(ωN ) ≥ (3/5)/ N
√
(0)
δ(ωN ) ≥ (7/4)/ N
√
(0)
δ(ωN ) ≥ 2/ N − 1
R-S-Z (1995)
Dubickas (1997)
D. (2002)
Proof.
• R-S-Z, Dubickas: Stereographical projection with South Pole
in ωN .
• Dragnev: Stereographical projection with North Pole in ωN . This
creates external field on projections of remaining N − 1
points {zk }. All weighted Fekete
√ points are contained in support
of continuous MEP, i.e. |zk | ≤ N − 2, which implies estimate.
OPCOP 2017, Peter Dragnev, IPFW
OPC separation on Sd and external fields
Approach for Sd
(s)
• Fix a point of ωN and consider external field QN it generates on
the remaining n = N − 1 points.
• Study continuous energy problem for this external field QN .
• Discrete energy points for QN are contained in CEP equilibrium
support.
Theorem (D-Saff 2007) OPC separation on Sd for d − 2 ≤ s < d
1/d
Ks,d
2B(d/2, 1/2)
(s,d)
δ(ωN ) ≥ 1/d , Ks,d :=
,
B(d/2, (d − s)/2)
N
where B(x, y ) denotes the Beta function. In particular,
p
Kd−1,d = 21/d , Ks,2 = 2 1 − s/2.
Remark: We need Principle of Domination, de la Valleè-Pousin type
theorem, and Riesz balayage, hence the restriction on s.
OPCOP 2017, Peter Dragnev, IPFW
Discrete MEP on Sd for d − 2 ≤ s < d
Q-optimal points
Let Q be an external field. Find Q-optimal
configuration of n points on Sd , that solve


n 
X
1
d
min
+
Q(x
)
+
Q(x
)
:
x
∈
S
k
j
k
s


|xj − xk |
j6=k
Q(x) =
q
|x − Rp|s
2007 Separation: q = 1/(N − 2), R = 1,
n = N − 1.
OPCOP 2017, Peter Dragnev, IPFW
Discrete MEP on Sd for d − 2 ≤ s < d
Q-optimal points
Let Q be an external field. Find Q-optimal
configuration of n points on Sd , that solve


n X

1
d
min
+
Q(x
)
+
Q(x
)
:
x
∈
S
j
k
k
s


|xj − xk |
j6=k
Q(x) =
q
|x − Rp|s
Key idea:
2007 Separation: q = 1/(N − 2), R = 1,
n = N − 1.
OPCOP 2017, Peter Dragnev, IPFW
Discrete MEP on Sd for d − 2 ≤ s < d
Q-optimal points
Let Q be an external field. Find Q-optimal
configuration of n points on Sd , that solve


n X

1
d
min
+
Q(x
)
+
Q(x
)
:
x
∈
S
j
k
k
s


|xj − xk |
j6=k
Q(x) =
q
|x − Rp|s
2007 Separation: q = 1/(N − 2), R = 1,
n = N − 1.
Key idea:
Theorem
Q-optimal points are contained in supp(µQ ).
OPCOP 2017, Peter Dragnev, IPFW
External field Continuous MEP on Sd for d − 2 ≤ s < d
K ⊂ Sd compact; M(K ) class of positive unit Borel measures µ
supported on K
Z
Z Z
−s
−s
µ
Us (x):= |x − y| d µ(y)
Is [µ]:=
|x − y| d µ(x) d µ(y)
Riesz s-potential of µ
Riesz s-energy of µ
Ws (K ) := inf {Is [µ] : µ ∈ M(K )}
Riesz s-energy of K
Extremal measure
Given an external field Q on K , there exists unique extremal
measure µQ that minimizes the weighted energy
Z
Is [µ] + 2 Q d µ,
µ ∈ M(K ),
characterized by Usµ (x) + Q(x) ≥ C on Sd with "=" on supp(µQ ).
OPCOP 2017, Peter Dragnev, IPFW
Physicist’s Problem (Signed Equilibrium)
Given compact K ⊂ Sd , Q external field on K , find a signed measure
ηQ s.t.
UsηQ (x) + Q(x) = const.
everywhere on K
ηQ (K ) = 1
Definition
ηQ = ηQ,K is called signed equilibrium on K associated with Q.
Proposition
If ηQ exists, then it is unique.
Theorem
−
+
+
Let ηQ,K = ηQ,K
− ηQ,K
. Then supp(µQ,K ) ⊆ supp(ηQ,K
)
OPCOP 2017, Peter Dragnev, IPFW
Mhaskar-Saff Fs -functional and signed equilibrium
Definition (Fs -Mhaskar-Saff functional for general Q)
Z
Fs (K ) := Ws (K ) + Q d µK ,
Ws (K ) is s-energy of K .
Theorem
If d − 2 ≤ s < d with s > 0, then Fs is minimized for SQ := supp(µQ ).
Proposition (Connection to signed equilibrium)
If d − 2 ≤ s < d with s > 0, Q : K → R continuous and Ws (K ) < ∞,
η
then Us Q,K + Q ≡ Fs (K ) on K .
OPCOP 2017, Peter Dragnev, IPFW
Example (Brauchart-Saff-D., 2009)
Η-Q
Η+Q
t
K = Sd ,
s
Qa (x) = q/ |x − a| ,
R = |a| ≥ 1
−
+
η Qa = η Q
− ηQ
a
a
Let Σt be spherical cap centered at South Pole of height −1 ≤ t ≤ 1
−
+
supp(ηQ
) = Σt(Qa ) , supp(ηQ
) = Sd \ Σt(Qa ) .
a
a
Remark
+
If ηQa ≥ 0, then µQa = ηQa . If not, then supp(µQa ) ⊆ supp(ηQ
).
a
OPCOP 2017, Peter Dragnev, IPFW
Finding µQa when supp(µQa ) = Sd ; BDS ’09
Gonchar’s Problem for Sd
Let q = 1, s = d − 1 (Newton potential).
Find R0 > 0 s.t. for Qa (x) = |x − a|1−d , a = Rp
(
= Sd if R ≥ R0 ,
supp(µQa )
( Sd if R < R0 .
Proposition
For s = d − 1,
"
d ηQa (x) = 1 +
1
R d−1
−
R2 − 1
|x − a|
d+1
#
d σd (x)
√
1+ 5
If d = 2, then R0 − 1 =
. When d = 4, R0 − 1 = Plastic
2
number from architecture (see Padovan sequence Pn+3 = Pn+1 + Pn ).
OPCOP 2017, Peter Dragnev, IPFW
Finding µQa when supp(µQa ) ( Sd ; BDS ’09
Theorem
s
Let d − 2 ≤ s < d. Qa (x) = q/ |x − a| . Signed equilibrium on Σt is
ηt := ηQa ,Σt =
t = Bals (δa , Σt ),
1 + qkt k
ν t − q t ,
kνt k
νt = Bals (σd , Σt ).
The weighted s-potential is
Usηt (z) + Qa (z) = Fs (Σt )
Usηt (z)
on Σt ,
+ Qa (z) = Fs (Σt ) + · · ·
on Sd \ Σt .
The equilibrium measure µQa = ηt0 , where t = t0 (by minimizing
Fs -functional) is the unique solution of
d−s
Ws (Sd )
1 + q kt k
q (R + 1)
=
.
d/2
kνt k
(R 2 − 2Rt + 1)
OPCOP 2017, Peter Dragnev, IPFW
t
t0
-1
1
t > t0 ,
Η't
Ηt
Us +Qa
Usηt (z)
Usηt (z)
t0 t
-1
Ηt
t=t0
-1
Η't
-1
Usηt (z)
Usηt (z)
1
t t0
+ Qa (z) = Fs (Σt ) on Σt ,
on Σt .
t < t0 ,
Ηt
Us +Qa
t
+ Qa (z) ≥ Fs (Σt ) on Sd \ Σt ,
ηt0 ≥ 0
1
t0
-1
on Σt .
t = t0 ,
Usηt (z)
Usηt (z)
t=t0 1
-1
+ Qa (z) = Fs (Σt ) on Σt ,
ηt0 0
1
Us +Qa
Η't
+ Qa (z) ≥ Fs (Σt ) on Sd \ Σt ,
1
+ Qa (z) Fs (Σt )
on Sd \ Σt ,
+ Qa (z) = Fs (Σt ) on Σt ,
ηt0 ≥ 0
on Σt .
OPCOP 2017, Peter Dragnev, IPFW
Separation of Q-optimal OPC on Sd ; BDS ’14
s
Set Q(x) := q/ |x − b| , |b| > 1, let {x1 , x2 , . . . , xN } be a Q-optimal
e
OPC. If xN is the fixed, then {x1 , x2 , . . . , xN−1 } is a Q-optimal
OPC
−s
e
with Q(x)
= Q(x) + |x − xN | /(N − 2).
Theorem
e
• If d − 2 ≤ s < d, then all Q-Fekete
points are in supp(µQe ).
+
• In addition, supp(µQ
e ) ⊆ supp(η e ) for any compact
Q,K
supp(µQe ) ⊆ K ⊆ Sd .
Theorem
If d − 2 < s < d, then
(s)
δ(ωQ,N )
≥
2B(d/2, 1/2)
(1 + q)B(d/2, (d − s)/2)
1/d
N −1/d .
OPCOP 2017, Peter Dragnev, IPFW
Multiple source Log external fields on S2 - BDSW ’17
Q(x) :=
k
X
i=1
qi log
p
1
, ai ∈ S2 , q := q1 + · · · + qk , ri := 2 qi /(1 + q).
|x − ai |
Theorem
Let qi be small enough, such that Σi := {x : |x − ai | < ri },
i = 1, . . . , k , are non-intersecting. Then
supp(µQ ) = S2 \ ∪Σi ,
µQ = (1 + q)σ2|supp(µ ) .
Moreover, all Q-optimal OPC are contained in S2 \ ∪Σi .
Q
OPCOP 2017, Peter Dragnev, IPFW
Multiple source Log external fields on S2 - BDSW ’17
The same theorem holds if we substitute point masses at ai with
localized rotational measures dφi , namely let
Qφ (x) :=
k
X
U0φi (x)
=
i=1
k Z
X
i=1
log
1
dφi (y)
|x − y|
where dφi = fi (hx, ai i)dσ2 (x), fi (u) ≡ 0, u ∈ [−1,
qi := kφi k.
q
1 − ri2 /2],
Theorem
Let qi be small enough, such that Σi := {x : |x − ai | < ri },
i = 1, . . . , k , are non-intersecting. Then
supp(µQφ ) = S2 \ ∪Σi ,
µQφ = (1 + q)σ2|supp(µ
Qφ )
Moreover, all Qφ -optimal OPC are contained in S2 \ ∪Σi .
.
OPCOP 2017, Peter Dragnev, IPFW
Multiple source Log external fields on C - BDSW ’17
Fix ak at the north pole, projection K : S2 → C, wi = K(ai ),
k −1
q
X
1
e
Q(z)
:=
qi log
+ (1 + q) log 1 + |z|2 ,
z ∈ C.
|z − wi |
i=1
e
e
Q(z)
is admissible as lim|z|→∞ Q(z)
− log |z| = ∞.
Theorem
For small enough qi there are open discs D1 , . . . , Dm−1 in C with
wi ∈ Di = K(Σi,i ), 1 ≤ i ≤ m − 1, such that
s
(
) m−1
[
1 + q − qm
SQe = z ∈ C : |z| ≤
\
Di .
qm
i=1
e is given by
The extremal measure µQe associated with Q
dµQe (z) =
1+q
2
π (1 + |z|2 )
dA(z).
(1)
OPCOP 2017, Peter Dragnev, IPFW
Multiple source (d − 2)-external fields - BDSW ’17
Q(x) :=
k
X
i=1
qi
,
|x − ai |d−2
ai ∈ Sd .
In this case, there is a mass loss in the balayage process and the
radii of the spherical caps Σi are determined implicitly.
Theorem
For qi small enough, there are unique ρi such that
Σi := {x : |x − ai | < ρi }, i = 1, . . . , k , are non-intersecting and
supp(µQ ) = Sd \ ∪Σi ,
µQ = Cσd |supp(µ ) .
Q
Moreover, all Q-optimal OPC are contained in Sd \ ∪Σi .
Remark (Connection to Facility Allocation Problem on Sd )
In [FoCM ’15] Carlos Beltran reformulated the Log OPC problem on
S2 as facility allocation problem. Possible generalization to Sd .
OPCOP 2017, Peter Dragnev, IPFW
Regions of electrostatic influence and s-OPC
Qs (x) :=
m
X
i=1
qi
,
|ai − x|s
q i :=
qi
.
1 + q − qi
Let Φs (ti ) be the Fs -functional associated with q i /|ai − xi |s evaluated
for Σi . Let γ i denote the unique solution of the equation
Φs (ti ) =
2d−s q i
,
γid
1 ≤ i ≤ m.
(2)
Theorem
Let d − 2 ≤ s < d, d ≥ 2, and let γ = (γ 1 , . . . , γ m ) be the vector of
solutions of (2). Then the support SQs of the s-extremal
measure µQs
Tm
associated with Qs is contained in the set Σγ = i=1 Σi,γ i .
Furthermore, no point of an optimal N-point configuration w.r.t. Qs
lies in Σi,γ i , 1 ≤ i ≤ m.
OPCOP 2017, Peter Dragnev, IPFW
Regions of electrostatic influence and s-OPC
Figure: Approximate Coulomb-optimal points
for m = 2, N = 4000,
√
√
3
3
q1 = q2 = 41 , a1 = (0, 0, 1) and a2 = (0, 1091 , − 10
) or a2 = (0, 1091 , 10
)
OPCOP 2017, Peter Dragnev, IPFW
Regions of electrostatic influence and s-OPC
Problem
The two images below compare approximate log-optimal
configurations with 4000 and 8000 points - overlapping case. We
pose as an open problem the precise determination of the support in
such a case.
OPCOP 2017, Peter Dragnev, IPFW
THANK YOU!