SHAPE OPTIMIZATION PROBLEMS COMING FROM HILBERTIAN FUNCTIONAL - - PowerPoint PPT Presentation

SHAPE OPTIMIZATION PROBLEMS

COMING FROM

HILBERTIAN FUNCTIONAL CALCULUS

Michel Crouzeix Universit´ e de Rennes

NUMERICAL RANGE Let A ∈ Cd,d be a square matrix. Its numerical range is defined by W(A) := {A v, v ; v ∈ Cd, v = 1}.

NUMERICAL RANGE Let A ∈ Cd,d be a square matrix. Its numerical range is defined by W(A) := {A v, v ; v ∈ Cd, v = 1}.

• W(A) is a convex subset of C (Toeplitz, Hausdorff),
• W(A) contains the spectrum σ(A),

NUMERICAL RANGE Let A ∈ Cd,d be a square matrix. Its numerical range is defined by W(A) := {A v, v ; v ∈ Cd, v = 1}.

• W(A) is a convex subset of C (Toeplitz, Hausdorff),
• W(A) contains the spectrum σ(A),
• ∂W(A) is an algebraic curve of degree ≤ d(d−1) and of class d,

NUMERICAL RANGE Let A ∈ Cd,d be a square matrix. Its numerical range is defined by W(A) := {A v, v ; v ∈ Cd, v = 1}.

• W(A) is a convex subset of C (Toeplitz, Hausdorff),
• W(A) contains the spectrum σ(A),
• ∂W(A) is an algebraic curve of degree ≤ d(d−1) and of class d,
• W(A) is the intersection of the half-planes Πθ,

Πθ := {x+iy ; x cos θ + y sin θ ≤ µ(θ)}, where µ(θ) is the largest eigenvalue of

1 2 (eiθA + e−iθA∗).

Let Ω = ∅ be a convex domain of C We consider the function C(Ω) defined by : C(Ω) := sup

d,r,A

{r(A) ; W(A) ⊂ Ω, |r(z)| ≤ 1 in Ω}, In this definition d ∈ N∗, r is a rational function r : C → C and A a square matrice A ∈ Cd,d.

Let Ω = ∅ be a convex domain of C We consider the function C(Ω) defined by : C(Ω) := sup

d,r,A

{r(A) ; W(A) ⊂ Ω, |r(z)| ≤ 1 in Ω}, In this definition d ∈ N∗, r is a rational function r : C → C and A a square matrice A ∈ Cd,d.

• Remark. C(Ω) only depends on the shape of Ω.

i.e. C(λ Ω + µ) = C(Ω) for all λ = 0 and µ ∈ C.

Let Ω = ∅ be a convex domain of C We consider the functions C(Ω, d) and C(Ω) defined by : C(Ω, d) := sup

r,A

{r(A) ; W(A) ⊂ Ω, |r(z)| ≤ 1 in Ω}, C(Ω) := sup

d

C(Ω, d). In the first definition r is a rational function r : C → C and A a square matrice A ∈ Cd,d.

Let Ω = ∅ be a convex domain of C We consider the functions C(Ω, d) and C(Ω) defined by : C(Ω, d) := sup

r,A

{r(A) ; W(A) ⊂ Ω, |r(z)| ≤ 1 in Ω}, C(Ω) := sup

d

C(Ω, d). In the first definition r is a rational function r : C → C and A a square matrice A ∈ Cd,d. In other words C(Ω) is the best constant such that the inequality r(A) ≤ C(Ω) sup

z∈Ω

|r(z)|, holds for all rational functions r and all matrices A with W(A) ⊂ Ω.

What is the interest ? The estimate r(A) ≤ C(Ω) sup

z∈Ω

|r(z)|, plays for non self-adjoint operators a similar role to the inequality r(A) ≤ sup

z∈σ(A)

|r(z)|, which is well-known for selfadjoint (or normal) operators.

What is the interest ? The estimate r(A) ≤ C(Ω) sup

z∈Ω

|r(z)|, plays for non self-adjoint operators a similar role to the inequality r(A) ≤ sup

z∈σ(A)

|r(z)|, which is well-known for selfadjoint (or normal) operators. This allows to develop a functional calculus along the framework

• f Alan Mc Intosh.

(An excellent review is provided by a book of Markus Haase).

What is known ?

• (J. von Neumann, 1951) C(Π) = 1, if Π is a half-plane,

What is known ?

• (J. von Neumann, 1951) C(Π) = 1, if Π is a half-plane,
• (B.&F. Delyon, 1999) C(Ω) < +∞, if Ω is bounded,

What is known ?

• (J. von Neumann, 1951) C(Π) = 1, if Π is a half-plane,
• (B.&F. Delyon, 1999) C(Ω) < +∞, if Ω is bounded,
• (M.C.& B. Delyon, 2003) C(S) < 2 + 2/

√ 3, if S is a strip of a convex sector,

What is known ?

• (J. von Neumann, 1951) C(Π) = 1, if Π is a half-plane,
• (B.&F. Delyon, 1999) C(Ω) < +∞, if Ω is bounded,
• (M.C.& B. Delyon, 2003) C(S) < 2 + 2/

√ 3, if S is a strip of a convex sector,

• (M.C., 2003) C(Ω, 2) ≤ 2, and C(Ω, 2) = 2 iff Ω is a disk,

What is known ?

• (J. von Neumann, 1951) C(Π) = 1, if Π is a half-plane,
• (B.&F. Delyon, 1999) C(Ω) < +∞, if Ω is bounded,
• (M.C.& B. Delyon, 2003) C(S) < 2 + 2/

√ 3, if S is a strip of a convex sector,

• (M.C., 2003) C(Ω, 2) ≤ 2, and C(Ω, 2) = 2 iff Ω is a disk,
• (C. Badea, 2003) C(D) = 2, if D is a disk,

see also (K. Okubo & T. Ando, 1975)

What is known ?

• (J. von Neumann, 1951) C(Π) = 1, if Π is a half-plane,
• (B.&F. Delyon, 1999) C(Ω) < +∞, if Ω is bounded,
• (M.C.& B. Delyon, 2003) C(S) < 2 + 2/

√ 3, if S is a strip of a convex sector,

• (M.C., 2003) C(Ω, 2) ≤ 2, and C(Ω, 2) = 2 iff Ω is a disk,
• (C. Badea, 2003) C(D) = 2, if D is a disk,

see also (K. Okubo & T. Ando, 1975)

• (M.C., 2004) C(Ω) < 33.75, in any case.

Some applications.

• proof of a Burkholder conjecture in ergodic theory,

Some applications.

• proof of a Burkholder conjecture in ergodic theory,
• characterization of similarities of ω-accretive operators,

Some applications.

• proof of a Burkholder conjecture in ergodic theory,
• characterization of similarities of ω-accretive operators,
• characterization for generators of cosine functions,

Some applications.

• proof of a Burkholder conjecture in ergodic theory,
• characterization of similarities of ω-accretive operators,
• characterization for generators of cosine functions,
• simplification of the proof of a Boyadzhiev-de Laubenfels theo-

rem, concerning decomposition for group generators,

Some applications.

• proof of a Burkholder conjecture in ergodic theory,
• characterization of similarities of ω-accretive operators,
• characterization for generators of cosine functions,
• simplification of the proof of a Boyadzhiev-de Laubenfels theo-

rem,

• improvement of the convergence estimates for Krylov methods,

in computational linear algebra,

Some applications.

• proof of a Burkholder conjecture in ergodic theory,
• characterization of similarities of ω-accretive operators,
• characterization for generators of cosine functions,
• simplification of the proof of a Boyadzhiev-de Laubenfels theo-

rem,

• improvement of the convergence estimates for Krylov methods,
• stability and convergence estimates for time discretizations of

evolutive problems.

A first shape optimization problem It corresponds to Q = sup C(Ω), with the constraint Ω convex subset of C.

A first shape optimization problem It corresponds to Q = sup C(Ω), with the constraint Ω convex subset of C. The only known results are Q ≤ 33.75 and C(Ω) is lower semi-continuous w.r.t. Ω. Conjecture Q = 2.

A second shape optimization problem It corresponds to, for fixed d ≥ 3, Qd = sup C(Ω, d), with the constraint Ω convex subset of C.

A second shape optimization problem It corresponds to, for fixed d ≥ 3, Qd = sup C(Ω, d), with the constraint Ω convex subset of C. We know much more facts

• There exists Ωo such that Qd = C(Ωo, d), and Ωo is bounded.

A second shape optimization problem It corresponds to, for fixed d ≥ 3, Qd = sup C(Ω, d), with the constraint Ω convex subset of C. We know much more facts

• There exists Ωo such that Qd = C(Ωo, d), and Ωo is bounded.
• There exists a matrix A ∈ Cd,d such that W(A) = Ωo and a

holomorphic function f in Ωo, bounded by 1, continuous up to the boundary, such that C(Ωo, d) = f(A).

• Let a be a conforming map from Ωo onto the unit disk. The

function f has the form (Blaschke product) f(z) =

d−1

• j=1

a(z) − a(ζj) 1 − a(ζj) a(z) , with ζj ∈ Ω.

• Let a be a conforming map from Ωo onto the unit disk. The

function f has the form (Blaschke product) f(z) =

d−1

• j=1

a(z) − a(ζj) 1 − a(ζj) a(z) , with ζj ∈ Ω.

• But, even in the simplest case d = 3, I have not been able

to obtain more information concerning the optimal domain Ωo, (unable to deduce some symmetry properties)... Conjecture Qd = 2.

In order to prove the bound Q ≤ 33.75, I have first looked for an estimate of C(Ω) which only depends on the geometry of Ω. This estimate is given in the next slide

C(Ω) ≤ 2 +

2π

G(θ−ψ) dψ G(α) := max(α, π − α) π sin α θ−ψ = π 2 − arctan ρ′(ψ) ρ(ψ) , where σ(ψ) = ρ(ψ) eiψ. ∂Ω ψ θ

• σ

The estimate can be written also C(Ω) ≤ J(Ω) := 2 +

2π

g

ρ′(ψ)

ρ(ψ)

• dψ.

with g(t) =

π 2 + | arctan t|

π

• 1 + t2.

The estimate can be written also C(Ω) ≤ J(Ω) := 2 +

2π

g

ρ′(ψ)

ρ(ψ)

• dψ.

with g(t) =

π 2 + | arctan t|

π

• 1 + t2.

Noticing that g(t) ≤ g1(t) := 1

2 + |t|, we have also

C(Ω) ≤ J1(Ω) := 2 + π + TV (log ρ).

• Remark. The functions g and g1 are convex and even.

A last shape optimization problem This problem is (for γ ∈ [0, π/2) given) Qγ = max{J(Ω) ; Ω is convex and min ρ(ψ) max ρ(ψ) ≥ cos γ}.

A last shape optimization problem This problem is (for γ ∈ [0, π/2) given) Qγ = max{J(Ω) ; Ω is convex and min ρ(ψ) max ρ(ψ) ≥ cos γ}. Recall that g is even and convex on R. J(Ω) =

2π

g

ρ′(ψ)

ρ(ψ)

• dψ,

Ω = {r eiψ ; 0 ≤ r < ρ(ψ), ψ ∈ [0, 2π]}. We can assume, without loss of generality, that max ρ(ψ) = 1.

For each value of γ there is a optimal domain. All optimal domains are polygonals with n sides, n−1 < π

γ ≤ n, and

satisfy

• either “all vertices are on the unit circle and all but one apothems

are equal to cos γ”,

• or “all vertices but one are on the unit circle and all apothems

are equal to cos γ”.

I am mainly interested by the situation cos γ ≤ 1/2. In these case the optimal domains Ω are the two triangles γ γ γ γ γ γ γ γ That shows the estimate C(Ω) ≤ 33.75 if cos γ ≥ 0.0004.

• Proof. We start from a polygonal situation with a fixed number

m of sides. α3/2 α2 αm−1/2 αm α1/2 α1 σ1 σ2 σ3 σ0 = σm σ1/2 σ3/2 |σ1/2| is assumed minimum

With the previous notations we have J(Ω) =

2m

• k=1

αk/2

g(tan ϕ) dϕ. The constraints on Ω are periodicity

2m

• k=1

αk/2 = 2π,

m

• j=1

log cos αj−1/2 cos αj = 0, convexity αj−1/2 + αj > 0, αj + αj+1/2 > 0, j = 1, . . . , m, flatness |σj| ≤ 1, |σj+1/2| ≥ |σ1/2| = cos γ, j = 1, . . . , m.

Translated only in angle terms, we have J(Ω) =

2m

• k=1

αk/2

g(tan ϕ) dϕ. The constraints on Ω become

2m

• k=1

αk/2 = 2π,

m

• j=1

log cos αj−1/2 cos αj = 0, αj−1/2 + αj > 0, αj + αj+1/2 > 0, j = 1, . . . , m log 1 cos α1 +

k−1

• j=1

log cos αj+1/2 cos αj+1 ≤ log 1 cos γ, k = 1, . . . , m

k

• j=1

log cos αj+1/2 cos αj ≥ 0, k = 1, . . . , m − 1.