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

NUMERICAL RANGE Let A ∈ C d,d be a square matrix. Its numerical range is defined by W ( A ) := {� A v, v � ; v ∈ C d , � v � = 1 } . • W ( A ) is a convex subset of C (Toeplitz, Hausdorff), • W ( A ) contains the spectrum σ ( A ),

NUMERICAL RANGE Let A ∈ C d,d be a square matrix. Its numerical range is defined by W ( A ) := {� A v, v � ; v ∈ C d , � 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 ∈ C d,d be a square matrix. Its numerical range is defined by W ( A ) := {� A v, v � ; v ∈ C d , � 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 Π θ , Π θ := { x + iy ; x cos θ + y sin θ ≤ µ ( θ ) } , 1 2 ( e iθ A + e − iθ A ∗ ) . where µ ( θ ) is the largest eigenvalue of

Let Ω � = ∅ be a convex domain of C We consider the function C (Ω) defined by : {� r ( A ) � ; W ( A ) ⊂ Ω , | r ( z ) | ≤ 1 in Ω } , C (Ω) := sup d,r,A In this definition d ∈ N ∗ , r is a rational function r : C → C and A a square matrice A ∈ C d,d .

Let Ω � = ∅ be a convex domain of C We consider the function C (Ω) defined by : C (Ω) := sup {� r ( A ) � ; W ( A ) ⊂ Ω , | r ( z ) | ≤ 1 in Ω } , d,r,A In this definition d ∈ N ∗ , r is a rational function r : C → C and A a square matrice A ∈ C d,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 : {� r ( A ) � ; W ( A ) ⊂ Ω , | r ( z ) | ≤ 1 in Ω } , C (Ω , d ) := sup r,A C (Ω) := sup C (Ω , d ) . d In the first definition r is a rational function r : C → C and A a square matrice A ∈ C d,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 ) � ; W ( A ) ⊂ Ω , | r ( z ) | ≤ 1 in Ω } , r,A C (Ω) := sup C (Ω , d ) . d In the first definition r is a rational function r : C → C and A a square matrice A ∈ C d,d . In other words C (Ω) is the best constant such that the inequality � r ( A ) � ≤ C (Ω) sup | r ( z ) | , z ∈ Ω 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 | r ( z ) | , z ∈ Ω plays for non self-adjoint operators a similar role to the inequality � r ( A ) � ≤ | r ( z ) | , sup z ∈ σ ( A ) which is well-known for selfadjoint (or normal) operators.

What is the interest ? The estimate � r ( A ) � ≤ C (Ω) sup | r ( z ) | , z ∈ Ω plays for non self-adjoint operators a similar role to the inequality � r ( A ) � ≤ sup | r ( z ) | , z ∈ σ ( A ) which is well-known for selfadjoint (or normal) operators. This allows to develop a functional calculus along the framework of 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, Q d = sup C (Ω , d ) , with the constraint Ω convex subset of C .

A second shape optimization problem It corresponds to, for fixed d ≥ 3, Q d = sup C (Ω , d ) , with the constraint Ω convex subset of C . We know much more facts • There exists Ω o such that Q d = C (Ω o , d ), and Ω o is bounded.

A second shape optimization problem It corresponds to, for fixed d ≥ 3, Q d = sup C (Ω , d ) , with the constraint Ω convex subset of C . We know much more facts • There exists Ω o such that Q d = C (Ω o , d ), and Ω o is bounded. • There exists a matrix A ∈ C d,d such that W ( A ) = Ω o and a holomorphic function f in Ω o , bounded by 1, continuous up to the boundary, such that C (Ω o , d ) = � f ( A ) � .

• Let a be a conforming map from Ω o onto the unit disk. The function f has the form (Blaschke product) d − 1 a ( z ) − a ( ζ j ) � f ( z ) = with ζ j ∈ Ω . , 1 − a ( ζ j ) a ( z ) j =1

• Let a be a conforming map from Ω o onto the unit disk. The function f has the form (Blaschke product) d − 1 a ( z ) − a ( ζ j ) � f ( z ) = with ζ j ∈ Ω . , 1 − a ( ζ j ) a ( z ) j =1 • But, even in the simplest case d = 3, I have not been able to obtain more information concerning the optimal domain Ω o , (unable to deduce some symmetry properties)... Q d = 2. Conjecture

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

� 2 π C (Ω) ≤ 2 + G ( θ − ψ ) dψ 0 G ( α ) := max( α, π − α ) π sin α θ • ∂ Ω σ ψ • 0 2 − arctan ρ ′ ( ψ ) θ − ψ = π σ ( ψ ) = ρ ( ψ ) e iψ . ρ ( ψ ) , where

The estimate can be written also � 2 π � ρ ′ ( ψ ) � C (Ω) ≤ J (Ω) := 2 + g dψ. ρ ( ψ ) 0 with π 2 + | arctan t | � 1 + t 2 . g ( t ) = π

The estimate can be written also � 2 π � ρ ′ ( ψ ) � C (Ω) ≤ J (Ω) := 2 + g dψ. ρ ( ψ ) 0 with π 2 + | arctan t | � 1 + t 2 . g ( t ) = π Noticing that g ( t ) ≤ g 1 ( t ) := 1 2 + | t | , we have also C (Ω) ≤ J 1 (Ω) := 2 + π + TV (log ρ ) . Remark. The functions g and g 1 are convex and even.