Optimized Schwarz Methods and Best Approximation Problem. · Optimized Schwarz Methods and Best...

Optimized Schwarz Methods andBest Approximation Problem.

Laurence Halpern

LAGA-Institut Galilée -Université Paris 13

JJJ III × DD16, january 2005 – p. 1/26

Table of contents

Presentation of the problem;

Optimized Schwarz methods for theconvection-diffusion equation;A non-standard problem of bestapproximation.Relevance of the optimization


Table of contents

Presentation of the problem;Optimized Schwarz methods for theconvection-diffusion equation;

A non-standard problem of bestapproximation.Relevance of the optimization


Table of contents

Presentation of the problem;Optimized Schwarz methods for theconvection-diffusion equation;A non-standard problem of bestapproximation.

Relevance of the optimization


The toy problem

Domain of interest

ΙΩ

Ω2Ω1

original problem domain decomposition


The Schwarz algorithm

Ω1 = (−∞, L)× Rn , Ω2 = (0,∞)× Rn.

L(uk+11 ) = f in Ω1 × (0, T )

uk+11 (·, 0) = u0 in Ω1

B1uk+11 (L, ·) = B1u

k2(L, ·) in (0, T )

L(uk+12 ) = f in Ω2 × (0, T )

uk+12 (·, 0) = u0 in Ω2

B2uk+12 (0, ·) = B2u

k1(0, ·) in (0, T )


The convection-diffusion equation

L(u) := ut − ν∆u+ a∂1u+ b · ∇u+ cu = f

u(·, 0) = u0

ν > 0, a > 0, b ∈ Rn, c > 0.

Fourier transform in time (t↔ ω)and in the tangential variable (y ↔ k)


The convection-diffusion equation

L(u) := ut − ν∆u+ a∂1u+ b · ∇u+ cu = f

u(·, 0) = u0

ν > 0, a > 0, b ∈ Rn, c > 0.Fourier transform in time (t↔ ω)

and in the tangential variable (y ↔ k)


The optimal Schwarz method

THEOREM The Schwarz method converges in twoiterations with or without overlap when theoperators Bi are given by :

B1 = ∂1 − Λ−, B2 = ∂1 − Λ+

Λ± Pseudo Differential Operator of order 1 in∂t, ∂y.

λ± =a∓ (a2 + 4ν(i(ω + b · k) + ν|k|2 + c))1/2

2ν


Approximate transmission conditions and convergence rate

Ba1 = ∂1 − Λ−a , Ba2 = ∂1 − Λ+a

λ− + λ+ =a

ν→ λ−a + λ+

a =a

νCONVERGENCE RATE

ρ(ω, k, λ−a , L) =

(λ− − λ−aλ+ − λ−a

)2

e(λ−−λ+)L

ek+2j (ω, 0, k) = ρ(ω, k, λ−a , L)ekj (ω, 0, k)


Variations of the convergence rate

δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c)

λ− =a− δ1/2

2ν;λ−a =

a− P2ν

;

ρ(ω, k, λ−a , L) =

(P − δ1/2

P + δ1/2

)2

e−2δ1/2L

REMARK :lim

(ω,k)→+∞

∣∣∣∣P − δ1/2

P + δ1/2

∣∣∣∣ = 1


Approximate transmission conditions

P = p ∈ P0 →B1 ≡ ∂1 −

a− p2ν

,

P = p+ qx ∈ P1 →

B1 ≡ ∂1 −a− p

2ν+ q(∂t + b · ∇− ν∆S + cI)

.

THEOREM(V. Martin) For p > 0, (resp. p, q > 0, p > a2

4νq ), the

algorithm converges with and without overlap.



P = p ∈ P0 →B1 ≡ ∂1 −

a− p2ν

,

P = p+ qx ∈ P1 →

B1 ≡ ∂1 −a− p

2ν+ q(∂t + b · ∇− ν∆S + cI)

.


4νq ), the




P = p ∈ P0 →B1 ≡ ∂1 −

a− p2ν

,

P = p+ qx ∈ P1 →

B1 ≡ ∂1 −a− p

2ν+ q(∂t + b · ∇− ν∆S + cI)

.


4νq ), the



Choice of the coefficients

δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c)

ρ(ω, k, λ−a , L) =

(P − δ1/2

P + δ1/2

)2

e−2δ1/2L



δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c)

ρ(ω, k, λ−a , L) =

(P − δ1/2

P + δ1/2

)2

e−2δ1/2L

* Taylor approximation,δ1/2 ≈ x0 =

√a2 + 4νc,

δ1/2 ≈ x0 + 2ν((i(ω + b · k) + ν|k|2 + c)/x0.



δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c)

ρ(ω, k, λ−a , L) =

(P − δ1/2

P + δ1/2

)2

e−2δ1/2L

* Discrete frequencies : IT = ( πT, π

∆t), Ij = ( π

Xj, π

∆xj).

For a given n, find P in Pn minimizing

supω∈IT ,kj∈Ij

|ρ(ω, k, λ−a , L)|


A new best approximation problem


The problemK is a compact set in C. f is continuous on K, such that

f(K) ⊂ z ∈ C : Re z > 0.

δn(l) = infp∈Pn

supz∈K

∣∣∣∣p(z)− f(z)

p(z) + f(z)e−lf(z)

∣∣∣∣ ,

Find p∗n such that supz∈K

∣∣∣∣p∗n(z)− f(z)

p∗n(z) + f(z)e−lf(z)

∣∣∣∣ = δn(l)

The classical problem (De la Vallée-Poussin) ♣

infp∈Pn‖p− f‖Lq([a,b])

.JJJ III × DD16, january 2005 – p. 12/26

Existence result

A KEY-GEOMETRICAL LEMMA

D(z0, δ) = z ∈ C,∣∣∣∣z − z0

z + z0

∣∣∣∣ < δ

δ < 1

PSfrag replacementsz0

δ > 1

Z0


Existence result

THEOREM For any n ≥ 0, there exists a bestapproximation polynomial p∗n.

Proof :δn = infp∈Pn

supz∈K

∣∣∣∣p(z)− f(z)

p(z) + f(z)

∣∣∣∣ ,

1. δn < 1,2. a minimizing sequence is such that

pk

f (z) ∈ D(1, δn + ε), thus is bounded in Pnby the lemma.


Equioscillation result

THEOREM For any n ≥ 0, there exist at least n+ 2points z1, · · · , zn+2 in K such that

∣∣∣∣p∗n(zi)− f(zi)

p∗n(zi) + f(zi)

∣∣∣∣ =

∥∥∥∥p∗n − fp∗n + f

∥∥∥∥∞.

Proof : by contradiction, using that the derivativeof

Aw : Pn → Cm, p 7→(p(wi)− f(wi)

p(wi) + f(wi)

)

1≤i≤m

is continuous and onto.


Uniqueness

THEOREM For any n ≥ 0, the best approximationpolynomial p∗n is unique.Proof :

1. The set of best approximation polynomials isconvex,

2. As in the classical case, if P 1 et P 2 are twobest approximation polynomials, so isP = (P 1 + P 2)/2, and equioscillates in atleast n+ 2 points zj in K.


Uniqueness

PSfrag replacements

P 1(zk)

f(zk)

P 2(zk)

f(zk)

P (zk)

f(zk)

1

2

P 1(zk) + P 2(zk)

f(zk)

D(1, δn)

P = (P 1 + P 2)/2


A local result without overlap

THEOREM

For any n ≥ 0, any strict local minimum forδn =

∣∣∣∣∣∣∣∣p− fp+ f

∣∣∣∣∣∣∣∣ is a global minimum.


The 1d case

n = 0 : calculations by hand (with maple!!).n = 1 :1. we calculate by hand a solution

equioscillating in 3 real points,2. asymptotic expansions in 1/∆t prove it to be

a strict local minimum,3. conclusion by previous theorem.And the constrains on the coefficients forwell-posedness are satisfied.


Asymptotic convergence ratesL ≈ C1∆x and ∆t ≈ C2∆xβ , β ≥ 1.

method β = 1 β = 2

Taylor order 0 1−O(∆x1/2) 1−O(∆x1/2)

P0 approximation 1−O(∆x1/4) 1−O(∆x1/3)

Taylor order 1 1−O(∆x1/2) 1−O(∆x1/2)

P1 approximation 1−O(∆x1/8) 1−O(∆x1/5)

WITH OVERLAPmethod asymptotic convergence rate

Taylor order 0 1−O(∆t1/2)

P0 approximation 1−O(∆t1/4)

Taylor order 1 1−O(∆t1/2)

P1 approximation 1−O(∆t1/8)

WITHOUT OVERLAP


Comparison

The best method: P1 approximation,

∆t ≈ ∆x : performances with or without overlapare comparable,

∆t ≈ ∆x2 : performances with overlap are higher.


Numerical validation in 1-D

Dataν = 0.2, a = 1, c = 0.

u(0, t) = 0 and u(6, t) = 0.Ω = (0, 6), T = 2.5. u(x, 0) = e−3(1.2−x)2.Ω1 = (0, 3.04), Ω2 = (2.96, 6), L = 0.08.

Numerical scheme: upwind in space +backwardEuler in time.

∆x = 0.02,∆t = 0.005.Initial guess : random.


Comparison

0 1 2 3 4 5 6 7 8 9 1010−12

10−10

10−8

10−6

10−4

10−2

100

102

iteration

erro

r

ClassicalTaylor 0Optimized 0Taylor 1Optimized 1


Relevance of the optimization

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

q

p

−9

−8 −8

−7

−7

−7

−6

−6

−6

−6

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−2

−2

−10

Error after 5 iterations as a function of p and q.


Conclusionperspectives


Extensions

The 2D case for P0 and P1 optimization,

Approximation by rational fractions as forabsorbing boundary conditions, ♣The wave equation,The case of systems of equations,etc..


Extensions

The 2D case for P0 and P1 optimization,Approximation by rational fractions as forabsorbing boundary conditions, ♣

The wave equation,The case of systems of equations,etc..


Extensions

The 2D case for P0 and P1 optimization,Approximation by rational fractions as forabsorbing boundary conditions, ♣The wave equation,

The case of systems of equations,etc..


Extensions

The 2D case for P0 and P1 optimization,Approximation by rational fractions as forabsorbing boundary conditions, ♣The wave equation,The case of systems of equations,

etc..


Extensions

The 2D case for P0 and P1 optimization,Approximation by rational fractions as forabsorbing boundary conditions, ♣The wave equation,The case of systems of equations,etc..


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