High-Order Shifted Laplace Preconditioners for Wave EquationsIbrahim.Zangre/files/ECCOMAS... ·...

High-Order Shifted Laplace Preconditionersfor Wave Equations

Ibrahim Zangre

Institut Elie Cartan Nancy (IECN), Universite de LorraineApplied and Computational Electromagnetics (ACE), University of Liege

[email protected]

PhD advisors: Xavier Antoine (IECN) & Christophe Geuzaine (ACE)

(funded by Fondation EADS and ANR Microwave)

ECCOMAS 2012, September 10th 2012, Vienna.

ÉLIEInst

itu

t

CARTANNancy

I. Zangre (IECN/ACE) Preconditioning Scattering Problems ECCOMAS, Sept. 10th 2012 1 / 19

mailto:[email protected]

Outline

1 The scattering problem

2 Preconditioners

3 Shifted Laplace Preconditioners (SLP)

4 High-Order Shifted Laplace Preconditioners (HSLP)

5 Numerical results

6 Conclusion


The scattering problem

Truncated sound-soft acoustic scattering problem

−∆u − k2u = 0 in Ω,

u = f on Γ,

∂nu + Bu = 0 on Γ∞,

(1)

where Ω is the bounded domain enclosed by the fictitious boundary Γ∞ and Γand where the operator B represents an approximation of the DtN operator(for example B = −ık) on Γ∞. The vector n is the outwardly directed unitnormal to Γ∞.

Sound-hard acoustic scattering problem

The same but with the Neumann BC on Γ: does not fundamentally changethe talk


The scattering problem: an example of 2D configuration

x = 3, y = 1, e = 10−1

(Γ∞)

y

xe

(Ω)

uinc

Typically: f = −uinc = −e−ıkα·x

x = (x1, x2, x3) ∈ R3, ı =√

−1.Incidence angle α: normalized on theunit sphere (|α| = 1)k: wavenumber related tothe wavelength λ by: k = 2π/λ.



The FEM discretization of the weak form leads to the following linear system

Hhuh :=(Sh − k2

Mh − ıkBh

)uh = bh

h: mesh size

nh : total number of DOFs

Sh : stiffness matrix

Mh : mass matrix

Bh : Γ∞-surface mass matrix

nλ = λ/h: density of discretization points per wavelength

uh , bh ∈ Cnh



Difficulties (among others)

high frequency regime: we want to consider small wavelength problems:λ ≪ size(scatterer): computation of a highly oscillating solution

related to the geometry: singularities, resonances, guiding structures...

Hh is large sized, complex-valued, highly indefinite

Goals: solve efficiently the linear system

limitation of direct solvers: nh is very large for large k

we use preconditioned iterative Krylov solvers:e.g. GMRES [Saad & Schultz 87]

P−1Hhuh := P−1

(Sh − k2

Mh − ıkBh

)uh = P−1bh


A few preconditioning techniques

Classical algebraic approaches: discretize and approximate

ILU-like factorizations

pARMS (algebraic recursive multilevel)

Linear methods (diagonal, SOR, SSOR)

Multigrid...

Limitations1

These classes of preconditioning techniques are not sufficient in practice:indeed, the solver diverges at some point for large enough values of k

But: work much better for dissipative media:k has a sufficiently large positive imaginary part

ReasonThinking only in terms of matrix is not sufficient because we loose theinformation related to the true physics and included into the operator (6=positive definite matrices which are much easier to solve)

1O. Ernst and M.J. Gander, Why it is Difficult to Solve Helmholtz Problems

with Classical Iterative Methods, 2011.


A few preconditioning techniques

Analytic approaches: approximate and discretize

The idea based on an operator point of view: build a preconditioner based onthe operator approximation, and possibly hybridize with algebraic approaches

A popular approach is

SLP: Shifted Laplace Preconditioners [Erlangga & al. 2004]:

idea: introduce artificial dissipation

A(α) := −∆ − αk2 = −∆ − (a + ıb)k2

the discrete SLP (by FEM) is:

P = Aık(α),h := Sh − αk2

Mh − ıkBh

then SLP can be solved by ILU-like, ARMS, Multigrid, ...we typically use ILUT here to solve SLP


Shifted Laplace Preconditioners (SLP)

1D analysis of the spectral properties:

discrete eigenvalues of H = −∆ − k2: µn = k2n − k2 (kn = nπ, n ≥ 1)

discrete eigenvalues of A(α) = −∆ − αk2: νn = k2n − αk2

discrete eigenvalues of A−1(α)H:

σn =k2

n − k2

k2n − αk2

=1 + zn

α + zn

, zn = −k2n/k2

→ σn lie on a circular arc in the right half plane.

Choice of α = a + ıb:

in general: a = 1

two conflicting arguments for b:

b is too large: solvers (ILUT, Multigrid ...) behave well, but SLP is nomore a good approximationb is too small: SLP is a good candidate, but: solvers (ILUT, Multigrid ...)encounter the same difficulties as for the original operator!


SLP: eigenvalues clustering - 1D example

1D Example:

Ω = (0, 1), Γ = 0, Γ∞ = 1, k = 20π, nλ = 80, α = 1 + 0.5ı

Real part

Imagin

ary

part

−800 −600 −400 −200 0 200 400 600 800

−600

−400

−200

0

200

400

600

H


SLP: eigenvalues clustering - 1D example

Real part

Imagin

ary

part

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.81D - k = 20π, nλ = 80, α = 1 + 0.5ı

A−1(α)H (analytic)

zone E

zone G

zone H

[A0(α),h]

−1[Hh]

[Aik

(α),h]−1[Hh]

[Ai√

αk

(α),h ]−1[Hh]


SLP: Gmres(50) convergence - 1D example

Here an ILUT(10−3) is used to invert SLP / nλ = 200, k ∈ [100π, 8000π]

0 0.5 1 1.5 2 2.5 3

x 104

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Wavenumber

GM

RE

SIt

erati

ons

1D - Acoustic

SLP[1+1i]SLP[1+0.17i]SLP[1+0.039i]SLP[1+0.009i]


HSLP: High-Order Shifted Laplace Preconditioners

+ SLP introduces dissipation and positive definiteness in the problem

+ SLP can be coupled with ILU-like, Multigrid, ARMS...

− The Gmres convergence remains wavenumber-dependent

Can we still improve SLP?

→ We propose a generalization of SLP concept to provide more robustnessand efficiency through high order approximations in the microlocalanalysis framework

IDEA: compute a pseudo-inverse P2 of H such that

P2H = Iwhere P is the nonlocal pseudodifferential operator of order −1

P := H− 12 =

[−∆ − k2

]− 12

with symbol (continuous eigenvalue-like)

σ[P] =(|ξ|2 − k2

)− 12 =

1

ık(1 + z)− 1

2 , z = −|ξ|2k2

∈ R−


HSLP: High-Order Shifted Laplace Preconditioners

Then (1 + z)−

12 is localized by Pade rational approximants:

Pℓ1−1(z)

Qℓ1(z),

Pℓ2 (z)

Qℓ2(z)

Using poles decomposition (for possible parallel implementation),

Pℓ1−1(z)/Qℓ1 (z) =

ℓ1∑

j=1

fj

z − αj

, Pℓ2 (z)/Qℓ2 (z) = g0 +

ℓ2∑

j=1

gj

z − βj

Indeed: HSLP takes the discrete form P−1 := P1P2 where

P1 :=

ℓ1∑

j=1

fj

(Sh + αjk

2Mh + ıkBh

)−1

,P2 := g0I−k2Mh

ℓ2∑

j=1

gj

(Sh + βjk

2Mh + ıkBh

)−1

Finally: code implementation of HSLP is straightforward from that of SLP.

1D example ofeigenvalue clustering:

ℓ1 = ℓ2 = 2 (left)

ℓ1 = ℓ2 = 4 (right)

−0.5 0 0.5 1 1.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8HSLP[2|2|π

2 ]

Real part

Imag

inar

ypar

t

analytic arc

HSLP[2|2|π2 ]

−0.5 0 0.5 1 1.5

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

HSLP[4|4|π2 ]

Real part

Imag

inar

ypar

t

analytic arc

HSLP[4|4|π2 ]


1D High frequencies examples: k ∈ [100π, 8000π]

Preconditioners are solved using an ILUT(10−3) / nλ = 200

0 2000 4000 6000 8000 10000 12000 140000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Wavenumber

GM

RE

SIt

erati

ons

1D - Acoustic – Npade = 1

SLP[1+1i]

HSLP[1]

0 0.5 1 1.5 2 2.5 3

x 104

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Wavenumber

GM

RE

SIt

erati

ons


SLP[1+0.17i]

HSLP[2]


1D High frequencies examples: k ∈ [100π, 8000π]

Preconditioners are solved using an ILUT(10−3) / nλ = 200

0 0.5 1 1.5 2 2.5 3

x 104

0

500

1000

1500

Wavenumber

GM

RE

SIt

erati

ons


SLP[1+0.039i]

HSLP[4]

0 1 2 3 4 5 6 7

x 104

0

200

400

600

800

1000

1200

Wavenumber

GM

RE

SIt

erati

ons


SLP[1+0.009i]

HSLP[8]


2D case: scattering by cavity: k = 15π

x = 3, y = 1, e = 10−1

(Γ∞)

y

xe

(Ω)

uinc

Preconditioners are solved using anILUT(10−3) / nλ = 10

GMRES(50) iterations

GM

RE

SR

esid

ual

2D Cavity, nλ = 10, k = 15π

0 50 100 150 200 250 30010−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SLP[1+0.009i]HSLP[8]


Conclusion

HSLPIs a Generalization of the concept of SLP

Based idea: Square-Root operator + Pade Approximation + ILU

The limitations of HSLP are similar to the limitations of the pole’simaginary part of SLP

Hence, one important perspective is to find good rational approximationswith the possible largest imaginary parts

Extension to Maxwell equations is straightforward and already done(computational tests under progress)


Conclusion

Merci pour votre attention!


Shifted Laplace Preconditioners (SLP)

Let us consider the principal symbol of H and A(α) (ξ is the Fourier variable)

σ[H](ξ) = |ξ|2 − k2 and σ[A(α)](ξ) = |ξ|2 − αk2, ∀ξ ∈ Rd .

Then the symbol of the preconditioned operator A−1(α)H is

σ[A−1(α)

H](z) =|ξ|2 − k2

|ξ|2 − αk2=

1 + z

α + z, ∀z = −|ξ|2

k2∈ R

−.

→ The “continuous eigenvalues” of A−1(α)H ly on a circular arc with endpoints:

α−1 related to the propagative modes:H :=

(k, ξ) ∈ R × Rd/|ξ| < k

(z → 0)

(1, 0) related to the evanescent modes:E :=

(k, ξ) ∈ R × Rd/|ξ| > k

(z → −∞)

(0, 0) related to the grazing modes:G :=

(k, ξ) ∈ R × Rd/|ξ| ≈ k

(z → −1)


SLP: FEM formulation

The discrete Shifted Laplace operator reads

If one considers the classic ABC ∂nu − ıku = 0 on Γ∞:

Aık(α),h := Sh − αk2

Mh − ıkBh .

or the well-suited ABC ∂nu − ı√

αku = 0 on Γ∞:

Aı√

αk

(α),h := Sh − αk2Mh − ı

√αkBh .

1D Example:

uinc = e−ıkx1 , Ω = (0, 1), Γ = 0, Γ∞ = 1k = 20π, nλ = 80, α = 1 + 0.5ıSLP preconditioners are solved by an exact LU.


SLP: continuous eigenvalues

Real part

Imagin

ary

part

−800 −600 −400 −200 0 200 400 600 800

−600

−400

−200

0

200

400

600

H


SLP: eigenvalues clustering

Real part

Imagin

ary

part

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

A−1(α)H (analytic)

zone E

zone G

zone H

[A0(α),h]−1[Hh]

[Aik

(α),h]−1[Hh]

[Ai√

αk

(α),h ]−1[Hh]


SLP: limitations

+ SLP can be solved efficiently by multigrids methodsIt is likewise well-suited for ILU-like preconditioning(provides factorization stability because of the dissipative term)It provides positive definiteness to the Helmholtz problem

− Many eigenvalues remain close to zero (grazing modes)Convergence remains dependent of the wavenumber (linearly)

→ We propose a generalization of SLP concept to provide robustness andefficiency through high order approximations in the microlocal analysisframework


High-Order Shifted Laplace Preconditioners (HSLP)

We propose to compute a pseudo-inverse P2 of H such that

P2H = I

where P is the nonlocal pseudodifferential operator of order −1

P := H− 12 =

[−∆ − k2

]− 12

with symbol

σ[P] =(|ξ|2 − k2

)− 12 =

1

ık(1 + z)− 1

2 , z = −|ξ|2k2

∈ R−

Then P can be localized by using Pade rational approximants of

fτ (z) = (1 + z)τ , τ ∈ R.


HSLP: Pade rational approximants

They are of the form

fτ (z) ≈ f[ℓ/m]τ (z) =

Pℓ(z)

Qm(z)

where Pℓ and Qm are polynomials of degrees ℓ and m; and provide an(ℓ + m + 1)-order approximation.

The coefficients of Pℓ and Qm can be computed by solving a certain linearsystem.

For numerical purposes, we use the following decomposition

f[ℓ/m]τ (z) =

Pℓ(z)

Qm(z)= r

[ℓ/m]0 (z) +

m∑

j=1

r[ℓ/m]j

z − q[ℓ/m]j

q[ℓ/m]j : j = 1, · · · , m are the zeros of Qm(z)

r[ℓ/m]j : j = 0, · · · , m are given from the zeros of Pℓ(z) and Qm(z).


HSLP: Pade rational approximants (τ = −1/2)

Applying a θ-rotation of the branch-cut:

f− 12(z) =

1√1 + z

=1

eı θ

2

√e−ıθ(1 + z)

=e−ı θ

2

√1 + [e−ıθ(1 + z) − 1]

≃ e−ı θ

2 f[ℓ/m]

− 12

(e−ıθ(1 + z) − 1

)= r

[ℓ/m]0 (z) +

m∑

j=1

r[ℓ/m]j

z − q[ℓ/m]j

This formulation is more accurate for both regions z > −1 and z < −1


HSLP: Implicit localization of HSLP via Pade

The exact pseudo-inverse of H can then be represented by the approximateprincipal symbol

σ[H−1] = σ[P2]

= − 1

k2

1

(1 + z)

≃ − 1

k2

[r[ℓ/m]0 +

m∑

j=1

r[ℓ/m]j

z − q[ℓ/m]j

] [r[ℓ/m]0 +

m∑

j=1

r[ℓ/m]j

z − q[ℓ/m]j

]

P2 is of order −2, and so must be its symbol. This is not the case here:2(ℓ − m) + 2(ℓ − m)→ thus we split the symmetric form above



We introduce P1 and P2 such that

σ[P1]

=1

ıke−ı θ

2 f[ℓ1−1/ℓ1]

− 12

(e−ıθ(1 + z) − 1

)=

1

ık

[ℓ1∑

j=1

r[ℓ1−1/ℓ1]j

z − q[ℓ1−1/ℓ1]j

]

and

σ[P2]

=1

ıke−ı θ

2 f[ℓ2/ℓ2]

− 12

(e−ıθ(1 + z) − 1

)=

1

ık

[r[ℓ2/ℓ2]0 +

ℓ2∑

j=1

r[ℓ2/ℓ2]j

z − q[ℓ2/ℓ2]j

].



Then the exact pseudo-inverse of H is approximated by H−1 ≃ P1P2 and itsprincipal symbol reads

σ[H−1] ≃ σ[P1P2]

= − 1

k2

[ℓ1∑

j=1

r[ℓ1−1/ℓ1]j

z − q[ℓ1−1/ℓ1]j

] [r[ℓ2/ℓ2]0 +

ℓ2∑

j=1

r[ℓ2/ℓ2]j

z − q[ℓ2/ℓ2]j

]

Then the global order of the approximation is

−2 := 2(ℓ1 − 1 − ℓ1) + 2(ℓ2 − ℓ2)

and the preconditioned operator P1P2H is a perturbation of the identityoperator. We designate this approximation as HSLP[ℓ1|ℓ2|θ]; then theHSLP[1|0|θ] case reads

σ[P1P2H]

= 2eı θ

21 + z

(1 − eıθ) + z

→ HSLP[1|0|θ] is exactly the same as SLP given in with α = 1 − eıθ.For example, for θ = π/2, one finds α = 1 + ı.


HSLP: algebraic formulation

Let us come back to the Helmholtz matrix

[Hh ] = [Sh ] − k2[Mh ] − ık[Bh ]

Setting [Sh ] = [Sh ] − ık[Bh ] . It follows that

[Hh ] = [Sh ] − k2[Mh ]

= −k2[Mh ]

([I] − 1

k2[Mh ]−1[Sh ]

).

Therefore, the inverse of [Hh ] writes

[Hh ]−1 = − 1

k2

([I] − 1

k2[Mh ]−1[Sh ]

)−1

[Mh ]−1.

([I] − 1

k2 [Mh ]−1[Sh ])−1

is evaluated using pseudo-representation of (1 + z)−1,

and one finds

[Hh ]−1 ≃ P1P2


HSLP: algebraic formulation

where P1 and P2 are given by:

P1 :=

[ℓ1∑

j=1

r[ℓ1−1/ℓ1]j

([Sh ] + q

[ℓ1−1/ℓ1]j k2[Mh ]

)−1]

P2 :=

[r[ℓ2/ℓ2]0 [I] − k2[Mh ]

ℓ2∑

j=1

r[ℓ2/ℓ2]j

([Sh ] + q

[ℓ2/ℓ2]j k2[Mh ]

)−1]

Then Helmholtz matrix is then preconditioned as follows:

P1P2[Hh ]


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