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Preconditioning the Helmholtz Equation

using approximate Dirichlet-to-Neumann

operators on optimal grids

P. N. Childs1,∗, V. Druskin2, L. Knizhnerman3

1Schlumberger Gould Research, Cambridge, UK

2Schlumberger Doll Research, Boston, USA

3Central Geophysical Expedition, Moscow, Russia

∗childs4@slb.com

In seismic exploration, we wish to solve an inverse prob-

lem to determine seismic velocities in the subsurface. This

is commonly done by a PDE-constrained optimization

procedure. In the frequency domain, the appropriate PDE

is the heterogeneous Helmholtz equation. For a waveﬁeld

u(x)∈Cand wavenumber k(x), for x∈R3, this can be

written:

Au(x) = k2(x)u(x) + ∇2u(x) = f(x), x ∈Ω,

with suitable boundary conditions on ∂Ω. This provides

a challenge for development of fast preconditioners, due

to the oscillatory and non-local nature of the Green func-

tions. A complex wavenumber k∈Cis typically used

in the preconditioner to accelerate convergence of Krylov

methods.

We precondition the Helmholtz equation via domain

decomposition, with appropriate boundary conditions on

each domain. The optimized Schwarz method (OSM) [1],

[2] uses Robin boundary conditions to construct a robust

domain decomposition preconditioner. Here we approxi-

mate Dirichlet-to-Neumann (DtN) operators to construct

the preconditioner via so-called ‘optimal grids’. The pre-

conditioner is then used within an iterative Krylov method

to solve the 3D Helmholtz equation for inversion applica-

tions.

For clarity, we partition our model into non-overlapping

domains Ωiwhich have pairwise common interfaces Γij =

∂Ωi

∩∂Ωj. At each stage of an alternating Schwarz method,

we solve the Helmholtz equation in each domain Ωi:

Aui=fiin Ωi;ui+Si

∂ui

∂ni=uj−Sj

∂uj

∂njon Γij .

Here, Sidenotes the inverse DtN map for a perfectly ab-

sorbing boundary condition on ∂Ωi, and niis the outward

normal on ∂Ωi. The exact DtN is a non-local operator on

Γ. Upon discretization, this will be approximated via a

local discrete operator analogous to a perfectly matched

layer (PML).

An optimal grid is used to construct the absorbing

boundary condition, by augmenting the system with addi-

tional planes, analogous to a PML. For constant wavenum-

ber kand a planar interface, we eliminate tangential coor-

dinates by a discrete Fourier transform to arrive at a 1D

model ﬁnite volume scheme. This is augmented by adding

nadditional planes (j≥0) for the PML to match the DtN

operator:

1

ˆ

hjuj+1 −uj

hj+1 −uj−uj−1

hj+k2λuj=0 (1 ≤j≤n−1) ,

1

ˆ

h0u1−u0

h1−γu0+k2λu0= 0 ,

with a Dirichlet condition un= 0. Here λ= 1 −ξ2/k2,

where ξis the dual Fourier variable for tangential coor-

dinates at the interface. The true DtN γtrue =k√−λ

for the 1D Helmholtz equation is approximated as γ=

γ(k, {hj,ˆ

hj}) by choosing primary (hj, j = 1, ..., n) and

dual steps ˆ

hj, j = 0, ..., n −1satisfying:

{hj,ˆ

hj}= argmin

{hj,ˆ

hj}

γk, {hj,ˆ

hj}

γtrue −1

L∞(Iev∪Ipr )

.

The range I=Iev ∪Ipr is split into evanescent Iev (λ >

0) and propagating components Ipr (λ < 0), following [3],

with coeﬃcients hjand ˆ

hjobtained through an optimal

rational approximation procedure.

Although the analysis here is based on a constant

wavenumber k, we discuss extensions for heterogeneous

wavenumbers kvarying in both the normal and tangen-

tial directions at an interface. To approximate the DtN

for a Schwarz preconditioner within a Krylov solver, each

domain Ω is extended with an optimal PML, which ap-

proximates an absorbing boundary condition (ABC), and

is discretized as above. The action of the DtN may be

obtained by a Schur complement of an augmented linear

system on each domain.

We use the Schwarz method with overlap as a precon-

ditioner for solving the Helmholtz equation, where the

boundary conditions on each domain are derived from ap-

proximate DtN operators. In practical application, we

may employ a cascade of multilevel preconditioners. We

will show convergence results for the preconditioner, and

the application to 3D seismic inverse problems arising in

oilﬁeld exploration.

Acknowledgements

Many thanks are due to Ivan Graham and Douglas

Shanks from Bath University.

References

[1] M. Gander, L. Halpern and F. Magoul`es: An opti-

mized Schwarz method with two-sided Robin trans-

mission conditions for the Helmholtz equation, Int. J.

For Num. Meth. In Fluids, 55 (2007), pp.163–175.

[2] J.D. Shanks, P.N. Childs, and I.G. Graham: Shifted

Laplace DDM preconditioners for the Helmholtz equa-

tion, Proc. WAVES 2013, Gammarth, Tunisia

[3] V. Druskin, S. G¨uttel and L, Knizhnerman Near-

optimal perfectly matched layers for indeﬁnite

Helmholtz problems, MIMS EPrint: 2013.53, Univer-

sity of Manchester, 2013.