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Preconditioning the Helmholtz Equation using approximate Dirichlet-to-Neumann operators on optimal grids

Authors:
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 xR3, this can be
written:
Au(x) = k2(x)u(x) + 2u(x) = f(x), x ,
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 kCis 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 =
i
j. At each stage of an alternating Schwarz method,
we solve the Helmholtz equation in each domain Ωi:
Aui=fiin Ωi;ui+Si
∂ui
∂ni=ujSj
∂uj
∂njon Γij .
Here, Sidenotes the inverse DtN map for a perfectly ab-
sorbing boundary condition on i, and niis the outward
normal on i. 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 (j0) for the PML to match the DtN
operator:
1
ˆ
hjuj+1 uj
hj+1 ujuj1
hj+k2λuj=0 (1 jn1) ,
1
ˆ
h0u1u0
h1γu0+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 =kλ
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(IevIpr )
.
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.
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Full-text available
A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included.
Article
Optimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behaviour. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2d case, which uses a different Robin condition for neighbouring subdomains at their common interface, and which we call two-sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency an asymptotic convergence factor of 1 – O(h1/4) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O(1/ω) for ⩾1, then the optimized asymptotic convergence factor is 1 – O(ω(1–2)/8). We illustrate our analysis with 2d numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.
Shifted Laplace DDM preconditioners for the Helmholtz equation
  • J D Shanks
  • P N Childs
  • I G Graham
J.D. Shanks, P.N. Childs, and I.G. Graham: Shifted Laplace DDM preconditioners for the Helmholtz equation, Proc. WAVES 2013, Gammarth, Tunisia