Fast solvers for seismic imagingMatthias TausPostdoctoral Associate,Department of MathematicsIn collaboration with Leonardo Zepeda-Núñez, Russell J. Hewett, Laurent Demanet

MIT Earth Resources Laboratory2017 Annual Founding Members MeetingMay 31, 2017

Motivation[Slide text – ok to change size]

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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• Inhomogeneous media

• Point sources

• High frequency

Model problem

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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��u� !2mu = f+A.B.C.

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! ... Frequency

m ... Squared Slowness

f ... Source term

Finite Element Methods (FEM):

• Discontinuous wave speed

• Point sources

• Pollution error

Discretization – existing techniques

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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Finite Difference Methods:

• Discontinuous wave speed

• Point sources

• Pollution error

ku� uhkL2 C(!)(!h)

ku� uhkL2 =?

C(!) ! 1 as ! ! 1

ku� uhkL2 C(!)(!h)2

ku� uhkL2 ! 0

C(!) ! 1 as ! ! 1

Discretization – new technique

Hybridizable Discontinuous Galerkin Method (HDG)

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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Standard FEM:• Discontinuous wave speed

• Complicated meshes• Point sources

• Pollution error

ku� uhkL2 C(!)(!h)2HDG:• Discontinuous wave speed

• Simple meshes• Point sources

• Pollution error

ku� uhkL2 C(!)(!h)2

ku� uhkL2 ! 0

C(!) ! 1 as ! ! 1 C(!) C as ! ! 1

ku� uhkL2 C(!)(!h)2

Discretization – Simple Meshes

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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Standard FEM: HDG:

Improved integration rule:• Appropriate for discontinuous functions• Requires knowledge of discontinuities

Simple mesh generation

Discretization – Point sourcesSingularity cancellation technique:

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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xS ... location of singularity

u(x) = ũ(x)� 12⇡

log |x� xS |

kũ� ũhkL2 C(!)(!h)2

Discretization – Pollution error

Increase order of approximation:

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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p = log(!)

C(!) C as ! ! 1

Numerical Example: BP 2004 model

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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Numerical Example: 2nd order accuracy

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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Numerical Example: Pollution error

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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Fast algorithms: Existing techniques

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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• Iterative methods: • Direct methods:

• Preconditioned iterative methods: - Low-order discretizations- Smooth wave speeds

O(!N)

1D 2D 3D

operations O(N) O(N32) O(N2)

memory O(N) O(N logN) O(N43)

O(N)

algorithm: Method of polarized traces

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

Slide 13O(N log↵ N)

Solution: Method of Polarized Traces (PhD thesis by Leonardo Zepeda-Nunez)

• Highly inhomogeneous (discontinuous) wave speeds

• If and :

O(p↵N)

p = log(!) ! = O

✓1

h

◆N =

⇣ ph

⌘d= !d logd !

O(!d log↵+d !)

Summary

MIT Earth Resources Laboratory2017 Annual Founding Members Meeting

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Fast and accurate algorithm for wave propagation problems in geophysical applications

• 2nd order accuracy and no pollution error in the presence of- Discontinuous wave speeds- Point sources

• complexity using the method of polarized tracesO(!d log↵+d !)