fast solvers for seismic imaging - mit erl matthias... · 2017. 6. 5. · fast solvers for seismic...
TRANSCRIPT
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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 !)
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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 !)