salvus io, adjoints & sensitivity kernels · computational seismology department of earth...
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| Computational Seismology Department of Earth Sciences, Institute of Geophysics
MESS 2017
1
Salvus IO, Adjoints & Sensitivity Kernels
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 2
Modular Design through Template Mixins
Wavefield I/O
Finite Elements
Basic Wave Equation
Additional Physics
Boundary Conditions
Tetrahedra Hexahedra
Attenuation Anisotropy
Acoustic Elastic
Absorbing
I/O
Quads Triangles
Dirichlet
Coupling
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 3
Modular Design through Template Mixins
Wavefield I/O
Finite Elements
Basic Wave Equation
Additional Physics
Boundary Conditions
Hexahedra
Elastic
I/O
template class WavefieldIO< IsotropicElastic<
TensorGll3D< Hexahedra<HexP1>,
ORDER>>>;
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 4
Modular Design through Template Mixins
Wavefield I/O
Finite Elements
Basic Wave Equation
Additional Physics
Boundary Conditions
Hexahedra
Scalar
Fluid-Solid Coupling
Absorbing BC
I/O
template class WavefieldIO< AbsorbingBoundary<
SolidToFluid< Scalar<
TensorGll3D< Hexahedra<HexP1>,
ORDER>>>;
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 5
Wavefield IO for Volume, Surface and Point Data
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 6
Wavefield IO for Volume, Surface and Point Data
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 7
Extensions
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 8
Extensions
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 9
Forward Problem Given the material properties of the Earth, simulate the propagation of seismic waves through the subsurface. Inverse Problem Given a set of seismograms and a description of the seismic sources, determine the material parameters of the Earth.
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 10
Nonlinear Optimization Problem
Inverse Problem
Reformulation as minimization problem
misfit regularization
§ model parameters § data § forward operator (the wave equation)
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 11
A Topography Map of Misfit Land
contour lines of misfit functional
minimum
mk
negative gradient of the misfit functional
w.r.t. the model
Iterative minimization based on local information (misfit, derivatives)
(i) Find a direction along which the misfit decreases. (ii) Determine a step-length to update the model and ensure
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 12
Full-Waveform Inversion
simplifying notation
misfit wave equation
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 13
Deriving Adjoints without Physics
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 14
easy
Deriving Adjoints without Physics
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 15
easy
Deriving Adjoints without Physics
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 16
easy
Deriving Adjoints without Physics
difficult
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 17
easy difficult
Deriving Adjoints without Physics
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 18
Abstraction from basic linear algebra
Deriving Adjoints without Physics
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 19
Abstraction from basic linear algebra
Deriving Adjoints without Physics
linear systems vector-vector products
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 20
Abstraction from basic linear algebra
Deriving Adjoints without Physics
linear systems vector-vector products
linear system vector-vector products
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 21
Abstraction from basic linear algebra
Deriving Adjoints without Physics
linear systems vector-vector products
linear system vector-vector products
adjoint equation
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 22
Back to the Wave Equation
§ The adjoint of a linear operator is the generalization of the transpose of a matrix
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 23
Back to the Wave Equation
§ The adjoint of a linear operator is the generalization of the transpose of a matrix
2x integration by parts in time + initial time conditions (forward) + final time conditions (adjoint)
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 24
Back to the Wave Equation
§ The adjoint of a linear operator is the generalization of the transpose of a matrix
symmetry of the elastic tensor
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 25
Sensitivity Kernels
-1 1
nothing happens
How does the misfit change (up to first order) if we increase the model parameters?
misfit decreases
misfit increases
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 26
Sensitivity Kernels
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 27
Kernel Computations
for (auto i=0; i<mStoreLocalFieldData.size(); i++ ) mStoreLocalFieldData[i] = &(mEpsilonVoigt[i]); template <typename Element>void IsotropicElastic3D<Element>::updateKernel() { this->mGradient += (this->mEpsilonVoigt[0]
+ this->mEpsilonVoigt[1] + this->mEpsilonVoigt[2]) *
(this->mAdjEpsilonVoigt[3] + this->mAdjEpsilonVoigt[4] + this->mAdjEpsilonVoigt[5]);
} template <typename Element>void IsotropicElastic3D<Element>::writeKernel() { Element::sumFieldsInExodusModel(kernelFields, Element::MapIntPtsToVtx(
Element::applyTestAndIntegrate(this->mGradient)));}
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 28
Kernel Computations
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 29
Kernel Computations
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 30
Unit Tests for Sensitivity Kernels
§ Example: Check for a consistently discretized adjoint equation and gradient by a finite-difference approximation
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 31
Full-Waveform Inversion Workflow
Compute Update
Smoothing
Clipping
Depth scaling
Compute Gradients
Model Compute Update
Compute Misfits
Update
Automation is possible if § Kernels and model parameterization are consistent § File I/O and data transfer are efficiently handled § A workflow management tool allows for reproducibility and user interaction
| Computational Seismology Department of Earth Sciences, Institute of Geophysics 32
Final Remarks
§ Salvus is work in progress
§ Watch http://salvus.io for updates
§ Sign up to salvus@googlegroups.com
§ Please report bugs and feature requests
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