Université de Paris VI &
Institut de Physique du Globe de Paris
Course: Inverse Problems

Monday,
September 29, 10H45 
Lesson I: Introduction.
The PopperBayes approach. Jeffreys quantities. Viewgraphs. A small numerical example about
Benford's law (small
explanatory text, mathematica
notebook, pdf of
mathematica notebook). Lesson II: Volumetric probabilities. Probability densities. Changing variables. Viewgraphs. Numerical example: changing from { Compressibility , Bulk modulus } to { Young modulus , Poisson's ratio }. Notebook. 
Monday, October 6, 10H45  Lesson III: Conditional
and marginal
probabilities. Independent uncertainties. Uncorrelated uncertainties.
Covariance. Viewgraphs.
Numerical example: representing a 2D Gaussian. Notebook. Lesson IV: Sampling a probability distribution. Rejection algorithm. Metropolis algorithm. Why the Metropolis algorithm is not panacea. Viewgraphs. Numerical example: Sampling 1D and 2D volumetric probabilities. Notebook 1, PDF explanation. Notebook 2, PDF explanation. Numerical example: Sampling the Fisher distribution. Notebook. PDF explanation. Numerical example: Sampling a Gaussian random function. Notebook (please help me in correcting the plot commands). PDF explanatiion. Numerical example: Sampling using the Metropolis algorithm. Notebook. Note: I should check that, everywhere, when the test point is rejected, the old point is taken again, as in this simple example. The ultimate example would be a random walk that uniformly samples the sphere, then samples the Fisher distribution. 
Monday, October 13, 10H45  Lesson V: Sets
and mappings. Viewgraphs. Numerical example: measuring the
aspect ratio of a screen. Notebook, PDF
explanation. Intersection of probabilities. Image
of a probability. Reciprocal image of a probability. Fundamental
theorem. Viewgraphs. Numerical
example:
transport of probabilities. Notebook, PDF
explanation. Lesson VI: General formulation of an inverse problem. Method I: Explicit plotting of the posterior volumetric probability. Viewgraphs. Numerical example: Estimation of a seismic epicenter. Notebooks: complete pdf document, EpicenterMathematica.nb, EpicenterVelocityMathematica.nb, EpicenterMathematicaDiffractors.nb. 
Monday, October 20, 10H45  Lesson VII: More
numerical examples (Viewgraphs).
A variant of the epicenter problem
above: the
probability density for the observations has a bimodal probability
density (complete
pdf document, mathematica
code). A second geological example concerns
the
estimation of the center of rotation,
and of the rotation velocity,
using as data the linear velocities of some points (pdf document, executable
notebook). Fitting a curve
to mass disintegration data (mathematica
notebook). Robust fitting of data: fitting a
logistic
curve to the variation of
Human population (complete
pdf document, mathematica
code). Lesson VIII: Method II: Sampling of the posterior volumetric probability (Monte Carlo methods). Rejection algorithm. Metropolis algorithm. Viewgraphs. Numerical example: the epicenter (rejection). Numerical example: the epicenter (Metropolis). Numerical example: the epicenter (again) but using a bimodal distribution for the arrival times (mathematica notebook). A pdf document on the epicenter exercise. Why we don't need an explicit expression for the prior probability. Why we don't need the simulated annealing algorithm. Why I don't like the genetic algorithms. Different kinds of data. Geostatistics and inverse problems. Can we solve complex problems? (No.) 
Monday, October 27, 10H45  Lesson IX: More numerical examples (viewgraphs). Numerical example: determining the
parameters of a fissure from geodetic data (pdf document, mathematica
notebook). Numerical
example: using the Metropolis algorithm to solve a waveform fitting problem (mathematica notebook
[pdf document not yet available]). Lesson X: Method III: Optimization (and approximate sampling of the posterior probability). Leastsquares. Leastabsolute values. Viewgraphs. 
Monday, November 3, 10H45  Lesson XI: Optimization
and
nonlinear problems (viewgraphs).
Numerical
example: Epicenter, gradient method (notebook, pdf
document). Numerical
example: measuring the
chlorophyll concentration of vegetal leafs (complete pdf
document, notebook
1, notebook
2, notebook
3). Lesson XII: Optimization and linear problems (viewgraphs). Numerical example: Xray tomography (notebook, pdf document). Note: I am in the process of changing this example, that now contains a random Gaussian field with exponential covariance, and sampling of the prior and the posterior distributions (pictures, notebook 1, pdf version of notebook 1, notebook 2, pdf version of notebook 2). 
Monday, November 10, 10H45  Lesson XIII: Viewgraphs. Numerical
example: using envisat
satellite data (notebook, pdf document). Lesson XIV: Viewgraphs. Numerical example: Regression lines when there are uncertainties in both axes (error crosses instead of error bars) (notebook, pdf document). Numerical example: Nonlinear leastabsolute values. 
Monday, November 17, 10H45  Lesson XV: The squareroot variable metric algorithm (efficient sampling of the posterior probability). Viewgraphs. A little bit of theory (pdf document). Numerical example: 1D Gaussian random function with some (linear) constraints (mathematica notebook). Numerical example: the example for gradientbased method of optimization based on the epicenter problem already contained an implementation of the (nonlinear) squareroot variable metric method (mathematica notebook again). Small text about the squarte root of the exponential covariance and two (1,2) notebooks. 
Monday, November 24, 10H45 
Lesson
XVI: Leastsquares
involving
functions (notion of random function, the mathematics of
leastsquares). Viewgraphs. Lesson XVII: Leastsquares involving functions (notions of functional analysis, the formulation of the inverse problem). Viewgraphs. 
Monday, December 1, 10H45 
Lesson XVIII: Viewgraphs.
Numerical example: Raytomography
without blocks (mathematica
notebook, pdf
document, notebook
2). Numerical
example: Building a smooth function given some of its points (mathematica notebook, pdf document). 
Monday, December 8, 10H45  Lesson XIX: Fitting
seismic waveforms
to retrieve the source and the medium properties (theory). Viewgraphs. A mathematica code for the simulation
of 1D acoustic wave propagation (small theory [pdf, viewgraphs], mathematica
code). Lesson XX: Numerical example: Fitting seismic waveforms to retrieve the source and the medium properties (viewgraphs, mathematica notebook, pdf document). 