Universidad de Chile, Facultad de
Ciencias Físicas y Matemáticas
Departmento de Geofísica 25 de agosto  15 de septiembre, 2009 Courso: Problemas Inversos

Martes,
25 de Agosto, 10H0012H00 
Lección
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). Lección II: Volumetric probabilities. Probability densities. Changing variables. Viewgraphs. Numerical example: changing from { Compressibility , Bulk modulus } to { Young modulus , Poisson's ratio }. Notebook. 
Jueves, 27
de Agosto, 10H0012H00 
Lección
III: Conditional
and marginal
probabilities. Independent uncertainties. Uncorrelated uncertainties.
Covariance. Viewgraphs.
Numerical example: representing a 2D Gaussian. Notebook. Lección 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 (using both, the rejection algorithm and the Metropolis algorithm). 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. Direct sampling of a Gaussian ndimensional distribution (Viewgraphs). A small exercise on the chisquared distribution (Notebook). Direct sampling of a onedimensional Gaussian random field (exponential and Gaussian covariance) (notebook [for a more direct comparison between exponential and Gaussian covariances, use this notebook]) and direct sampling of a twodimensional Gaussian random field (exponential covariance) (notebook). Lección V (a): Sets and mappings. Viewgraphs. Numerical example: measuring the aspect ratio of a screen. Notebook, PDF explanation. 
Martes, 1 de Septiembre, 10H0012H00  Lección V
(b):
Advanced probability notions (Intersection of probabilities. Image
of a probability. Reciprocal image of a probability. Fundamental
theorem.) New
viewgraphs. Old viewgraphs.
Numerical
example:
transport of probabilities. Notebook, PDF
explanation. Lección 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. Lección VII: More numerical examples (Viewgraphs). A variant of the epicenter problem above: the probability density for the observations has a bimodal probability density (mathematica code). An old example: hypocenter instead of epicenter (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). 
Jueves, 3 de Septiembre, 10H0012H00  Lección
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.) Lección 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]). 
Martes, 8 de Septiembre, 10H0012H00  Lección
X: Method
III: Optimization
(and approximate sampling of the posterior probability). Leastsquares.
Leastabsolute values. Viewgraphs. Lección 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). Lección 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, pdf version of notebook). Complement: a better way of passing from tables to vectors (notebook). Lección XIII: Viewgraphs. Numerical example: using envisat satellite data (notebook, pdf document). Lección 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. 
Jueves, 10 de Septiembre, 10H0012H00  Lección
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 square
root of the exponential covariance and two (1,2)
notebooks. Lección XVI: Leastsquares involving functions (notion of random function, the mathematics of leastsquares). Viewgraphs. Lección XVII: Leastsquares involving functions (notions of functional analysis, the formulation of the inverse problem). Viewgraphs. Lección 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). 
Martes 15 (?), 10H0012H00  Lección
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). Lección XX: Numerical example: Fitting seismic waveforms to retrieve the source and the medium properties (viewgraphs, mathematica notebook, pdf document). 