Lecture No. 1
Lecture Topics
Lecture No. 1
| Date | Wednesday, August 29, 2001. |
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| Topic | Course Objectives and Expectations. Road Map. |
Lecture No. 2
| Date | Friday, August 31, 2001. |
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| Topic | Some
Basic Definitions.
Abel's Problem. Linear vs. Nonlinear Systems. |
Lecture No. 3
| Date | Wednesday, September 5, 2001. |
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| Topic | Deterministic
vs. Non-Deterministic (Statistical Systems).
Linear vs. Nonlinear Inverse Problems. Existence, Uniqueness, and Stability. |
Lecture No. 4
| Date | Friday, September 7, 2001. |
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| Topic | Discrete
vs. Continuous Inverse Problems.
Over-determined, Even-determined, and Under-determined Inverse Problems. Introduction to Linear Inverse Problems. |
Lecture No. 5
| Date | Monday, September 10, 2001. |
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| Topic | Continuous
and Discrete Linear Inverse Problems.
Examples. Concepts of Data Space and Model Space. |
Lecture No. 6
| Date | Wednesday, September 12, 2001. |
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| Topic | The
Concept of the Pseudo Inverse.
Linear Inverse Problems in Matrix Form. Example of Linear Regression. |
Lecture No. 7
| Date | Friday, September 14, 2001. |
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| Topic | Example
of Linear Regression.
Introduction to the Concepts of Metric, and Norm. Inversion as Minimization of the Error Norm. Normal Equations. Matrix Form of the Solution. Equivalent Matrix Formulation. |
Lecture No. 8
| Date | Monday, September 17, 2001. |
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| Topic | The
Pseudoinverse of an Overdetermined Linear System.
Matrix Formulation of the Least-Squares Minimization Process. Examples. Computational Issues. Minimization in Model Space. |
Lecture No. 9
| Date | Wednesday, September 19, 2001. |
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| Topic | The
Concept of Null Space.
The Concept of Regularization. Computational Issues. Model Resolution Matrix. Data Resolution Matrix. |
Lecture No. 10
| Date | Friday, September 21, 2001. |
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| Topic | Weighted
Least Squares Solution.
Examples. |
Lecture No. 11
| Date | Monday, September 24, 2001. |
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| Topic | Discussion
and Questions on Homework Project No. 1.
Weighted Least Squares Solution, Part II. |
Lecture No. 12
| Date | Wednesday, September 26, 2001. |
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| Topic | Weighted
Least Squares Solution, Part III: Diagonal Preconditioning.
Generalized Inverse (Pseudoinverse) of the Underdetermined Linear Problem, Part I. |
Lecture No. 13
| Date | Friday, September 28, 2001. |
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| Topic | Comments
on the Application of Homework Project No. 1 to Practical Petrophysical
Problems in Well Logging: Technical Constraints and Practical Mathematical
Strategies.
Generalized
Inverse (Pseudoinverse) of the Underdetermined Linear Problem, Part II.
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Lecture No. 14
| Date | Monday, October 1, 2001. |
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| Topic | Minimization
in Model Space, Part II.
Introduction to Stochastic Variables and Processes, Part I. Probability Density Distributions. |
Lecture No. 15
| Date | Wednesday, October 3, 2001. |
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| Topic | No Lecture Scheduled Today in Observance of the Annual SPE Conference. |
Lecture No. 16
| Date | Friday, October 5, 2001. |
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| Topic | Introduction
to Stochastic Variables and Processes, Part II.
Basic Operations with Random Variables and Functions of Random Variables. Calculation of First and Second Moments. |
Lecture No. 17
| Date | Monday, October 8, 2001. |
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| Topic | Introduction
to Stochastic Variables and Processes, Part III.
Convolution Theorem and Central Limit Theorem. Examples. Joint Probability Density Functions. Linear Transformations. Covariance Matrix. |
Lecture No. 18
| Date | Wednesday, October 10, 2001. |
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| Topic | Introduction
to Stochastic Variables and Processes, Part IV.
Linear Transformations. Covariance Matrix. Stochastic Processes. First- and Second-Order Stationarity. Ergodocity. White and Colored Processes. Examples. |
Lecture No. 19
| Date | Friday, October 12, 2001. |
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| Topic | Introduction
to Stochastic Variables and Processes, Part V.
Stochastic Processes. First- and Second-Order Stationarity. Ergodocity. White and Colored Processes. Examples. The Deconvolution Problem, Part I. Interpretation in the Fourier Domain. |
Lecture No. 20
| Date | Monday, October 15, 2001. |
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| Topic | The
Deconvolution Problem, Part II. Interpretation in the Fourier Domain.
Low- and High-Pass Filters. The Wiener Filter. |
Lecture No. 21
| Date | Wednesday, October 17, 2001. |
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| Topic | The
Deconvolution Problem, Part III. Interpretation of the Wiener Filter.
Sampling Theorem. Nyquist Frequency. Aliasing. Comments on Homework Assignment No. 2. |
Lecture No. 22
| Date | Friday, October 19, 2001. |
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| Topic | Comments
on Homework Assignment No. 2.
Eigenvalue Decompositions. Eigenvalues and Eigenvectors. Interpretation. Analogy with Fourier Analysis. |
Lecture No. 23
| Date | Monday, October 22, 2001. |
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| Topic | Comments
on Homework Assignment No. 2.
Eigenvalue Analysis of the Overdetermined and Underdetermined Inverse Problems, Part I. |
Lecture No. 24
| Date | Wednesday, October 24, 2001. |
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| Topic | Eigenvalue Analysis of the Overdetermined and Underdetermined Inverse Problems, Part II. |
Lecture No. 25
| Date | Friday, October 26, 2001. |
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| Topic | Eigenvalue Analysis of the Overdetermined and Underdetermined Inverse Problems, Part III. Tickhonov Regularization. Eigenvalue Filtering. |
Lecture No. 26
| Date | Monday, October 29, 2001. |
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| Topic | Eigenvalue
Filtering. Data and Model Resolution Matrices.
Spectral Expansion Method. |
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Lecture No. 27
| Date | Wednesday, October 31, 2001. |
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| Topic | The Stochastic Inverse, Part I. |
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Lecture
No. 28
| Date | Friday, November 2, 2001. |
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| Topic | The
Stochastic Inverse, Part II.
Backus and Gilbert Procedure. |
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Lecture No. 29
| Date | Monday, November 5, 2001. |
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| Topic | Technical
Comments and Discussion on Homework Project No. 2.
Maximum Likelihood Inversion, Part I. |
Lecture No. 30
| Date | Wednesday, November 5, 2001. |
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| Topic | Maximum
Likelihood Inversion, Part II
Norm Optimality. Introduction to Nonlinear Inversion, Part I. |
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Lecture No. 31
| Date | Friday, November 9, 2001. |
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| Topic | FIELD
TRIP TO SCHLUMBERGER'S SUGAR LAND PRODUCT CENTER.
A
chartered bus will depart from the CPE building at 6:40AM.
Breakfast, refreshments, and snacks will be provided in the bus. We are planning to leave Houston back to Austin at about 4:00PM. |
Lecture No. 32
| Date | Monday, November 12, 2001. |
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| Topic | Introduction
to Nonlinear Inversion, Part II.
The zeros and stationary points of a nonlinear function. Numerical methods used to calculate the stationary points of a nonlinear function. |
Lecture No. 33
| Date | Wednesday, November 14, 2001. |
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| Topic | Minimization
of a scalar multivariate function. Taylor series expansion of a scalar
multivariate function.
The concepts of gradient vector and Hessian matrix. Necessary condition for the existence of a stationary point. Necessary conditions for the existence of an unconstrained minimum. Minimization strategies. |
Lecture No. 34
| Date | Friday, November 16, 2001. |
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| Topic | Minimization
of a scalar multivariate function. Method of steepest descent. Newton and
Quasi-Newton Methods.
Step-wise regularization strategies and relation between Newton and steepest descent methods of minimization. Convergence properties and algorithmic efficiency. |
Lecture No. 35
| Date | Monday, November 19, 2001. |
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| Topic | Minimization
of a vectorial multivariate function. Taylor series expansion of a vectorial
multivariate function.
The Jacobian Matrix. Steepest descent and Gauss-Newton minimization methods. Marquardt-Levenberg minimization strategy. Convergence properties and algorithmic efficiency. Examples. |
Lecture No. 36
| Date | Wednesday, November 21, 2001. |
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| Topic | GUEST
LECTURE
Dr.
Mrinal Sen, Institute for Geophysics:
|
Lecture No. 37
| Date | Monday, November 26, 2001. |
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| Topic | Steepest
descent and Gauss-Newton minimization methods.
Marquardt-Levenberg minimization strategy. Convergence properties and algorithmic efficiency. Strategies for solving the normal equations that include regularization. Examples. |
Lecture No. 38
| Date | Wednesday, November 28, 2001. |
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| Topic | Strategies
for solving the normal equations that include regularization.
Examples. |
Lecture No. 39
| Date | Friday, November 30, 2001. |
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| Topic | Strategies
for computing the Jacobian matrix.
Strategies for solving the normal equations that include regularization. Examples. |
Lecture No. 40
| Date | Monday, December 3, 2001. |
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| Topic | Students'
presentations of research projects.
Questions and answers. |
Lecture No. 41
| Date | Wednesday, December 5, 2001. |
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| Topic | Minimization
strategies based on integral equation formulations.
Fredholm integral equations of the second kind. Volterra equations. Born approximation and Neumann series expansions. Examples. Course Evaluation. |
Lecture No. 42
| Date | Friday, December 7, 2001. |
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| Topic | Born
approximation and Neumann series expansions, Part II.
Examples. |
FINIS
OPUS