Inverse problems in geophysics

Reading for students of the Department of Physics of the Earth.

Read at 8-th semester for students of the Department of Physics of the Earth.

2 hours of lectures per week

• Yagola Anatoly Grigorevich

1. Some concepts of functional analysis. Elements of convex programming. Convex and strongly convex functionals. Minimization methods: steepest descent, conjugate gradient method, Newton's method and others.
2. Correctness and incorrectness of the mathematical formulation of the problem. Examples of correct and incorrect problems. The concept of Tikhonov regularization algorithm on solving ill-posed problem. Classification of inverse problems. Basic properties regularizable ill-posed problems.
3. Ill-posed problems on compact sets. The concept quasisolution. Numerical methods for solving ill-posed problems on sets of monotone and convex functions. An error estimate for solving linear ill-posed problems on convex compact sets.
4. Ill-posed problems provided sourcewise representation of the desired solution. The method of extending compacts. A posteriori error estimate.
5. Tikhonov's approach to the construction of regularizing algorithms. The linear case. Priori and a posteriori methods for choosing the regularization parameter. Convergence theorems. Fredholm integral equations of the first kind. Equations of convolution type.
6. Nonlinear ill-posed problem. Regularization algorithms for their solution. Piecewise-uniform regularization.
7. Iterative regularization and other approaches.
8. The inverse problem of potential theory for geomagnetic and gravitational fields. Direct problem for some bodies (ball, straight stem, infinitely long circular or elliptical cylinder, horizontal or inclined plane).
9. Inverse kinematic problem of seismology (appeal hodograph).
10. The inverse problem of seismology (restoration of the true ground displacement by seismograph records).

References:

Summary:

1. Tikhonov, Goncharsky AV, Stepanov VV, Yagola AG Numerical methods for solving ill-posed problems. M .: Nauka, 1990.
2. AN Tikhonov, AS Leonov, AG Yagola Nonlinear ill-posed problem. M .: Nauka, 1995.
3. Bakushinskii AB, AV Goncharsky Ill-posed problems. Numerical methods and applications. M .: Univ. Univ, 1989.
4. Bakushinskii AB, AV Goncharsky Iterative methods for solving ill-posed problems. M .: Nauka, 1989.
5. Logatchev AA, VP Zakharov Magnetic survey. L .: Nedra, 1973.
6. Aki, P. Richards Quantitative seismology. M .: Mir, vol. 1, 2, 1983.
7. Novikov GF Radiometric prospecting. L .: Nedra, 1989.

MORE:

1. VK Ivanov, VV Vasin, VP Tanana Theory of linear ill-posed problems and its applications. M .: Nauka, 1978.
2. MM Lavrentiev, VG Romanov, SP Shishatskii Ill-posed problems of mathematical physics and analysis. M .: Nauka, 1980.