Geophysical and Environmental Modeling

1) Development and application of a model.
2) Prototypical equations: conservation principles and phenomenological laws; prototypical partial differential equations.
3) Finite differences and finite volume methods for the numerical solution of flow and transport equations. Accuracy and convergence; Von Neumann stability analysis; Courant-Friedrichs-Lewy condition); finite volume method (or integrated finite differences); staggered grids; methods of solution of large linear systems (Gaussian elimination; successive over relaxation method; preconditioned conjugate gradient); explicit, implicit and Crank-Nicolson schemes for parabolic equations; nested models.
4) Spectral analysis. Signals and systems; Fourier transform (representation of discrete-time signals and systems in the frequency domain; sampling); discrete Fourier transform (representation of periodic and finite-length discrete-time signals; power spectrum).
5) Basic properties of spectral and finite elements method for the solution of flow and transport equations: weighted residuals and variational approaches.
6) Basic statistic and theory of probability; basic theory of stochastic processes.
7) Modeling solute transport. Analytical solutions of transport equation. Lagrangian models: particle tracking method for convective transport (Runge-Kutta method for ordinary differential equations). Stochastic models (Monte Carlo methods for the advective-dispersive equation (Kolmogorov-Dmitriev theory of branching stochastic processes; discrete-time random walk).
8) Data mining and data analysis. Regression methods; multivariate analysis (Principal Component Analysis & Factor analysis); clustering algorithms; wavelets; applications of stochastic processes to geophysics and to environmental physics (Kriging; Kalman filter; Markov chains); artificial neural networks.
9) Model calibration by solution of inverse problems. General definition and properties of the discrete inverse problem; the null space; use of prior information and weighted least squares; Bayesian approach and the maximum likelihood method; gradient-based optimization algorithms (Newton's method; Levenberg-Marquardt's algorithm); global optimization methods (genetic algorithms; differential evolution; simulated annealing); sensitivity analysis (one-at-a-time approach; adjoint-state equation; first-order sensitivity).

Students' project. Each student will be asked to complete a project by developing original codes or by application of freely available software tools (e.g., Princeton Ocean Model, CALMET/CALPUFF/CALGRID package, MODFLOW/MODPATH package).

Optional short course
* A tutorial on Sobolev spaces and weak solutions of partial differential equations
- Weak derivatives and Wk,p Sobolev spaces;
- Weak solutions to elliptic partial differential equations/boundary value problems.
* The inverse problem of identifying the leading coefficient of the prototypical elliptic equation: 1. Classical results and difficulties
- Calderón's inverse conductivity problem: the state-of-the-art until 2000;
- Determining the conductivity in transport media: integration along characteristic lines.
* The inverse problem of the identification of the leading coefficient of the prototypical elliptic equation: 2. Recent results
- Results on Calderón's problem in the third millennium;
- Univalent sigma-harmonic mappings and their relevance to inverse problems.
* The inverse problem for the prototypical reduced wave equation
- The Helmholtz equation;
- Full waveform inversion.