I Part of the course: Numerical methods

The first part of the course will give the students the basic knowledge concerning the most important mathematical and numerical methods used to solve the equation of motion of fluids in porous media and to solve the elastodynamic problem.

 II Part of the course: Statistical methods

The second part of the course aims at providing students with statistical tools for the analysis of data typically encountered in geo-science and environmental applications. The course is organized in theoretical classes and laboratories. 

Concerning the Numerical part the students with respect to

1) Knowledge and understanding

are expected to: 

- know the fundamental principles of the mathematical and numerical methods for solving porous media, elastodynamics and poroelasticity equation in general situations;

- use proper terminology.

2)  Applying knowledge and understanding

are expected to:

- apply the acquired knowledge to engineering problems;

- be able to provide an abstract formalization of the phenomena of interest;

- use the proposed software to solve practical problems.

Concerning the Statistical part the students with respect to

1) Knowledge and understanding

are expected to: 

- know the fundamental principles of the mathematical and statistical methods for the analysis of non-standard data possibly with spatial and/or temporal dependence that are typically encountered in geo-science and environmental applications;

- use proper terminology.

2)  Applying knowledge and understanding

are expected to:

- apply the acquired knowledge to engineering problems;

- be able to provide an abstract formalization of the phenomena of interest;

- use the proposed software to perform statistical data analysis.

I Part of the course: Numerical methods

• Introduction to flow in porous media. Numerical methods for single phase flow in porous media: primal (pressure) and mixed formulation. Finite volumes, mixed finite elements and mimetic finite differencing. 

• Elastodynamics. Methods for second order hyperbolic equations (Newmark, leap-frog schemes); finite difference and finite element for wave propagation.

• Poroelasticity. Biot equations: Splitting strategy for the coupled problem of flow and structure.

Course organization

The course is organized in theoretical classes and laboratories.

 II Part of the course: Statistical methods

• Euclidean Multivariate Data (brief review). The Euclidean geometry in the real space, Principal Component Analysis, clustering, permutational one- and two-population tests.

• Compositional Data (e.g., chemical compounds, mineralogical compositions, atmospheric pollutants). The Aitchinson geometry in the simplex, transformations of compositional data, Principal Component Analysis of compositional data, clustering of compositional data, permutational one- and two-population tests for compositional data.

• Directional Data (e.g, winds, waves, geological fault directions). The geodesic distance, Principal Component Analysis of directional data, clustering of directional data, permutational one- and two-population tests for directional data.

• Tensor Data (e.g., diffusion of oil, water, vehicles and people). Distances between tensors, clustering of tensor data, permutational one- and two-population tests for tensor data.

• Network Data (e.g., river networks, oil and gas pipelines, mobility networks). Network representations (adjacency, Laplacian, and modularity matrix), distances between networks, clustering of network data, permutational one- and two-population tests for network data.

• Data with Spatial and/or Temporal Dependence. Measures of spatial dependence, covariogram and variogram, estimation and prediction via ordinary and universal Kriging, hidden Markov random fields.

Course organization

The course is made of theoretical lectures followed by lab sessions. During the theoretical lectures methods and algorithms will be presented in the proper mathematical framework. During the lab sessions methods and algorithms will be instead illustrated and tested through applications to real data sets. The analyses performed during the lab sessions will be carried out by means the opensource software R (www.r-project.org). Along the course, students are expected to work on a data analysis team project. 

Common prerequisites: basic calculus and linear algebra.

Numerical parts: basic knowledge of numerical methods for solving linear systems and ODEs.

Statistical part: basic knowledge in probability and statistics at bachelor level is suggested.

The students will be divided into teams and for each part of the course the examination will consist in:

1) reading, understanding and presenting a research paper assigned by the teacher (50% of the part-specific mark);

2) a team project on specific real world problem assigned by the teacher (50% of the part-specific mark).

As for the numerical part the students with respect to

1) Knowledge and understanding

are expected to show that they: 

- know the fundamental principles of the mathematical and numerical methods for solving porous media, elastodynamics and poroelasticity equation in general situations;

- are able to use a proper terminology.

2)  Applying knowledge and understanding

are expected to show that they:

- are able to apply the acquired knowledge to engineering problems;

- are able to provide an abstract formalization of the phenomena of interest;

- are able to use the proposed software to solve practical problems.

As for  the Statistical part the students with respect to

1) Knowledge and understanding

are expected to show that they: 

- know the fundamental principles of the mathematical and statistical methods for the analysis of non-standard data possibly with spatial and/or temporal dependence that are typically encountered in geo-science and environmental applications;

- are able to use proper terminology.

2)  Applying knowledge and understanding

are expected to show that they:

- are able to apply the acquired knowledge to engineering problems;

- are able be able to provide an abstract formalization of the phenomena of interest;

- are able to use the proposed software to perform statistical data analysis.

The final mark will be the weighted average of the marks achieved in the two parts with a weight of 6/10 for the numerical one and 4/10 for the statistical one.