# Stochastic Methods

## Course Description:

Simulation and its importance. Review of probability theory and statistics. Stochastic processes and time series. Spectral analysis. Stochastic differential equations. Stationary univariate stochastic models. Long term persistence and simple scaling processes. Cyclostationary models. Multi-variate stochastic models. Discretization methods. Series expansion methods. Stochastic analysis of simple structural systems.

### Prerequisite Knowledge

Probability and statistics

### Course Units

# Title Description Hours
1 Introduction General notions, uncertainty and its quantification, usufulness and problem types. 3
2 Simulation The notion of simulation, categories of simulation, use of stochastic simulation, simulation models, random numbers. Simple examples of simulation in problems of statistical induction, Monte Carlo integration and stochastic optimization. 3
3 Review of probability theory and statistics Random variables, statistical parameters, statistical estimation, probability distributions and their fitting. The notion of entropy and its maximization. Application to statistical analysis of geophysical time series. 3
4 Stochastic processes and time series Stochastic processes, stationarity, ergodicity. Autocorrelation, cross-correlation, climacogram. Stochastic processes in discrete and continuous time. Sampling and times series. White noise. 3
5 Spectral analysis Fourier transform and its usefulness for solving integral equations. Convolution. Fourier transform of the autocovariance function and power spectrum. Power spectral estimation from time series. Computational aspects of power spectrum. Example on identification of periodicity. More complex example of analysis of large scale geophysical time series. 3
6 Stochastic differential equations General notions, the Langevin equation and its application to the problem of outflow from linear reservoir with white noise inflow. Fokker–Planck equation. Markov processes, the Ornstein–Uhlenbeck process. 3
7 Univariate and stationary stochastic models Discrete time models. Models AR(1), AR(2), ARMA(1,1), and their generalizations. The general simulation method of any arbitrary process using the SMA method. Fitting of stochastic models based on time series data and generation of synthetic time series. 3
8 Long term persistence and simpe scaling processes Empirical validation of the existence of long term persistence. Theoretical derivation based on the maximization of entropy production. The Hurst-Kolmogorov process and simple methods of simulating it. Analysis of the effect of long-term persistence on the availability of water resources and on the design of water development projects. 3
9 Cyclostationary models Introduction to cyclostationarly models. The PAR and PARSMA models. Application to reservoir design of and stochastic reliability analysis. 3
10 Multivariate models Review of linear algebra topics. Vector random variables and their manipulation. Covariance matrices. The multivariate cyclostationary model PAR. Introduction to disaggregation models. Application to reservoir systems management. 3
11 Series expansion simulation methods Simulation of stochastic processes with point discretization methods and local averages. Spectral representation and Karhunen–Loève series expansion methods. Simulation of stationary stochastic processes. 3
12 Simulation of nonstationary processes Simulation of nonstationary stochastic processes and power spectra estimation from real data. Production of synthetic seismic ground motions - accelerograms. 3
13 Stochastic analysis of simple structures Introduction to stochastic virtual work principle. Solution of the stochastic static problem using analytic solutions as well as Monte Carlo simulation approximations. Estimation of response variability. 3

### Learning Objectives

• Familiarization with uncertainty, inherent in complex but also in simple systems.
• Understanding the potential of Monte Carlo simulation for stochastic numerical problems as well as in deterministic ones.
• Familiarization with stochastic calculus.
• Learning how to use simple stationary and cyclostationary stochastic models for simulation purposes.
• Understanding the notion of persistence and its effect on the uncertainty increase and the design of engineering projects.
• Familiarization with stochastic process simulation methods and their relation to the corresponding physical processes.
• Understanding the formulation and solution of structural mechanics problems in the presence of uncertain parameters.
• Understanding the effect of uncertainties in the analysis and design of structures.

### Teaching Methods

 Teaching methods Teaching of theory and its application in examples at the PC Lab (mostly using Excel and Visual Basic and Matlab). Use of power point presentations Use of PC Lab Formulation and solution of complex simulation and optimization problems in spreadsheets Simple classroom exercises Seven computational exercises which cover the topics of the course: 1. Emergence of uncertainty in simple deterministic models. 2. Fitting of probability distributions on time series data. 3. Fitting of stochastic models.on time series data. 4. Using stochastic simulation in reservoir design. 5. Using cyclostationary and multivariate models. 6. Synthetic accelerogram generation using Matlab. 7. Implementation of Monte Carlo simulation on simple beams with uncertainties in material properties. The students are encouraged to prepare and present their research work in the international conference of the European Geosciences Union at Vienna. This effort is supervised and guided from the teaching stuff of the course.

### Student Assessment

• Final written exam: 50%
• Assignments (projects, reports): 50%

### Textbooks - Bibliography

The full course material, which includes lecture notes, presentations and problems of past exams, as well as references to research publications and related websites, is available on the course website.

1. D. Koutsoyiannis, Statistical Hydrology, Edition 4, 312 pages, doi:10.13140/RG.2.1.5118.2325, National Technical University of Athens, Athens, 1997 (in Greek).

2. D. Koutsoyiannis, Probability and statistics for geophysical processes, doi:10.13140/RG.2.1.2300.1849/1, National Technical University of Athens, Athens, 2008.

3. V. Papadopoulos, and D.G. Giovanis, Stochastic Finite Elements - An Introduction, doi:10.1007/978-3-319-64528-5, Springer, 2018.

## Lecture Time - Place:

• Wednesday, 12:45 – 16:30,
Rooms:
• Κτ. Διοίκησης, PC Lab Σχολής

© 2017 School of Civil Engineering, ΕΜΠ