Sample space | EPFL Graph Search

In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite. A subset of the sample space is an event, denoted by E. If the outcome of an experiment is included in E, then event E has occurred. For example, if the experiment is tossing a single coin, the sample space is the set {H,T}, where the outcome H means that the coin is heads and the outcome T means that the coin is tails. Th

Frequency-Domain Non-intrusive Greedy Model Order Reduction Based on Minimal Rational Approximation

Fabio Nobile, Davide Pradovera

We present a technique for Model Order Reduction (MOR) of frequency-domain problems relying on rational interpolation of vector-valued functions. The selection of the sample points is carried out adaptively according to a greedy procedure. We describe several options for the choice of a posteriori error indicators, which are used to drive the greedy algorithm and define its termination condition. Namely, we illustrate a tradeoff between each indicator's accuracy and its "intrusiveness", i.e. how much information on the underlying high-fidelity model needs to be available. We test numerically the effectiveness of this technique in solving a non-Hermitian eigen-problem and a microwave frequency response analysis.

SPRINGER INTERNATIONAL PUBLISHING AG2021

Stable computation of probability densities for metastable dynamical systems

Lionel Walter

Whenever the invariant stationary density of metastable dynamical systems decomposes into almost invariant partial densities, its computation as eigenvector of some transition probability matrix is an ill-conditioned problem. In order to avoid this computational difficulty, we suggest applying an aggregation/disaggregation method which addresses only well-conditioned subproblems aud thus results in a stable algorithm. In contrast to existing methods, the aggregation step is done via a sampling algorithm which covers only small patches of the sampling space. Finally, the theoretical analysis is illustrated by two biomolecular examples.

Society for Industrial and Applied Mathematics2007

Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins

Fabio Nobile, Anna Scotti, Lorenzo Tamellini

In this work we propose an Uncertainty Quantification methodology for sedimentary basins evolution under mechanical and geochemical compaction processes, which we model as a coupled, time-dependent, non-linear, monodimensional (depth-only) system of PDEs with uncertain parameters. While in previous works (Formaggia et al. 2013, Porta et al. 2014) we assumed a simplified depositional history with only one material, in this work we consider multi-layered basins, in which each layer is characterized by a different material, and hence by different properties. This setting requires several improvements with respect to our earlier works, both concerning the deterministic solver and the stochastic discretization. On the deterministic side, we replace the previous fixed-point iterative solver with a more efficient Newton solver at each step of the time-discretization. On the stochastic side, the multi-layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters, that need an appropriate treatment for surrogate modeling techniques, such as sparse grids, to be effective. We propose an innovative methodology to this end which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space. The reference coordinate system is built upon exploiting physical features of the problem at hand. We employ the locations of material interfaces, which display a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations. We showcase the capabilities of our numerical methodologies through two synthetic test cases. In particular, we show that our methodology reproduces with high accuracy multi-modal probability density functions displayed by target state variables (e.g., porosity).

Elsevier2018

Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins

Fabio Nobile, Anna Scotti, Lorenzo Tamellini

Elsevier2018