Linear form | EPFL Graph Search

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted Hom(V, k), or, when the field k is understood, V^*; other notations are also used, such as V', V^{#} or V^{\vee}. When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by m

Many-body localization in a fragmented Hilbert space

Loïc Jean Pierre Herviou

We study many-body localization (MBL) in a pair-hopping model exhibiting strong fragmentation of the Hilbert space. We show that several Krylov subspaces have both ergodic statistics in the thermodynamic limit and a dimension that scales much slower than the full Hilbert space but still exponentially. Such a property allows us to study the MBL phase transition in systems including up to 64 spins. The different Krylov spaces that we consider show clear signatures of a many-body localization transition, both in the Kullback-Leibler divergence of the distribution of their level spacing ratio and their entanglement properties. However, they also present distinct scalings with the system size. Depending on the subspace, the critical disorder strength can be nearly independent of the system size or conversely show an approximately linear increase with the number of spins.

2021

One-Shot Approach for Enforcing Piecewise Linearity on Hybrid Functionals: Application to Band Gap Predictions

Stefano Falletta, Alfredo Pasquarello, Jing Yang

We present an efficient procedure for constructing nonempirical hybrid functionals to accurately predict band gaps of extended systems. We determine mixing parameters by enforcing the generalized Koopmans’ condition on localized electron states, which are achieved by inserting an optimized potential probe. Application of this scheme to a large set of materials yields band gaps with a mean error of 0.30 eV with respect to experiment. Next, we consider a perturbative one-shot approach in which the single-particle eigenvalues are calculated with the wave functions obtained at the semilocal level. In this way, the computational cost is reduced by ∼85% without loss of accuracy. The scheme is found to be robust upon consideration of different defect species and functional forms.

2022

Numerical Approximation of Flows in Random Porous Media

Francesco Tesei

The objective of this thesis is to develop efficient numerical schemes to successfully tackle problems arising from the study of groundwater flows in a porous saturated medium; we deal therefore with partial differential equations(PDE) having random coefficients and we are interested in computing statistics related to specific quantities of interest (QoI), e.g. a linear functional of the solution of the PDE or the solution itself. We mainly consider the approximation of the pressure in the medium through a stochastic Darcy problem with random lognormally distributed permeability relying on Matérn-type covariance functions to take into account a wide range of possible smoothness of the permeability field. Once the problem has been reformulated in terms of a countable number of random variables, we analyze sparse grid polynomial approximations of the QoI. We propose different strategies to exploit the anisotropicity of the QoI with respect to the different random entries; to this end we consider a priori'' and a posteriori'' strategies to drive the exploration of the multi-index set that defines the sparse grid, associating a profit to each multi-index either by using explicit theoretical estimates or by actually solving the PDE and computing on the fly the corresponding sparse grid interpolant. We show on several numerical examples the effectiveness of this strategy in treating the case of smooth permeability fields. In order to cover also the case of rough input permeabilities we consider, instead, Multi Level Monte Carlo techniques based on the use of a suitable control variate. Such a control variate is obtained from the solution of an auxiliary Darcy problem with a regularized input permeability which leads to pressure distributions that are smoother and less oscillatory than the original ones, but still highly correlated with them. We use then a sparse grid approximation to compute effectively the mean of the control variate and provide explicit bounds for the corresponding estimator as well as a complexity result. We also consider groundwater transport problems and focus, in particular, on arrival times properly defined starting from particle trajectories driven by the stochastic Darcy velocity and subject to molecular diffusion taking place at porous level. In this case, by using suitable PDEs whose solution can be linked to specific expectations (with respect to all Brownian motions) thanks to the famous Feynman-Kac formula, we compute statistics of such arrival times, e.g. their expected value or the probability of exiting the physical domain in a given time horizon. We discuss several scenarios and readapt the methodologies previously developed involving adaptive sparse grid stochastic collocation and Monte Carlo type schemes to this case.

EPFL2016

Numerical Approximation of Flows in Random Porous Media

Francesco Tesei

EPFL2016