Search results for "approksimointi"

Dynamically screened vertex correction to $GW$

2020

Diagrammatic perturbation theory is a powerful tool for the investigation of interacting many-body systems, the self-energy operator $\mathrm{\ensuremath{\Sigma}}$ encoding all the variety of scattering processes. In the simplest scenario of correlated electrons described by the $GW$ approximation for the electron self-energy, a particle transfers a part of its energy to neutral excitations. Higher-order (in screened Coulomb interaction $W$) self-energy diagrams lead to improved electron spectral functions (SFs) by taking more complicated scattering channels into account and by adding corrections to lower order self-energy terms. However, they also may lead to unphysical negative spectral f…

Approximation of W1,p Sobolev homeomorphism by diffeomorphisms and the signs of the Jacobian

2018

Let Ω ⊂ R n, n ≥ 4, be a domain and 1 ≤ p 0 on a set of positive measure and Jf < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) fk such that fk → f in W1,p . peerReviewed

Time-dependent weak rate of convergence for functions of generalized bounded variation

2016

Let $W$ denote the Brownian motion. For any exponentially bounded Borel function $g$ the function $u$ defined by $u(t,x)= \mathbb{E}[g(x{+}\sigma W_{T-t})]$ is the stochastic solution of the backward heat equation with terminal condition $g$. Let $u^n(t,x)$ denote the corresponding approximation generated by a simple symmetric random walk with time steps $2T/n$ and space steps $\pm \sigma \sqrt{T/n}$ where $\sigma > 0$. For quite irregular terminal conditions $g$ (bounded variation on compact intervals, locally H\"older continuous) the rate of convergence of $u^n(t,x)$ to $u(t,x)$ is considered, and also the behavior of the error $u^n(t,x)-u(t,x)$ as $t$ tends to $T$

Mean square rate of convergence for random walk approximation of forward-backward SDEs

2020

AbstractLet (Y,Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk$B^n$from the underlying Brownian motionBby Skorokhod embedding, one can show$L_2$-convergence of the corresponding solutions$(Y^n,Z^n)$to$(Y, Z).$We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in$C^{2,\alpha}$. The proof relies on an approximative representation of$Z^n$and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to t…

Donsker-Type Theorem for BSDEs: Rate of Convergence

2019

In this paper, we study in the Markovian case the rate of convergence in Wasserstein distance when the solution to a BSDE is approximated by a solution to a BSDE driven by a scaled random walk as introduced in Briand, Delyon and Mémin (Electron. Commun. Probab. 6 (2001) Art. ID 1). This is related to the approximation of solutions to semilinear second order parabolic PDEs by solutions to their associated finite difference schemes and the speed of convergence. peerReviewed

On Malliavin calculus and approximation of stochastic integrals for Lévy processes

2012

Space partitioning of exchange-correlation functionals with the projector augmented-wave method

2018

We implement a Becke fuzzy cells type space partitioning scheme for the purposes of exchange-correlation within the GPAW projector augmented-wave method based density functional theory code. Space partitioning is needed in the situation where one needs to treat different parts of a combined system with different exchange-correlation functionals. For example, bulk and surface regions of a system could be treated with functionals that are specifically designed to capture the distinct physics of those regions. Here, we use the space partitioning scheme to implement the quasi-nonuniform exchange-correlation scheme, which is a useful practical approach for calculating metallic alloys on the gene…

Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: dynamic ray tracing in Cartesian coordinates

2018

With a Hamilton–Jacobi equation in Cartesian coordinates as a starting point, it is common to use a system of ordinary differential equations describing the continuation of first-order derivatives of phase-space perturbations along a reference ray. Such derivatives can be exploited for calculating geometrical spreading on the reference ray and for establishing a framework for second-order extrapolation of traveltime to points outside the reference ray. The continuation of first-order derivatives of phase-space perturbations has historically been referred to as dynamic ray tracing. The reason for this is its importance in the process of calculating amplitudes along the reference ray. We exte…

Funktion approksimointi

2015

Attractor as a convex combination of a set of attractors

2021

This paper presents an effective approach to constructing numerical attractors of a general class of continuous homogenous dynamical systems: decomposing an attractor as a convex combination of a set of other existing attractors. For this purpose, the convergent Parameter Switching (PS) numerical method is used to integrate the underlying dynamical system. The method is built on a convergent fixed step-size numerical method for ODEs. The paper shows that the PS algorithm, incorporating two binary operations, can be used to approximate any numerical attractor via a convex combination of some existing attractors. Several examples are presented to show the effectiveness of the proposed method.…