Search results for "Estimates"

Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation

2020

We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson-Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp., 69:481-500, 2000] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computa…

New insights into the diversity dynamics of Triassic conodonts

2013

International audience; In this paper, we examine the diversity trends and the evolutionary patterns of Triassic conodonts through a newly powered large-scale data-set compiled directly from the primary literature. Paleodiversity dynamics analyses have been undertaken by working at the species level and using a system of time units based on biozone subdivisions for a fine temporal level resolution. The role of heterogeneous duration of taxa in diversity estimates has been evaluated through the probabilistic profiles. Results reveal three different stages in the diversity behaviour of Triassic conodonts from standing metrics delimited by two inflections at the mid-Anisian and mid-Carnian. Su…

Generalized dimension estimates for images of porous sets in metric spaces

2016

Trends in epidemiology: the role of denominator fluctuation in population based estimates

2021

Background Population estimates are of paramount importance for calculating occurrence and association measures although they can be affected by problems of accuracy and completeness. This study has performed a simulation of the impact of Italian population size variability on incidence rates. Methods Data have been obtained by the Italian National Institute of Statistics. For each year expected cases were calculated at increasing fixed rates (up to 1,000/100,000) and were considered constant in the “following year”, calculating statistical differences (P Results In Italy and in other regions, statistically significant higher RRs were found in 2012 vs. 2011 whereas statistically significant…

Asymptotic Hölder regularity for the ellipsoid process

2020

We obtain an asymptotic Hölder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.

Poincaré Type Inequalities for Vector Functions with Zero Mean Normal Traces on the Boundary and Applications to Interpolation Methods

2019

We consider inequalities of the Poincaré–Steklov type for subspaces of H1 -functions defined in a bounded domain Ω∈Rd with Lipschitz boundary ∂Ω . For scalar valued functions, the subspaces are defined by zero mean condition on ∂Ω or on a part of ∂Ω having positive d−1 measure. For vector valued functions, zero mean conditions are applied to normal components on plane faces of ∂Ω (or to averaged normal components on curvilinear faces). We find explicit and simply computable bounds of constants in the respective Poincaré type inequalities for domains typically used in finite element methods (triangles, quadrilaterals, tetrahedrons, prisms, pyramids, and domains composed of them). The second …

A posteriori error identities for nonlinear variational problems

2015

A posteriori error estimation methods are usually developed in the context of upper and lower bounds of errors. In this paper, we are concerned with a posteriori analysis in terms of identities, i.e., we deduce error relations, which holds as equalities. We discuss a general form of error identities for a wide class of convex variational problems. The left hand sides of these identities can be considered as certain measures of errors (expressed in terms of primal/dual solutions and respective approximations) while the right hand sides contain only known approximations. Finally, we consider several examples and show that in some simple cases these identities lead to generalized forms of the …

Estimates for the Differences of Certain Positive Linear Operators

2020

The present paper deals with estimates for differences of certain positive linear operators defined on bounded or unbounded intervals. Our approach involves Baskakov type operators, the kth order Kantorovich modification of the Baskakov operators, the discrete operators associated with Baskakov operators, Meyer&ndash

Decay estimates for time-fractional and other non-local in time subdiffusion equations in R^d

2016

We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in R d . An important special case is the timefractional diffusion equation, which has seen much interest during the last years, mostly due to its applications in the modeling of anomalous diffusion processes. We follow three different approaches and techniques to study this particular case: (A) estimates based on the fundamental solution and Young’s inequality, (B) Fourier multiplier methods, and (C) the energy method. It turns out that the decay behaviour is markedly different from the heat equation case, in particular there occurs a critical dimension phenom…

Guaranteed error bounds for linear algebra problems and a class of Picard-Lindelöf iteration methods

2012

This study focuses on iteration methods based on the Banach fixed point theorem and a posteriori error estimates of Ostrowski. Their application for systems of linear simultaneous equations, bounded linear operators, as well as integral and differential equations is considered. The study presents a new version of the Picard–Lindelöf method for ordinary differential equations (ODEs) supplied with guaranteed and explicitly computable upper bounds of the approximation error. The estimates derived in the thesis take into account interpolation and integration errors and, therefore, provide objective information on the accuracy of computed approximations.