793 research outputs found
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM
Elliptic boundary value problems which are posed on a random domain can be
mapped to a fixed, nominal domain. The randomness is thus transferred to the
diffusion matrix and the loading. While this domain mapping method is quite
efficient for theory and practice, since only a single domain discretisation is
needed, it also requires the knowledge of the domain mapping.
However, in certain applications, the random domain is only described by its
random boundary, while the quantity of interest is defined on a fixed,
deterministic subdomain. In this setting, it thus becomes necessary to compute
a random domain mapping on the whole domain, such that the domain mapping is
the identity on the fixed subdomain and maps the boundary of the chosen fixed,
nominal domain on to the random boundary.
To overcome the necessity of computing such a mapping, we therefore couple
the finite element method on the fixed subdomain with the boundary element
method on the random boundary. We verify the required regularity of the
solution with respect to the random domain mapping for the use of multilevel
quadrature, derive the coupling formulation, and show by numerical results that
the approach is feasible
Some inequalities for mappings whose derivatives are bounded and applications to special means of real numbers
AbstractIn this paper, we shall introduce two new inequalities of Hermite-Hadamard type for convex functions with bounded derivatives. Some applications to special means of real numbers are also included
Analyticity in spaces of convergent power series and applications
We study the analytic structure of the space of germs of an analytic function
at the origin of \ww C^{\times m} , namely the space \germ{\mathbf{z}} where
\mathbf{z}=\left(z\_{1},\cdots,z\_{m}\right) , equipped with a convenient
locally convex topology. We are particularly interested in studying the
properties of analytic sets of \germ{\mathbf{z}} as defined by the vanishing
locus of analytic maps. While we notice that \germ{\mathbf{z}} is not Baire we
also prove it enjoys the analytic Baire property: the countable union of proper
analytic sets of \germ{\mathbf{z}} has empty interior. This property underlies
a quite natural notion of a generic property of \germ{\mathbf{z}} , for which
we prove some dynamics-related theorems. We also initiate a program to tackle
the task of characterizing glocal objects in some situations
Optimal Algorithms for Numerical Integration: Recent Results and Open Problems
We present recent results on optimal algorithms for numerical integration and
several open problems. The paper has six parts:
1. Introduction
2. Lower Bounds
3. Universality
4. General Domains
5. iid Information
6. Concluding RemarksComment: Survey written for the MCQMC conference in Linz, 26 pages. arXiv
admin note: text overlap with arXiv:2108.0205
The Virtual Element Method with curved edges
In this paper we initiate the investigation of Virtual Elements with curved
faces. We consider the case of a fixed curved boundary in two dimensions, as it
happens in the approximation of problems posed on a curved domain or with a
curved interface. While an approximation of the domain with polygons leads, for
degree of accuracy , to a sub-optimal rate of convergence, we show
(both theoretically and numerically) that the proposed curved VEM lead to an
optimal rate of convergence
Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems
In this paper we present a rigorous cost and error analysis of a multilevel
estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for
lognormal diffusion problems. These problems are motivated by uncertainty
quantification problems in subsurface flow. We extend the convergence analysis
in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite
element discretizations and give a constructive proof of the
dimension-independent convergence of the QMC rules. More precisely, we provide
suitable parameters for the construction of such rules that yield the required
variance reduction for the multilevel scheme to achieve an -error
with a cost of with , and in
practice even , for sufficiently fast decaying covariance
kernels of the underlying Gaussian random field inputs. This confirms that the
computational gains due to the application of multilevel sampling methods and
the gains due to the application of QMC methods, both demonstrated in earlier
works for the same model problem, are complementary. A series of numerical
experiments confirms these gains. The results show that in practice the
multilevel QMC method consistently outperforms both the multilevel MC method
and the single-level variants even for non-smooth problems.Comment: 32 page
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