2,303 research outputs found
On the existence of optimal affine methods for approximating linear functionals
AbstractThe existence of an optimal affine method using linear information is established for the approximation of a linear functional on a convex set. This is a generalization of a result of S. A. Smolyak (“On Optimal Restoration of Functions and Functionals of Them,” Candidate Dissertation, Moscow State University, 1965)
Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
We analyze the convergence of compressive sensing based sampling techniques
for the efficient evaluation of functionals of solutions for a class of
high-dimensional, affine-parametric, linear operator equations which depend on
possibly infinitely many parameters. The proposed algorithms are based on
so-called "non-intrusive" sampling of the high-dimensional parameter space,
reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a
functional of the parametric solution is then computed via compressive sensing
methods from samples of functionals of the solution. A key ingredient in our
analysis of independent interest consists in a generalization of recent results
on the approximate sparsity of generalized polynomial chaos representations
(gpc) of the parametric solution families, in terms of the gpc series with
respect to tensorized Chebyshev polynomials. In particular, we establish
sufficient conditions on the parametric inputs to the parametric operator
equation such that the Chebyshev coefficients of the gpc expansion are
contained in certain weighted -spaces for . Based on this we
show that reconstructions of the parametric solutions computed from the sampled
problems converge, with high probability, at the , resp.
convergence rates afforded by best -term approximations of the parametric
solution up to logarithmic factors.Comment: revised version, 27 page
Approximating gradients with continuous piecewise polynomial functions
Motivated by conforming finite element methods for elliptic problems of
second order, we analyze the approximation of the gradient of a target function
by continuous piecewise polynomial functions over a simplicial mesh. The main
result is that the global best approximation error is equivalent to an
appropriate sum in terms of the local best approximations errors on elements.
Thus, requiring continuity does not downgrade local approximability and
discontinuous piecewise polynomials essentially do not offer additional
approximation power, even for a fixed mesh. This result implies error bounds in
terms of piecewise regularity over the whole admissible smoothness range.
Moreover, it allows for simple local error functionals in adaptive tree
approximation of gradients.Comment: 21 pages, 1 figur
Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems
A new class of non-homogeneous state-affine systems is introduced for use in
reservoir computing. Sufficient conditions are identified that guarantee first,
that the associated reservoir computers with linear readouts are causal,
time-invariant, and satisfy the fading memory property and second, that a
subset of this class is universal in the category of fading memory filters with
stochastic almost surely uniformly bounded inputs. This means that any
discrete-time filter that satisfies the fading memory property with random
inputs of that type can be uniformly approximated by elements in the
non-homogeneous state-affine family.Comment: 41 page
A Variational Approach to Particles in Lipid Membranes
A variety of models for the membrane-mediated interaction of particles in
lipid membranes, mostly well-established in theoretical physics, is reviewed
from a mathematical perspective. We provide mathematically consistent
formulations in a variational framework, relate apparently different modelling
approaches in terms of successive approximation, and investigate existence and
uniqueness. Numerical computations illustrate that the new variational
formulations are directly accessible to effective numerical methods
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