1,015 research outputs found
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
In this paper, we consider the problem of recovering a compactly supported
multivariate function from a collection of pointwise samples of its Fourier
transform taken nonuniformly. We do this by using the concept of weighted
Fourier frames. A seminal result of Beurling shows that sample points give rise
to a classical Fourier frame provided they are relatively separated and of
sufficient density. However, this result does not allow for arbitrary
clustering of sample points, as is often the case in practice. Whilst keeping
the density condition sharp and dimension independent, our first result removes
the separation condition and shows that density alone suffices. However, this
result does not lead to estimates for the frame bounds. A known result of
Groechenig provides explicit estimates, but only subject to a density condition
that deteriorates linearly with dimension. In our second result we improve
these bounds by reducing the dimension dependence. In particular, we provide
explicit frame bounds which are dimensionless for functions having compact
support contained in a sphere. Next, we demonstrate how our two main results
give new insight into a reconstruction algorithm---based on the existing
generalized sampling framework---that allows for stable and quasi-optimal
reconstruction in any particular basis from a finite collection of samples.
Finally, we construct sufficiently dense sampling schemes that are often used
in practice---jittered, radial and spiral sampling schemes---and provide
several examples illustrating the effectiveness of our approach when tested on
these schemes
Fractal homogenization of multiscale interface problems
Inspired by continuum mechanical contact problems with geological fault
networks, we consider elliptic second order differential equations with jump
conditions on a sequence of multiscale networks of interfaces with a finite
number of non-separating scales. Our aim is to derive and analyze a description
of the asymptotic limit of infinitely many scales in order to quantify the
effect of resolving the network only up to some finite number of interfaces and
to consider all further effects as homogeneous. As classical homogenization
techniques are not suited for this kind of geometrical setting, we suggest a
new concept, called fractal homogenization, to derive and analyze an asymptotic
limit problem from a corresponding sequence of finite-scale interface problems.
We provide an intuitive characterization of the corresponding fractal solution
space in terms of generalized jumps and gradients together with continuous
embeddings into L2 and Hs, s<1/2. We show existence and uniqueness of the
solution of the asymptotic limit problem and exponential convergence of the
approximating finite-scale solutions. Computational experiments involving a
related numerical homogenization technique illustrate our theoretical findings
A dissipative time reversal technique for photo-acoustic tomography in a cavity
We consider the inverse source problem arising in thermo- and photo-acoustic
tomography. It consists in reconstructing the initial pressure from the
boundary measurements of the acoustic wave. Our goal is to extend versatile
time reversal techniques to the case of perfectly reflecting boundary of the
domain. Standard time reversal works only if the solution of the direct problem
decays in time, which does not happen in the setup we consider. We thus propose
a novel time reversal technique with a non-standard boundary condition. The
error induced by this time reversal technique satisfies the wave equation with
a dissipative boundary condition and, therefore, decays in time. For larger
measurement times, this method yields a close approximation; for smaller times,
the first approximation can be iteratively refined, resulting in a convergent
Neumann series for the approximation
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