325 research outputs found
Weighted eigenfunction estimates with applications to compressed sensing
Using tools from semiclassical analysis, we give weighted L^\infty estimates
for eigenfunctions of strictly convex surfaces of revolution. These estimates
give rise to new sampling techniques and provide improved bounds on the number
of samples necessary for recovering sparse eigenfunction expansions on surfaces
of revolution. On the sphere, our estimates imply that any function having an
s-sparse expansion in the first N spherical harmonics can be efficiently
recovered from its values at m > s N^(1/6) log^4(N) sampling points.Comment: 25 pages, 5 figure
Compressive Wave Computation
This paper considers large-scale simulations of wave propagation phenomena.
We argue that it is possible to accurately compute a wavefield by decomposing
it onto a largely incomplete set of eigenfunctions of the Helmholtz operator,
chosen at random, and that this provides a natural way of parallelizing wave
simulations for memory-intensive applications.
This paper shows that L1-Helmholtz recovery makes sense for wave computation,
and identifies a regime in which it is provably effective: the one-dimensional
wave equation with coefficients of small bounded variation. Under suitable
assumptions we show that the number of eigenfunctions needed to evolve a sparse
wavefield defined on N points, accurately with very high probability, is
bounded by C log(N) log(log(N)), where C is related to the desired accuracy and
can be made to grow at a much slower rate than N when the solution is sparse.
The PDE estimates that underlie this result are new to the authors' knowledge
and may be of independent mathematical interest; they include an L1 estimate
for the wave equation, an estimate of extension of eigenfunctions, and a bound
for eigenvalue gaps in Sturm-Liouville problems.
Numerical examples are presented in one spatial dimension and show that as
few as 10 percents of all eigenfunctions can suffice for accurate results.
Finally, we argue that the compressive viewpoint suggests a competitive
parallel algorithm for an adjoint-state inversion method in reflection
seismology.Comment: 45 pages, 4 figure
The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch
Recent and forthcoming advances in instrumentation, and giant new surveys,
are creating astronomical data sets that are not amenable to the methods of
analysis familiar to astronomers. Traditional methods are often inadequate not
merely because of the size in bytes of the data sets, but also because of the
complexity of modern data sets. Mathematical limitations of familiar algorithms
and techniques in dealing with such data sets create a critical need for new
paradigms for the representation, analysis and scientific visualization (as
opposed to illustrative visualization) of heterogeneous, multiresolution data
across application domains. Some of the problems presented by the new data sets
have been addressed by other disciplines such as applied mathematics,
statistics and machine learning and have been utilized by other sciences such
as space-based geosciences. Unfortunately, valuable results pertaining to these
problems are mostly to be found only in publications outside of astronomy. Here
we offer brief overviews of a number of concepts, techniques and developments,
some "old" and some new. These are generally unknown to most of the
astronomical community, but are vital to the analysis and visualization of
complex datasets and images. In order for astronomers to take advantage of the
richness and complexity of the new era of data, and to be able to identify,
adopt, and apply new solutions, the astronomical community needs a certain
degree of awareness and understanding of the new concepts. One of the goals of
this paper is to help bridge the gap between applied mathematics, artificial
intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in
Astronomy, special issue "Robotic Astronomy
Compressive Sensing with Redundant Dictionaries and Structured Measurements
Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary D. This problem is now understood to be well-posed and efficiently solvable under suitable assumptions on the measurements and dictionary, if the number of measurements scales roughly with the sparsity level. One sufficient condition for such is the D-restricted isometry property (D-RIP), which asks that the sampling matrix approximately preserve the norm of all signals which are sufficiently sparse in D. While many classes of random matrices are known to satisfy such conditions, such matrices are not representative of the structural constraints imposed by practical sensing systems. We close this gap in the theory by demonstrating that one can subsample a fixed orthogonal matrix in such a way that the D-RIP will hold, provided this basis is sufficiently incoherent with the sparsifying dictionary D. We also extend this analysis to allow for weighted sparse expansions. Consequently, we arrive at compressive sensing recovery guarantees for structured measurements and redundant dictionaries, opening the door to a wide array of practical applications
Data-driven model reduction and transfer operator approximation
In this review paper, we will present different data-driven dimension
reduction techniques for dynamical systems that are based on transfer operator
theory as well as methods to approximate transfer operators and their
eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out
similarities and differences between methods developed independently by the
dynamical systems, fluid dynamics, and molecular dynamics communities such as
time-lagged independent component analysis (TICA), dynamic mode decomposition
(DMD), and their respective generalizations. As a result, extensions and best
practices developed for one particular method can be carried over to other
related methods
Jacobi Polynomials, Bernstein-type Inequalities and Dispersion Estimates for the Discrete Laguerre Operator
The present paper is about Bernstein-type estimates for Jacobi polynomials
and their applications to various branches in mathematics. This is an old topic
but we want to add a new wrinkle by establishing some intriguing connections
with dispersive estimates for a certain class of Schr\"odinger equations whose
Hamiltonian is given by the generalized Laguerre operator. More precisely, we
show that dispersive estimates for the Schr\"odinger equation associated with
the generalized Laguerre operator are connected with Bernstein-type
inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi
polynomials to establish some new dispersive estimates. In turn, the optimal
dispersive decay estimates lead to new Bernstein-type inequalities.Comment: 25 page
From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere
We prove a sharp multiplier theorem of Mihlin-H\"ormander type for the
Grushin operator on the unit sphere in , and a corresponding
boundedness result for the associated Bochner-Riesz means. The proof hinges on
precise pointwise bounds for spherical harmonics.Comment: 32 page
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