627 research outputs found
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
Noncommutative Bennett and Rosenthal inequalities
In this paper we extend the Bernstein, Prohorov and Bennett inequalities to
the noncommutative setting. In addition we provide an improved version of the
noncommutative Rosenthal inequality, essentially due to Nagaev, Pinelis and
Pinelis, Utev for commutative random variables. We also present new best
constants in Rosenthal's inequality. Applying these results to random Fourier
projections, we recover and elaborate on fundamental results from compressed
sensing, due to Candes, Romberg and Tao.Comment: Published in at http://dx.doi.org/10.1214/12-AOP771 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Deterministic Sparse Fourier Transform with an ?_{?} Guarantee
In this paper we revisit the deterministic version of the Sparse Fourier
Transform problem, which asks to read only a few entries of and design a recovery algorithm such that the output of the
algorithm approximates , the Discrete Fourier Transform (DFT) of .
The randomized case has been well-understood, while the main work in the
deterministic case is that of Merhi et al.\@ (J Fourier Anal Appl 2018), which
obtains samples and a similar runtime
with the guarantee. We focus on the stronger
guarantee and the closely related problem of incoherent
matrices. We list our contributions as follows.
1. We find a deterministic collection of samples for the
recovery in time , and a deterministic
collection of samples for the sparse
recovery in time .
2. We give new deterministic constructions of incoherent matrices that are
row-sampled submatrices of the DFT matrix, via a derandomization of Bernstein's
inequality and bounds on exponential sums considered in analytic number theory.
Our first construction matches a previous randomized construction of Nelson,
Nguyen and Woodruff (RANDOM'12), where there was no constraint on the form of
the incoherent matrix.
Our algorithms are nearly sample-optimal, since a lower bound of is known, even for the case where the sensing matrix can be
arbitrarily designed. A similar lower bound of is
known for incoherent matrices.Comment: ICALP 2020--presentation improved according to reviewers' comment
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
Multidimensional quantum entanglement with large-scale integrated optics
The ability to control multidimensional quantum systems is key for the
investigation of fundamental science and for the development of advanced
quantum technologies. Here we demonstrate a multidimensional integrated quantum
photonic platform able to robustly generate, control and analyze
high-dimensional entanglement. We realize a programmable bipartite entangled
system with dimension up to on a large-scale silicon-photonics
quantum circuit. The device integrates more than 550 photonic components on a
single chip, including 16 identical photon-pair sources. We verify the high
precision, generality and controllability of our multidimensional technology,
and further exploit these abilities to demonstrate key quantum applications
experimentally unexplored before, such as quantum randomness expansion and
self-testing on multidimensional states. Our work provides a prominent
experimental platform for the development of multidimensional quantum
technologies.Comment: Science, (2018
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