1,090 research outputs found
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
Perfect sampling algorithm for Schur processes
We describe random generation algorithms for a large class of random
combinatorial objects called Schur processes, which are sequences of random
(integer) partitions subject to certain interlacing conditions. This class
contains several fundamental combinatorial objects as special cases, such as
plane partitions, tilings of Aztec diamonds, pyramid partitions and more
generally steep domino tilings of the plane. Our algorithm, which is of
polynomial complexity, is both exact (i.e. the output follows exactly the
target probability law, which is either Boltzmann or uniform in our case), and
entropy optimal (i.e. it reads a minimal number of random bits as an input).
The algorithm encompasses previous growth procedures for special Schur
processes related to the primal and dual RSK algorithm, as well as the famous
domino shuffling algorithm for domino tilings of the Aztec diamond. It can be
easily adapted to deal with symmetric Schur processes and general Schur
processes involving infinitely many parameters. It is more concrete and easier
to implement than Borodin's algorithm, and it is entropy optimal.
At a technical level, it relies on unified bijective proofs of the different
types of Cauchy and Littlewood identities for Schur functions, and on an
adaptation of Fomin's growth diagram description of the RSK algorithm to that
setting. Simulations performed with this algorithm suggest interesting limit
shape phenomena for the corresponding tiling models, some of which are new.Comment: 26 pages, 19 figures (v3: final version, corrected a few misprints
present in v2
Dynamics of nodal points and the nodal count on a family of quantum graphs
We investigate the properties of the zeros of the eigenfunctions on quantum
graphs (metric graphs with a Schr\"odinger-type differential operator). Using
tools such as scattering approach and eigenvalue interlacing inequalities we
derive several formulas relating the number of the zeros of the n-th
eigenfunction to the spectrum of the graph and of some of its subgraphs. In a
special case of the so-called dihedral graph we prove an explicit formula that
only uses the lengths of the edges, entirely bypassing the information about
the graph's eigenvalues. The results are explained from the point of view of
the dynamics of zeros of the solutions to the scattering problem.Comment: 34 pages, 12 figure
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