101,292 research outputs found
Chebyshev pseudospectral methods for conservation laws with source terms and applications to multiphase flow
Pseudospectral methods are well known to produce superior results for the solution of partial differential equations whose solutions have a certain amount of regularity. Recent advances have made possible the use of spectral methods for the solution of conservation laws whose solutions may contain shocks. We use a recently described Super Spectral Viscosity method to obtain stable approximations of Systems of Nonlinear Hyperbolic Conservation Laws. A recently developed postprocessing method, which is theoretically capable of completely removing the Gibbs phenomenon from the Super Spectral Viscosity approximation, is examined. The postprocessing method has shown great promise when applied in some simple cases. We discuss its application to more complicated problems and examine the possibility of the method being used as a black box postprocessing method. Applications to multiphase fluid flow are made
Acceleration Methods
This monograph covers some recent advances in a range of acceleration
techniques frequently used in convex optimization. We first use quadratic
optimization problems to introduce two key families of methods, namely momentum
and nested optimization schemes. They coincide in the quadratic case to form
the Chebyshev method. We discuss momentum methods in detail, starting with the
seminal work of Nesterov and structure convergence proofs using a few master
templates, such as that for optimized gradient methods, which provide the key
benefit of showing how momentum methods optimize convergence guarantees. We
further cover proximal acceleration, at the heart of the Catalyst and
Accelerated Hybrid Proximal Extragradient frameworks, using similar algorithmic
patterns. Common acceleration techniques rely directly on the knowledge of some
of the regularity parameters in the problem at hand. We conclude by discussing
restart schemes, a set of simple techniques for reaching nearly optimal
convergence rates while adapting to unobserved regularity parameters.Comment: Published in Foundation and Trends in Optimization (see
https://www.nowpublishers.com/article/Details/OPT-036
Quantile calculus and censored regression
Quantile regression has been advocated in survival analysis to assess
evolving covariate effects. However, challenges arise when the censoring time
is not always observed and may be covariate-dependent, particularly in the
presence of continuously-distributed covariates. In spite of several recent
advances, existing methods either involve algorithmic complications or impose a
probability grid. The former leads to difficulties in the implementation and
asymptotics, whereas the latter introduces undesirable grid dependence. To
resolve these issues, we develop fundamental and general quantile calculus on
cumulative probability scale in this article, upon recognizing that probability
and time scales do not always have a one-to-one mapping given a survival
distribution. These results give rise to a novel estimation procedure for
censored quantile regression, based on estimating integral equations. A
numerically reliable and efficient Progressive Localized Minimization (PLMIN)
algorithm is proposed for the computation. This procedure reduces exactly to
the Kaplan--Meier method in the -sample problem, and to standard uncensored
quantile regression in the absence of censoring. Under regularity conditions,
the proposed quantile coefficient estimator is uniformly consistent and
converges weakly to a Gaussian process. Simulations show good statistical and
algorithmic performance. The proposal is illustrated in the application to a
clinical study.Comment: Published in at http://dx.doi.org/10.1214/09-AOS771 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Stationary multivariate subdivision: Joint spectral radius and asymptotic similarity
In this paper we study scalar multivariate non-stationary subdivision schemes with
a general integer dilation matrix. We present a new numerically efficient method for
checking convergence and H ̈older regularity of such schemes. This method relies on the
concepts of approximate sum rules, asymptotic similarity and the so-called joint spectral
radius of a finite set of square matrices. The combination of these concepts allows us to
employ recent advances in linear algebra for exact computation of the joint spectral radius
that have had already a great impact on studies of stationary subdivision schemes. We
also expose the limitations of non-stationary schemes in their capability to reproduce and
generate certain function spaces. We illustrate our results with several examples
On the Complexity of Solving Quadratic Boolean Systems
A fundamental problem in computer science is to find all the common zeroes of
quadratic polynomials in unknowns over . The
cryptanalysis of several modern ciphers reduces to this problem. Up to now, the
best complexity bound was reached by an exhaustive search in
operations. We give an algorithm that reduces the problem to a combination of
exhaustive search and sparse linear algebra. This algorithm has several
variants depending on the method used for the linear algebra step. Under
precise algebraic assumptions on the input system, we show that the
deterministic variant of our algorithm has complexity bounded by
when , while a probabilistic variant of the Las Vegas type
has expected complexity . Experiments on random systems show
that the algebraic assumptions are satisfied with probability very close to~1.
We also give a rough estimate for the actual threshold between our method and
exhaustive search, which is as low as~200, and thus very relevant for
cryptographic applications.Comment: 25 page
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