537 research outputs found
The Stability of One-Step Schemes for First-Order Two-Point Boundary Value Problems
The stability of a finite difference scheme is related explicitly to the stability of the continuous problem being solved. At times, this gives materially better estimates for the stability constant than those obtained by the standard process of appealing to the stability of the numerical scheme for the associated initial value problem
Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants
We present two new adaptive quadrature routines. Both routines differ from
previously published algorithms in many aspects, most significantly in how they
represent the integrand, how they treat non-numerical values of the integrand,
how they deal with improper divergent integrals and how they estimate the
integration error. The main focus of these improvements is to increase the
reliability of the algorithms without significantly impacting their efficiency.
Both algorithms are implemented in Matlab and tested using both the "families"
suggested by Lyness and Kaganove and the battery test used by Gander and
Gautschi and Kahaner. They are shown to be more reliable, albeit in some cases
less efficient, than other commonly-used adaptive integrators.Comment: 32 pages, submitted to ACM Transactions on Mathematical Softwar
Multivariate Anisotropic Interpolation on the Torus
We investigate the error of periodic interpolation, when sampling a function
on an arbitrary pattern on the torus. We generalize the periodic Strang-Fix
conditions to an anisotropic setting and provide an upper bound for the error
of interpolation. These conditions and the investigation of the error
especially take different levels of smoothness along certain directions into
account
Bivariate spline interpolation with optimal approximation order
Let be a triangulation of some polygonal domain f c R2 and let S9 (A) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to A. We develop the first Hermite-type interpolation scheme for S9 (A), q >_ 3r + 2, whose approximation error is bounded above by Kh4+i, where h is the maximal diameter of the triangles in A, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and nearsingular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of Sr, (A). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [71 and [181
Deep Learning for ECG Segmentation
We propose an algorithm for electrocardiogram (ECG) segmentation using a
UNet-like full-convolutional neural network. The algorithm receives an
arbitrary sampling rate ECG signal as an input, and gives a list of onsets and
offsets of P and T waves and QRS complexes as output. Our method of
segmentation differs from others in speed, a small number of parameters and a
good generalization: it is adaptive to different sampling rates and it is
generalized to various types of ECG monitors. The proposed approach is superior
to other state-of-the-art segmentation methods in terms of quality. In
particular, F1-measures for detection of onsets and offsets of P and T waves
and for QRS-complexes are at least 97.8%, 99.5%, and 99.9%, respectively.Comment: 10 pages, 7 figure
Kernel Approximation on Manifolds II: The -norm of the -projector
This article addresses two topics of significant mathematical and practical
interest in the theory of kernel approximation: the existence of local and
stable bases and the L_p--boundedness of the least squares operator. The latter
is an analogue of the classical problem in univariate spline theory, known
there as the "de Boor conjecture". A corollary of this work is that for
appropriate kernels the least squares projector provides universal near-best
approximations for functions f\in L_p, 1\le p\le \infty.Comment: 25 pages; minor revision; new proof of Lemma 3.9; accepted for
publication in SIAM J. on Math. Ana
Data analysis of gravitational-wave signals from spinning neutron stars. V. A narrow-band all-sky search
We present theory and algorithms to perform an all-sky coherent search for
periodic signals of gravitational waves in narrow-band data of a detector. Our
search is based on a statistic, commonly called the -statistic,
derived from the maximum-likelihood principle in Paper I of this series. We
briefly review the response of a ground-based detector to the
gravitational-wave signal from a rotating neuron star and the derivation of the
-statistic. We present several algorithms to calculate efficiently
this statistic. In particular our algorithms are such that one can take
advantage of the speed of fast Fourier transform (FFT) in calculation of the
-statistic. We construct a grid in the parameter space such that
the nodes of the grid coincide with the Fourier frequencies. We present
interpolation methods that approximately convert the two integrals in the
-statistic into Fourier transforms so that the FFT algorithm can
be applied in their evaluation. We have implemented our methods and algorithms
into computer codes and we present results of the Monte Carlo simulations
performed to test these codes.Comment: REVTeX, 20 pages, 8 figure
A goodness-of-fit test for the functional linear model with scalar response
In this work, a goodness-of-fit test for the null hypothesis of a functional
linear model with scalar response is proposed. The test is based on a
generalization to the functional framework of a previous one, designed for the
goodness-of-fit of regression models with multivariate covariates using random
projections. The test statistic is easy to compute using geometrical and matrix
arguments, and simple to calibrate in its distribution by a wild bootstrap on
the residuals. The finite sample properties of the test are illustrated by a
simulation study for several types of basis and under different alternatives.
Finally, the test is applied to two datasets for checking the assumption of the
functional linear model and a graphical tool is introduced. Supplementary
materials are available online.Comment: Paper: 17 pages, 2 figures, 3 tables. Supplementary material: 8
pages, 6 figures, 10 table
Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation
We apply multivariate Lagrange interpolation to synthesize polynomial
quantitative loop invariants for probabilistic programs. We reduce the
computation of an quantitative loop invariant to solving constraints over
program variables and unknown coefficients. Lagrange interpolation allows us to
find constraints with less unknown coefficients. Counterexample-guided
refinement furthermore generates linear constraints that pinpoint the desired
quantitative invariants. We evaluate our technique by several case studies with
polynomial quantitative loop invariants in the experiments
Numerical shadows: measures and densities on the numerical range
For any operator acting on an -dimensional Hilbert space we
introduce its numerical shadow, which is a probability measure on the complex
plane supported by the numerical range of . The shadow of at point
is defined as the probability that the inner product is equal to ,
where stands for a random complex vector from , satisfying .
In the case of N=2 the numerical shadow of a non-normal operator can be
interpreted as a shadow of a hollow sphere projected on a plane. A similar
interpretation is provided also for higher dimensions. For a hermitian its
numerical shadow forms a probability distribution on the real axis which is
shown to be a one dimensional -spline. In the case of a normal the
numerical shadow corresponds to a shadow of a transparent solid simplex in
onto the complex plane. Numerical shadow is found explicitly for
Jordan matrices , direct sums of matrices and in all cases where the
shadow is rotation invariant. Results concerning the moments of shadow measures
play an important role. A general technique to study numerical shadow via the
Cartesian decomposition is described, and a link of the numerical shadow of an
operator to its higher-rank numerical range is emphasized.Comment: 37 pages, 8 figure
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