9,956 research outputs found
Numerical computation of periodic orbits and isochrons for state-dependent delay perturbation of an ODE in the plane
We present algorithms and their implementation to compute limit cycles and
their isochrons for state-dependent delay equations (SDDE's) which are
perturbed from a planar differential equation with a limit cycle.
Note that the space of solutions of an SDDE is infinite dimensional. We
compute a two parameter family of solutions of the SDDE which converge to the
solutions of the ODE as the perturbation goes to zero in a neighborhood of the
limit cycle.
The method we use formulates functional equations among periodic functions
(or functions converging exponentially to periodic). The functional equations
express that the functions solve the SDDE. Therefore, rather than evolving
initial data and finding solutions of a certain shape, we consider spaces of
functions with the desired shape and require that they are solutions.
The mathematical theory of these invariance equations is developed in a
companion paper, which develops "a posteriori" theorems. They show that, if
there is a sufficiently approximate solution (with respect to some explicit
condition numbers), then there is a true solution close to the approximate one.
Since the numerical methods produce an approximate solution, and provide
estimates of the condition numbers, we can make sure that the numerical
solutions we consider approximate true solutions.
In this paper, we choose a systematic way to approximate functions by a
finite set of numbers (Taylor-Fourier series) and develop a toolkit of
algorithms that implement the operators -- notably composition -- that enter
into the theory. We also present several implementation results and present the
results of running the algorithms and their implementation in some
representative cases.Comment: 32 page
Metric Gaussian Variational Inference
Solving Bayesian inference problems approximately with variational approaches
can provide fast and accurate results. Capturing correlation within the
approximation requires an explicit parametrization. This intrinsically limits
this approach to either moderately dimensional problems, or requiring the
strongly simplifying mean-field approach. We propose Metric Gaussian
Variational Inference (MGVI) as a method that goes beyond mean-field. Here
correlations between all model parameters are taken into account, while still
scaling linearly in computational time and memory. With this method we achieve
higher accuracy and in many cases a significant speedup compared to traditional
methods. MGVI is an iterative method that performs a series of Gaussian
approximations to the posterior. We alternate between approximating the
covariance with the inverse Fisher information metric evaluated at an
intermediate mean estimate and optimizing the KL-divergence for the given
covariance with respect to the mean. This procedure is iterated until the
uncertainty estimate is self-consistent with the mean parameter. We achieve
linear scaling by avoiding to store the covariance explicitly at any time.
Instead we draw samples from the approximating distribution relying on an
implicit representation and numerical schemes to approximately solve linear
equations. Those samples are used to approximate the KL-divergence and its
gradient. The usage of natural gradient descent allows for rapid convergence.
Formulating the Bayesian model in standardized coordinates makes MGVI
applicable to any inference problem with continuous parameters. We demonstrate
the high accuracy of MGVI by comparing it to HMC and its fast convergence
relative to other established methods in several examples. We investigate
real-data applications, as well as synthetic examples of varying size and
complexity and up to a million model parameters.Comment: Code is part of NIFTy5 release at
https://gitlab.mpcdf.mpg.de/ift/NIFT
Semantics, Representations and Grammars for Deep Learning
Deep learning is currently the subject of intensive study. However,
fundamental concepts such as representations are not formally defined --
researchers "know them when they see them" -- and there is no common language
for describing and analyzing algorithms. This essay proposes an abstract
framework that identifies the essential features of current practice and may
provide a foundation for future developments.
The backbone of almost all deep learning algorithms is backpropagation, which
is simply a gradient computation distributed over a neural network. The main
ingredients of the framework are thus, unsurprisingly: (i) game theory, to
formalize distributed optimization; and (ii) communication protocols, to track
the flow of zeroth and first-order information. The framework allows natural
definitions of semantics (as the meaning encoded in functions), representations
(as functions whose semantics is chosen to optimized a criterion) and grammars
(as communication protocols equipped with first-order convergence guarantees).
Much of the essay is spent discussing examples taken from the literature. The
ultimate aim is to develop a graphical language for describing the structure of
deep learning algorithms that backgrounds the details of the optimization
procedure and foregrounds how the components interact. Inspiration is taken
from probabilistic graphical models and factor graphs, which capture the
essential structural features of multivariate distributions.Comment: 20 pages, many diagram
Operator learning approach for the limited view problem in photoacoustic tomography
In photoacoustic tomography, one is interested to recover the initial
pressure distribution inside a tissue from the corresponding measurements of
the induced acoustic wave on the boundary of a region enclosing the tissue. In
the limited view problem, the wave boundary measurements are given on the part
of the boundary, whereas in the full view problem, the measurements are known
on the whole boundary. For the full view problem, there exist various fast and
robust reconstruction methods. These methods give severe reconstruction
artifacts when they are applied directly to the limited view data. One approach
for reducing such artefacts is trying to extend the limited view data to the
whole region boundary, and then use existing reconstruction methods for the
full view data. In this paper, we propose an operator learning approach for
constructing an operator that gives an approximate extension of the limited
view data. We consider the behavior of a reconstruction formula on the extended
limited view data that is given by our proposed approach. Approximation errors
of our approach are analyzed. We also present numerical results with the
proposed extension approach supporting our theoretical analysis
Non-Archimedean Ergodic Theory and Pseudorandom Generators
The paper develops techniques in order to construct computer programs,
pseudorandom number generators (PRNG), that produce uniformly distributed
sequences. The paper exploits an approach that treats standard processor
instructions (arithmetic and bitwise logical ones) as continuous functions on
the space of 2-adic integers. Within this approach, a PRNG is considered as a
dynamical system and is studied by means of the non-Archimedean ergodic theory.Comment: Submitted to The Computer Journa
Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes
The long-term dynamics of many dynamical systems evolve on an attracting,
invariant "slow manifold" that can be parameterized by a few observable
variables. Yet a simulation using the full model of the problem requires
initial values for all variables. Given a set of values for the observables
parameterizing the slow manifold, one needs a procedure for finding the
additional values such that the state is close to the slow manifold to some
desired accuracy. We consider problems whose solution has a singular
perturbation expansion, although we do not know what it is nor have any way to
compute it. We show in this paper that, under some conditions, computing the
values of the remaining variables so that their (m+1)-st time derivatives are
zero provides an estimate of the unknown variables that is an mth-order
approximation to a point on the slow manifold in sense to be defined. We then
show how this criterion can be applied approximately when the system is defined
by a legacy code rather than directly through closed form equations.Comment: 21 pages, 1 figur
PBBFMM3D: a parallel black-box algorithm for kernel matrix-vector multiplication
We introduce \texttt{PBBFMM3D}, a parallel black-box method for computing
kernel matrix-vector multiplication, where the underlying kernel is a
non-oscillatory function in three dimensions. While a naive method requires
\O(N^2) computation, \texttt{PBBFMM3D} reduces the cost to \O(N) work. In
particular, our algorithm requires only the ability to evaluate the kernel
function, and is thus a black-box method. To further accelerate the computation
on shared-memory machines, a parallel algorithm is presented and implemented
using \verb|OpenMP|, which achieved at most speedup on 32 cores in
our numerical experiments. A real-world application in geostatistics is also
presented, where \texttt{PBBFMM3D} is used in computing the truncated
eigen-decomposition (a.k.a., principle component analysis) of a covariance
matrix (a.k.a., graph Laplacian)
Discrete Symbol Calculus
This paper deals with efficient numerical representation and manipulation of
differential and integral operators as symbols in phase-space, i.e., functions
of space and frequency . The symbol smoothness conditions obeyed by
many operators in connection to smooth linear partial differential equations
allow to write fast-converging, non-asymptotic expansions in adequate systems
of rational Chebyshev functions or hierarchical splines. The classical results
of closedness of such symbol classes under multiplication, inversion and taking
the square root translate into practical iterative algorithms for realizing
these operations directly in the proposed expansions. Because symbol-based
numerical methods handle operators and not functions, their complexity depends
on the desired resolution very weakly, typically only through
factors. We present three applications to computational problems related to
wave propagation: 1) preconditioning the Helmholtz equation, 2) decomposing
wavefields into one-way components and 3) depth-stepping in reflection
seismology.Comment: 32 page
Symbolic-numeric interface: A review
A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach
Time-Limited Toeplitz Operators on Abelian Groups: Applications in Information Theory and Subspace Approximation
Toeplitz operators are fundamental and ubiquitous in signal processing and
information theory as models for linear, time-invariant (LTI) systems. Due to
the fact that any practical system can access only signals of finite duration,
time-limited restrictions of Toeplitz operators are naturally of interest. To
provide a unifying treatment of such systems working on different signal
domains, we consider time-limited Toeplitz operators on locally compact abelian
groups with the aid of the Fourier transform on these groups. In particular, we
survey existing results concerning the relationship between the spectrum of a
time-limited Toeplitz operator and the spectrum of the corresponding
non-time-limited Toeplitz operator. We also develop new results specifically
concerning the eigenvalues of time-frequency limiting operators on locally
compact abelian groups. Applications of our unifying treatment are discussed in
relation to channel capacity and in relation to representation and
approximation of signals
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