Dagstuhl Seminar Proceedings, Volume 4401
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
4401
2005
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-4401
04401 Abstracts Collection – Algorithms and Complexity for Continuous
From 26.09.04 to 01.10.04, the Dagstuhl Seminar ``Algorithms and Complexity for Continuous Problems'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
Complexity and regularization of ill-posed problems
nonlinear approximation
tractability of high-dimensional numerical problems quantum computing
stochastic computation and quantization
global optimization
differential and integral equations
1-21
Regular Paper
Thomas
Müller-Gronbach
Thomas Müller-Gronbach
Erich
Novak
Erich Novak
Knut
Petras
Knut Petras
Joseph F.
Traub
Joseph F. Traub
10.4230/DagSemProc.04401.1
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04401 Summary – Algorithms and Complexity for Continuous Problems
The goal of this workshop was to bring together
researchers from different communities working on
computational aspects of continuous problems.
Continuous computational problems arise in many
areas of science and engineering. Examples include
path and multivariate integration, function
approximation, optimization, as well as
differential, integral, and operator equations.
Understanding the complexity of such problems and
constructing efficient algorithms is both
important and challenging.
The workshop was of a very interdisciplinary
nature with invitees from, e.g., computer science,
numerical analysis, discrete, applied, and pure
mathematics, physics, statistics, and scientific
computation. Many of the lectures were presented
by Ph.D. students.
Compared to earlier meetings, several very active
research areas received more emphasis. These
include Quantum Computing, Complexity and
Tractability of high-dimensional problems,
Stochastic Computation, and Quantization, which
was an entirely new field for this workshop.
Due to strong connections between the topics
treated at this workshop many of the participants
initiated new cooperations and research projects.
For more details, see the pdf-file with the same
title.
Complexity and Regularization of Ill-Posed Problems
Non-Linear Approximation
Tractability of High-Dimensional Numerical Problems Quasi-Monte Carlo Methods
Quantum Computing
Stochastic Computation and Quantization
Global Optimization
Differential and Integral Equation
0-0
Regular Paper
Thomas
Müller-Gronbach
Thomas Müller-Gronbach
Erich
Novak
Erich Novak
Knut
Petras
Knut Petras
Joseph F.
Traub
Joseph F. Traub
10.4230/DagSemProc.04401.2
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Fast Component-By-Component Construction of Rank-1 Lattice Rules for (Non-)Primes (Part II)
Part I: (this part of the talk by Ronald Cools)
We restate our previous result which showed that it is possible to
construct the generating vector of a rank-1 lattice rule in a fast way,
i.e. O(s n log(n)), with s the number of dimensions and n the number of
points assumed to be prime. Here we explicitly use basic facts from
algebra to exploit the structure of a matrix – which introduces the
crucial cost in the construction – to get a matrix-vector
multiplication in time O(n log(n)) instead of O(n^2). We again stress
the fact that the algorithm works for any tensor product reproducing
kernel Hilbert space.
Part II: (this part of the talk by Dirk Nuyens)
In the second part we generalize the tricks used for primes to
non-primes, by basically falling back to algebraic group theory. In
this way it can be shown that also for a non-prime number of points,
this crucial matrix-vector multiplication can be done in time O(n
log(n)). We conclude that the construction of rank-1 lattice rules in
an arbitrary r.k.h.s. for an arbitrary amount of points can be done in a
fast way of O(s n log(n)).
numerical integration
cubature/quadrature
rank-1 lattice
component-by-component construction
fast algorithm
1-26
Regular Paper
Dirk
Nuyens
Dirk Nuyens
Ronald
Cools
Ronald Cools
10.4230/DagSemProc.04401.3
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Functional Quantization and Entropy for Stochastic Processes
Let X be a Gaussian process and let U denote the
Strassen ball of X. A precise link between the
L^2-quantization error of X and the Kolmogorov
(metric) entropy of U in a Hilbert space setting
is established. In particular, the sharp
asymptotics of the Kolmogorov entropy problem is
derived. The condition imposed is regular
variation of the eigenvalues of the covariance
operator. Good computable quantizers for Gaussian
and diffusion processes and their numerical
efficieny are discussed.
This is joint work with G. PagÃƒÂ¨s, UniversitÃƒÂ© de Paris 6.
Functional quantization
entropy
product quantizers
1-15
Regular Paper
Harald
Luschgy
Harald Luschgy
Gilles
Pagès
Gilles Pagès
10.4230/DagSemProc.04401.4
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Information-Based Nonlinear Approximation: An Average Case Setting
Nonlinear approximation has usually been studied
under deterministic assumption and complete
information about the underlying functions.
We assume only partial information and we are
interested in the average case error and
complexity of approximation. It turns out that
the problem can be essentially split into two
independent problems related to average case
nonlinear (restricted) approximation from
complete information, and average case
unrestricted approximation from partial
information. The results are then applied to
average case piecewise polynomial approximation,
and to average case approximation of real
sequences.
average case setting
nonlinear approximation
information-based comlexity
1-1
Regular Paper
Leszek
Plaskota
Leszek Plaskota
10.4230/DagSemProc.04401.5
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Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations
We study algorithms for approximation of the mild solution
of stochastic heat equations on the spatial domain ]0,1[^d.
The error of an algorithm is defined in L_2-sense.
We derive lower bounds for the error of every algorithm
that uses a total of N evaluations of one-dimensional components
of the driving Wiener process W. For equations with additive
noise we derive matching upper bounds and we construct
asymptotically optimal algorithms. The error bounds depend on
N and d, and on the decay of eigenvalues of the covariance of W
in the case of nuclear noise. In the latter case the use of
non-uniform time discretizations is crucial.
Stochastic heat equation
Non-uniform time discretization
minimal errors
upper and lower bounds
1-37
Regular Paper
Klaus
Ritter
Klaus Ritter
Thomas
Müller-Gronbach
Thomas Müller-Gronbach
10.4230/DagSemProc.04401.6
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Monte Carlo solution for the Poisson equation on the base of spherical processes with shifted centres
We consider a class of spherical processes rapidly
converging to the boundary (so called Decentred
Random Walks on Spheres or spherical processes
with shifted centres) in comparison with the
standard walk on spheres. The aim is to compare
costs of the corresponding Monte Carlo estimates
for the Poisson equation. Generally, these costs
depend on the cost of simulation of one trajectory
and on the variance of the estimate.
It can be proved that for the Laplace equation the
limit variance of the estimation doesn't depend on
the kind of spherical processes. Thus we have very
effective estimator based on the decentred random
walk on spheres. As for the Poisson equation, it
can be shown that the variance is bounded by a
constant independent of the kind of spherical
processes (in standard form or with shifted
centres). We use simulation for a simple model
example to investigate variance behavior in more
details.
Poisson equation
Laplace operator
Monte Carlo solution
spherical process
random walk on spheres
rate of convergence
1-8
Regular Paper
Nina
Golyandina
Nina Golyandina
10.4230/DagSemProc.04401.7
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Numerical Approximation of Parabolic Stochastic Partial Differential Equations
The topic of the talk were the time approximation
of quasi linear stochastic partial differential
equations of parabolic type. The framework were
in the setting of stochastic evolution equations.
An error bounds for the implicit Euler scheme was
given and the stability of the scheme were considered.
Stochastic Partial Differential Equations
Stochastic evolution Equations
Numerical Approximation
implicit Euler scheme
1-2
Regular Paper
Erika
Hausenblas
Erika Hausenblas
10.4230/DagSemProc.04401.8
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On the Complexity of Computing Multi-Homogeneous BÃƒÂ©zout Numbers
We study the question how difficult it is to compute the multi-homogeneous
B\'ezout number for a polynomial system of given number $n$ of variables
and given support $A$ of monomials with non-zero coefficients.
We show that this number is NP-hard to compute. It cannot even be efficiently
approximated within an arbitrary, fixed factor unless P = NP.
This is joint work with Gregorio Malajovich.
multi-homogeneous BÃƒÂ©zout numbers
number of roots of polynomials
approximation algorithms
1-0
Regular Paper
Klaus
Meer
Klaus Meer
Gregorio
Malajovich
Gregorio Malajovich
10.4230/DagSemProc.04401.9
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On the Complexity of Parabolic Initial Value Problems with Variable Drift
We consider linear parabolic initial value
problems of second order in several dimensions.
The initial condition is supposed to be fixed
and we investigate the comutational complexity if
the coefficients of the parabolic equations
may vary in certain function spaces. Using the
parametrix method (or Neumann series), we prove
that lower bounds for the error of numerical
methods are related to lower bounds for
integration problems. On the other hand,
approximating the Neumann series with Smolyak's
method, we show that the problem is not much
harder than a certain approximation problem. For
HÃƒÂ¶lder classes on compact sets, e.g., lower and
upper bounds are close together, such that we have
an almost optimal method.
Partial differential equations
parabolic problems
Smolyak method
optimal methods
1-24
Regular Paper
Knut
Petras
Knut Petras
Klaus
Ritter
Klaus Ritter
10.4230/DagSemProc.04401.10
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Optimal algorithms for global optimization in case of unknown Lipschitz constant
We consider a family of function classes which
allow functions with several minima and which
demand only Lipschitz continuity for smoothness.
We present an algorithm almost optimal for each of
these classes.
Global optimization
Lipschitz functions
optimal rate of convergence
complexity
1-22
Regular Paper
Matthias U.
Horn
Matthias U. Horn
10.4230/DagSemProc.04401.11
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Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings
We study the optimal approximation of the solution
of an operator equation Au=f by linear mappings of
rank n and compare this with the best n-term
approximation with respect to an optimal Riesz
basis. We consider worst case errors, where f
is an element of the unit ball of a Hilbert space.
We apply our results to boundary value problems
for elliptic PDEs on an arbitrary bounded
Lipschitz domain. Here we prove that approximation
by linear mappings is as good as the best n-term
approximation with respect to an optimal Riesz
basis. Our results are concerned with
approximation, not with computation.
Our goal is to understand better the possibilities
of nonlinear approximation.
elliptic operator equation
worst case error
linear approximation method
nonlinear approximation method
best n-term approximation Bernstein widths
manifold widths
1-0
Regular Paper
Erich
Novak
Erich Novak
Stephan
Dahlke
Stephan Dahlke
Winfried
Sickel
Winfried Sickel
10.4230/DagSemProc.04401.12
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Optimal Approximation of Elliptic Problems II: Wavelet Methods
This talk is concerned with optimal approximations
of the solutions of elliptic boundary value
problems. After briefly recalling the fundamental
concepts of optimality, we shall especially
discuss best n-term approximation schemes based
on wavelets. We shall mainly be concerned with
the Poisson equation in Lipschitz domains. It
turns out that wavelet schemes are suboptimal
in general, but nevertheless they are superior to
the usual uniform approximation methods.
Moreover, for specific domains, i.e., for
polygonal domains, wavelet methods are
in fact optimal. These results are based on
regularity estimates of the exact solution
in a specific scale of Besov spaces.
Elliptic operator equations
worst case error
best n-term approximation
wavelets
Besov regularity
1-4
Regular Paper
Stephan
Dahlke
Stephan Dahlke
Erich
Novak
Erich Novak
Winfried
Sickel
Winfried Sickel
10.4230/DagSemProc.04401.13
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Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights; Worst Case Setting
We present polynomial-time algorithms for multivariate problems defined over tensor product
spaces of finite order. That is, algorithms that solve the problem with error not exceeding \epsilon
at cost bounded by C d^q \epsilon^{-1}, where
d denotes the number of variables.
Multivariate integration
multivariate approximation
complexity
polynomial-time algorithms
finite-order weights
1-0
Regular Paper
Gregorz W.
Wasilkowski
Gregorz W. Wasilkowski
Henryk
Wozniakowski
Henryk Wozniakowski
10.4230/DagSemProc.04401.14
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Quantization of self-similar Probabilities
The asymptotic behaviour of the quantization errors for self-similar probabilities is determined.
quantization
self-similar probabilities
1-0
Regular Paper
Siegfried
Graf
Siegfried Graf
Harald
Luschgy
Harald Luschgy
10.4230/DagSemProc.04401.15
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Upper Error Bounds for Approximations of Stochastic Differential Equations with Markovian Switching
We consider stochastic differential equations with
Markovian switching (SDEwMS). An SDEwMS is a
stochastic differential equation with drift and
diffusion coefficients depending not only on the
current state of the solution but also on the
current state of a right-continuous Markov chain
taking values in a finite state space.
Consequently, an SDEwMS can be viewed as the
result of a finite number of different scenarios
switching from one to the other according to the
movement of the Markov chain. The generator of the
Markov chain is given by transition probabilities
involving a parameter which controls the intensity
of switching from one state to another. We
construct numerical schemes for the approximation
of SDE'swMS and present upper error bounds for
these schemes. Our numerical schemes are based on
a time discretization with constant step-size and
on the values of a discrete Markov chain at the
discretization points. It turns out that for the
Euler scheme a similar upper bound as in the case
of stochastic ordinary differential equations can
be obtained, while for the Milstein scheme there
is a strong connection between the power of the
step-size appearing in the upper bound and the
intensity of the switching.
stochastic differential equations with Markovian switching
Markov chains
numerical methods
Euler scheme
Milstein scheme
1-2
Regular Paper
Norbert
Hofmann
Norbert Hofmann
10.4230/DagSemProc.04401.16
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