41,443 research outputs found
Concave Quadratic Cuts for Mixed-Integer Quadratic Problems
The technique of semidefinite programming (SDP) relaxation can be used to
obtain a nontrivial bound on the optimal value of a nonconvex quadratically
constrained quadratic program (QCQP). We explore concave quadratic inequalities
that hold for any vector in the integer lattice , and show that
adding these inequalities to a mixed-integer nonconvex QCQP can improve the
SDP-based bound on the optimal value. This scheme is tested using several
numerical problem instances of the max-cut problem and the integer least
squares problem.Comment: 24 pages, 1 figur
A shape optimization algorithm for cellular composites
We propose and investigate a mesh deformation technique for PDE constrained
shape optimization. Introducing a gradient penalization to the inner product
for linearized shape spaces, mesh degeneration can be prevented within the
optimization iteration allowing for the scalability of employed solvers. We
illustrate the approach by a shape optimization for cellular composites with
respect to linear elastic energy under tension. The influence of the gradient
penalization is evaluated and the parallel scalability of the approach
demonstrated employing a geometric multigrid solver on hierarchically
distributed meshes
Recent Advances in Denoising of Manifold-Valued Images
Modern signal and image acquisition systems are able to capture data that is
no longer real-valued, but may take values on a manifold. However, whenever
measurements are taken, no matter whether manifold-valued or not, there occur
tiny inaccuracies, which result in noisy data. In this chapter, we review
recent advances in denoising of manifold-valued signals and images, where we
restrict our attention to variational models and appropriate minimization
algorithms. The algorithms are either classical as the subgradient algorithm or
generalizations of the half-quadratic minimization method, the cyclic proximal
point algorithm, and the Douglas-Rachford algorithm to manifolds. An important
aspect when dealing with real-world data is the practical implementation. Here
several groups provide software and toolboxes as the Manifold Optimization
(Manopt) package and the manifold-valued image restoration toolbox (MVIRT)
The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
We list the so-called Askey-scheme of hypergeometric orthogonal polynomials.
In chapter 1 we give the definition, the orthogonality relation, the three term
recurrence relation and generating functions of all classes of orthogonal
polynomials in this scheme. In chapeter 2 we give all limit relation between
different classes of orthogonal polynomials listed in the Askey-scheme.
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme.
We give their definition, orthogonality relation, three term recurrence
relation and generating functions. In chapter 4 we give the limit relations
between those basic hypergeometric orthogonal polynomials. Finally in chapter 5
we point out how the `classical` hypergeometric orthogonal polynomials of the
Askey-scheme can be obtained from their q-analogues
Optimal control of a rate-independent evolution equation via viscous regularization
We study the optimal control of a rate-independent system that is driven by a
convex, quadratic energy. Since the associated solution mapping is non-smooth,
the analysis of such control problems is challenging. In order to derive
optimality conditions, we study the regularization of the problem via a
smoothing of the dissipation potential and via the addition of some viscosity.
The resulting regularized optimal control problem is analyzed. By driving the
regularization parameter to zero, we obtain a necessary optimality condition
for the original, non-smooth problem
A stroll in the jungle of error bounds
The aim of this paper is to give a short overview on error bounds and to
provide the first bricks of a unified theory. Inspired by the works of [8, 15,
13, 16, 10], we show indeed the centrality of the Lojasiewicz gradient
inequality. For this, we review some necessary and sufficient conditions for
global/local error bounds, both in the convex and nonconvex case. We also
recall some results on quantitative error bounds which play a major role in
convergence rate analysis and complexity theory of many optimization methods
Finite difference method for a Volterra equation with a power-type nonlinearity
In this work we prove that a family of explicit numerical finite-difference
methods is convergent when applied to a nonlinear Volterra equation with a
power-type nonlinearity. In that case the kernel is not of Lipschitz type,
therefore the classical analysis cannot be applied. We indicate several
difficulties that arise in the proofs and show how they can be remedied. The
tools that we use consist of variations on discreet Gronwall's lemmas and
comparison theorems. Additionally, we give an upper bound on the convergence
order. We conclude the paper with a construction of a convergent method and
apply it for solving some examples
A Conservative Flux Optimization Finite Element Method for Convection-Diffusion Equations
This article presents a new finite element method for convection-diffusion
equations by enhancing the continuous finite element space with a flux space
for flux approximations that preserve the important mass conservation locally
on each element. The numerical scheme is based on a constrained flux
optimization approach where the constraint was given by local mass conservation
equations and the flux error is minimized in a prescribed topology/metric. This
new scheme provides numerical approximations for both the primal and the flux
variables. It is shown that the numerical approximations for the primal and the
flux variables are convergent with optimal order in some discrete Sobolev
norms. Numerical experiments are conducted to confirm the convergence theory.
Furthermore, the new scheme was employed in the computational simulation of a
simplified two-phase flow problem in highly heterogeneous porous media. The
numerical results illustrate an excellent performance of the method in
scientific computing
A linear domain decomposition method for partially saturated flow in porous media
The Richards equation is a nonlinear parabolic equation that is commonly used
for modelling saturated/unsaturated flow in porous media. We assume that the
medium occupies a bounded Lipschitz domain partitioned into two disjoint
subdomains separated by a fixed interface . This leads to two problems
defined on the subdomains which are coupled through conditions expressing flux
and pressure continuity at . After an Euler implicit discretisation of
the resulting nonlinear subproblems a linear iterative (-type) domain
decomposition scheme is proposed. The convergence of the scheme is proved
rigorously. In the last part we present numerical results that are in line with
the theoretical finding, in particular the unconditional convergence of the
scheme. We further compare the scheme to other approaches not making use of a
domain decomposition. Namely, we compare to a Newton and a Picard scheme. We
show that the proposed scheme is more stable than the Newton scheme while
remaining comparable in computational time, even if no parallelisation is being
adopted. Finally we present a parametric study that can be used to optimize the
proposed scheme.Comment: 34 pages, 13 figures, 7 table
Projection Methods: An Annotated Bibliography of Books and Reviews
Projections onto sets are used in a wide variety of methods in optimization
theory but not every method that uses projections really belongs to the class
of projection methods as we mean it here. Here projection methods are iterative
algorithms that use projections onto sets while relying on the general
principle that when a family of (usually closed and convex) sets is present
then projections (or approximate projections) onto the given individual sets
are easier to perform than projections onto other sets (intersections, image
sets under some transformation, etc.) that are derived from the given family of
individual sets. Projection methods employ projections (or approximate
projections) onto convex sets in various ways. They may use different kinds of
projections and, sometimes, even use different projections within the same
algorithm. They serve to solve a variety of problems which are either of the
feasibility or the optimization types. They have different algorithmic
structures, of which some are particularly suitable for parallel computing, and
they demonstrate nice convergence properties and/or good initial behavior
patterns. This class of algorithms has witnessed great progress in recent years
and its member algorithms have been applied with success to many scientific,
technological, and mathematical problems. This annotated bibliography includes
books and review papers on, or related to, projection methods that we know
about, use, and like. If you know of books or review papers that should be
added to this list please contact us.Comment: Revised version. Accepted for publication in the journal
"Optimization
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