71,175 research outputs found
Derivative-free optimization methods
In many optimization problems arising from scientific, engineering and
artificial intelligence applications, objective and constraint functions are
available only as the output of a black-box or simulation oracle that does not
provide derivative information. Such settings necessitate the use of methods
for derivative-free, or zeroth-order, optimization. We provide a review and
perspectives on developments in these methods, with an emphasis on highlighting
recent developments and on unifying treatment of such problems in the
non-linear optimization and machine learning literature. We categorize methods
based on assumed properties of the black-box functions, as well as features of
the methods. We first overview the primary setting of deterministic methods
applied to unconstrained, non-convex optimization problems where the objective
function is defined by a deterministic black-box oracle. We then discuss
developments in randomized methods, methods that assume some additional
structure about the objective (including convexity, separability and general
non-smooth compositions), methods for problems where the output of the
black-box oracle is stochastic, and methods for handling different types of
constraints
On the optimal control of some nonsmooth distributed parameter systems arising in mechanics
Variational inequalities are an important mathematical tool for modelling
free boundary problems that arise in different application areas. Due to the
intricate nonsmooth structure of the resulting models, their analysis and
optimization is a difficult task that has drawn the attention of researchers
for several decades. In this paper we focus on a class of variational
inequalities, called of the second kind, with a twofold purpose. First, we aim
at giving a glance at some of the most prominent applications of these types of
variational inequalities in mechanics, and the related analytical and numerical
difficulties. Second, we consider optimal control problems constrained by these
variational inequalities and provide a thorough discussion on the existence of
Lagrange multipliers and the different types of optimality systems that can be
derived for the characterization of local minima. The article ends with a
discussion of the main challenges and future perspectives of this important
problem class
Exact determination of the volume of an inclusion in a body having constant shear modulus
We derive an exact formula for the volume fraction of an inclusion in a body
when the inclusion and the body are linearly elastic materials with the same
shear modulus. Our formula depends on an appropriate measurement of the
displacement and traction around the boundary of the body. In particular, the
boundary conditions around the boundary of the body must be such that they
mimic the body being placed in an infinite medium with an appropriate
displacement applied at infinity
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
A Multiscale Method for Model Order Reduction in PDE Parameter Estimation
Estimating parameters of Partial Differential Equations (PDEs) is of interest
in a number of applications such as geophysical and medical imaging. Parameter
estimation is commonly phrased as a PDE-constrained optimization problem that
can be solved iteratively using gradient-based optimization. A computational
bottleneck in such approaches is that the underlying PDEs needs to be solved
numerous times before the model is reconstructed with sufficient accuracy. One
way to reduce this computational burden is by using Model Order Reduction (MOR)
techniques such as the Multiscale Finite Volume Method (MSFV).
In this paper, we apply MSFV for solving high-dimensional parameter
estimation problems. Given a finite volume discretization of the PDE on a fine
mesh, the MSFV method reduces the problem size by computing a
parameter-dependent projection onto a nested coarse mesh. A novelty in our work
is the integration of MSFV into a PDE-constrained optimization framework, which
updates the reduced space in each iteration. We also present a computationally
tractable way of differentiating the MOR solution that acknowledges the change
of basis. As we demonstrate in our numerical experiments, our method leads to
computational savings particularly for large-scale parameter estimation
problems and can benefit from parallelization.Comment: 22 pages, 4 figures, 3 table
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
An Integral Equation Method for the Cahn-Hilliard Equation in the Wetting Problem
We present an integral equation approach to solving the Cahn-Hilliard
equation equipped with boundary conditions that model solid surfaces with
prescribed Young's angles. The discretization of the system in time using
convex splitting leads to a modified biharmonic equation at each time step. To
solve it, we split the solution into a volume potential computed with free
space kernels, plus the solution to a second kind integral equation (SKIE). The
volume potential is evaluated with the help of a box-based volume-FMM method.
For non-box domains, source density is extended by solving a biharmonic
Dirichlet problem. The near-singular boundary integrals are computed using
quadrature by expansion (QBX) with FMM acceleration. Our method has linear
complexity in the number of surface/volume degrees of freedom and can achieve
high order convergence with adaptive refinement to manage error from function
extension
Identification of a chemotactic sensitivity in a coupled system
Chemotaxis is the process by which cells behave in a way that follows the
chemical gradient. Applications to bacteria growth, tissue inflammation, and
vascular tumors provide a focus on optimization strategies. Experiments can
characterize the form of possible chemotactic sensitivities. This paper
addresses the recovery of the chemotactic sensitivity from these experiments
while allowing for nonlinear dependence of the parameter on the state
variables. The existence of solutions to the forward problem is analyzed. The
identification of a chemotactic parameter is determined by inverse problem
techniques. Tikhonov regularization is investigated and appropriate convergence
results are obtained. Numerical results of concentration dependent chemotactic
terms are explored
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)
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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