73,809 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
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
Suitable Spaces for Shape Optimization
The differential-geometric structure of certain shape spaces is investigated
and applied to the theory of shape optimization problems constrained by partial
differential equations and variational inequalities. Furthermore, we define a
diffeological structure on a new space of so-called -shapes. This can
be seen as a first step towards the formulation of optimization techniques on
diffeological spaces. The -shapes are a generalization of smooth
shapes and arise naturally in shape optimization problems
Numerical method for the time-fractional porous medium equation
This papers deals with a construction and convergence analysis of a finite
difference scheme for solving time-fractional porous medium equation. The
governing equation exhibits both nonlocal and nonlinear behaviour making the
numerical computations challenging. Our strategy is to reduce the problem into
a single one-dimensional Volterra integral equation for the self-similar
solution and then to apply the discretization. The main difficulty arises due
to the non-Lipschitzian behaviour of the equation's nonlinearity. By the
analysis of the recurrence relation for the error we are able to prove that
there exists a family of finite difference methods that is convergent for a
large subset of the parameter space. We illustrate our results with a concrete
example of a method based on the midpoint quadrature
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 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
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
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
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 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
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