80 research outputs found
Smoothed analysis of the simplex method
In this chapter, we give a technical overview of smoothed analyses of the shadow vertex simplex method for linear programming (LP). We first review the properties of the shadow vertex simplex method and its associated geometry. We begin the smoothed analysis discussion with an analysis of the successive shortest path algorithm for the minimum-cost maximum-flow problem under objective perturbations, a classical instantiation of the shadow vertex simplex method. Then we move to general linear programming and give an analysis of a shadow vertex based algorithm for linear programming under Gaussian constraint perturbations
Geometric aspects of linear programming : shadow paths, central paths, and a cutting plane method
Most everyday algorithms are well-understood; predictions made theoretically
about them closely match what we observe in practice. This is not the case for
all algorithms, and some algorithms are still poorly understood on a theoretical level.
This is the case for many algorithms used for solving optimization problems from operations reserach.
Solving such optimization problems is essential in many industries and is done every day.
One important example of such optimization problems are Linear Programming problems.
There are a couple of different algorithms that are popular in practice,
among which is one which has been in use for almost 80 years.
Nonetheless, our theoretical understanding of these algorithms is limited.
This thesis makes progress towards a better understanding of these key algorithms
for lineair programming, among which are the simplex method, interior point methods,
and cutting plane methods
Three real-space discretization techniques in electronic structure calculations
A characteristic feature of the state-of-the-art of real-space methods in
electronic structure calculations is the diversity of the techniques used in
the discretization of the relevant partial differential equations. In this
context, the main approaches include finite-difference methods, various types
of finite-elements and wavelets. This paper reports on the results of several
code development projects that approach problems related to the electronic
structure using these three different discretization methods. We review the
ideas behind these methods, give examples of their applications, and discuss
their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status
solidi (b) - basic solid state physics" devoted to the CECAM workshop "State
of the art developments and perspectives of real-space electronic structure
techniques in condensed matter and molecular physics". v2: Minor stylistic
and typographical changes, partly inspired by referee comment
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
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