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A comparison of general-purpose optimization algorithms forfinding optimal approximate experimental designs
Several common general purpose optimization algorithms are compared for findingA- and D-optimal designs for different types of statistical models of varying complexity,including high dimensional models with five and more factors. The algorithms of interestinclude exact methods, such as the interior point method, the Nelder–Mead method, theactive set method, the sequential quadratic programming, and metaheuristic algorithms,such as particle swarm optimization, simulated annealing and genetic algorithms.Several simulations are performed, which provide general recommendations on theutility and performance of each method, including hybridized versions of metaheuristicalgorithms for finding optimal experimental designs. A key result is that general-purposeoptimization algorithms, both exact methods and metaheuristic algorithms, perform wellfor finding optimal approximate experimental designs
Numerically optimized Markovian coupling and mixing in one-dimensional maps
Algorithms are introduced that produce optimal Markovian couplings for large finite-state-space discrete-time Markov chains with sparse transition matrices; these algorithms are applied to some toy models motivated by fluid-dynamical mixing problems at high Peclét number. An alternative definition of the time-scale of a mixing process is suggested. Finally, these algorithms are applied to the problem of coupling diffusion processes in an acute-angled triangle, and some of the simplifications that occur in continuum coupling problems are discussed
Optimal stability polynomials for numerical integration of initial value problems
We consider the problem of finding optimally stable polynomial approximations
to the exponential for application to one-step integration of initial value
ordinary and partial differential equations. The objective is to find the
largest stable step size and corresponding method for a given problem when the
spectrum of the initial value problem is known. The problem is expressed in
terms of a general least deviation feasibility problem. Its solution is
obtained by a new fast, accurate, and robust algorithm based on convex
optimization techniques. Global convergence of the algorithm is proven in the
case that the order of approximation is one and in the case that the spectrum
encloses a starlike region. Examples demonstrate the effectiveness of the
proposed algorithm even when these conditions are not satisfied
Solving Large-Scale Optimization Problems Related to Bell's Theorem
Impossibility of finding local realistic models for quantum correlations due
to entanglement is an important fact in foundations of quantum physics, gaining
now new applications in quantum information theory. We present an in-depth
description of a method of testing the existence of such models, which involves
two levels of optimization: a higher-level non-linear task and a lower-level
linear programming (LP) task. The article compares the performances of the
existing implementation of the method, where the LPs are solved with the
simplex method, and our new implementation, where the LPs are solved with a
matrix-free interior point method. We describe in detail how the latter can be
applied to our problem, discuss the basic scenario and possible improvements
and how they impact on overall performance. Significant performance advantage
of the matrix-free interior point method over the simplex method is confirmed
by extensive computational results. The new method is able to solve problems
which are orders of magnitude larger. Consequently, the noise resistance of the
non-classicality of correlations of several types of quantum states, which has
never been computed before, can now be efficiently determined. An extensive set
of data in the form of tables and graphics is presented and discussed. The
article is intended for all audiences, no quantum-mechanical background is
necessary.Comment: 19 pages, 7 tables, 1 figur
Introducing the sequential linear programming level-set method for topology optimization
The authors would like to thank Numerical Analysis Group at the Rutherford Appleton Laboratory for their FORTRAN HSL packages (HSL, a collection of Fortran codes for large-scale scientific computation. See http://www.hsl.rl.ac.uk/). Dr H Alicia Kim acknowledges the support from Engineering and Physical Sciences Research Council, grant number EP/M002322/1Peer reviewedPublisher PD
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