151,292 research outputs found
Convergence analysis of Riemannian Gauss-Newton methods and its connection with the geometric condition number
We obtain estimates of the multiplicative constants appearing in local
convergence results of the Riemannian Gauss-Newton method for least squares
problems on manifolds and relate them to the geometric condition number of [P.
B\"urgisser and F. Cucker, Condition: The Geometry of Numerical Algorithms,
2013]
A comparative study of simulated and experimental results for an extruding elastomeric component
With ever advancing simulation techniques and algorithms being introduced to commercial software, the importance of validation remains a priority. An experimental rig was designed to study the effects of rubber extrusion consisting of a compression testing system and a transparent extrusion barrel, of similar geometry to that used in a forming process. Through visual and numerical comparison, the experimental results would be compared to those obtained through Finite Element Analysis (FEA). To remedy the convergence difficulties of the complexity of the simulation, due to large deformations, a recent Nonlinear Adaptive Remeshing boundary condition was applied to the model
Sampling algebraic sets in local intrinsic coordinates
Numerical data structures for positive dimensional solution sets of
polynomial systems are sets of generic points cut out by random planes of
complimentary dimension. We may represent the linear spaces defined by those
planes either by explicit linear equations or in parametric form. These
descriptions are respectively called extrinsic and intrinsic representations.
While intrinsic representations lower the cost of the linear algebra
operations, we observe worse condition numbers. In this paper we describe the
local adaptation of intrinsic coordinates to improve the numerical conditioning
of sampling algebraic sets. Local intrinsic coordinates also lead to a better
stepsize control. We illustrate our results with Maple experiments and
computations with PHCpack on some benchmark polynomial systems.Comment: 13 pages, 2 figures, 2 algorithms, 2 table
Galois groups of Schubert problems via homotopy computation
Numerical homotopy continuation of solutions to polynomial equations is the
foundation for numerical algebraic geometry, whose development has been driven
by applications of mathematics. We use numerical homotopy continuation to
investigate the problem in pure mathematics of determining Galois groups in the
Schubert calculus. For example, we show by direct computation that the Galois
group of the Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes
non-trivially is the full symmetric group S_6006.Comment: 17 pages, 4 figures. 3 references adde
Computing Dynamic Output Feedback Laws
The pole placement problem asks to find laws to feed the output of a plant
governed by a linear system of differential equations back to the input of the
plant so that the resulting closed-loop system has a desired set of
eigenvalues. Converting this problem into a question of enumerative geometry,
efficient numerical homotopy algorithms to solve this problem for general
Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While
dynamic feedback laws offer a wider range of use, the realization of the output
of the numerical homotopies as a machine to control the plant in the time
domain has not been addressed before. In this paper we present symbolic-numeric
algorithms to turn the solution to the question of enumerative geometry into a
useful control feedback machine. We report on numerical experiments with our
publicly available software and illustrate its application on various control
problems from the literature.Comment: 20 pages, 3 figures; the software described in this paper is publicly
available via http://www.math.uic.edu/~jan/download.htm
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