68,769 research outputs found
Robust Dropping Criteria for F-norm Minimization Based Sparse Approximate Inverse Preconditioning
Dropping tolerance criteria play a central role in Sparse Approximate Inverse
preconditioning. Such criteria have received, however, little attention and
have been treated heuristically in the following manner: If the size of an
entry is below some empirically small positive quantity, then it is set to
zero. The meaning of "small" is vague and has not been considered rigorously.
It has not been clear how dropping tolerances affect the quality and
effectiveness of a preconditioner . In this paper, we focus on the adaptive
Power Sparse Approximate Inverse algorithm and establish a mathematical theory
on robust selection criteria for dropping tolerances. Using the theory, we
derive an adaptive dropping criterion that is used to drop entries of small
magnitude dynamically during the setup process of . The proposed criterion
enables us to make both as sparse as possible as well as to be of
comparable quality to the potentially denser matrix which is obtained without
dropping. As a byproduct, the theory applies to static F-norm minimization
based preconditioning procedures, and a similar dropping criterion is given
that can be used to sparsify a matrix after it has been computed by a static
sparse approximate inverse procedure. In contrast to the adaptive procedure,
dropping in the static procedure does not reduce the setup time of the matrix
but makes the application of the sparser for Krylov iterations cheaper.
Numerical experiments reported confirm the theory and illustrate the robustness
and effectiveness of the dropping criteria.Comment: 27 pages, 2 figure
Tolerancing and Sheet Bending in Small Batch Part Manufacturing
Tolerances indicate geometrical limits between which a component is expected to perform its function adequately. They are used for instance for set-up selection in process planning and for inspection. Tolerances must be accounted for in sequencing and positioning procedures for bending of sheet metal parts. In bending, the shape of a part changes not only locally, but globally as well. Therefore, sheet metal part manufacturing presents some specific problems as regards reasoning about tolerances. The paper focuses on the interpretation and conversion of tolerances as part of a sequencing procedure for bending to be used in an integrated CAPP system
Formulation, existence, and computation of boundedly rational dynamic user equilibrium with fixed or endogenous user tolerance
This paper analyzes dynamic user equilibrium (DUE) that incorporates the notion of boundedly rational (BR) user behavior in the selection of departure times and routes. Intrinsically, the boundedly rational dynamic user equilibrium (BR-DUE) model we present assumes that travelers do not always seek the least costly route-and-departure-time choice. Rather, their perception of travel cost is affected by an indifference band describing travelers’ tolerance of the difference between their experienced travel costs and the minimum travel cost. An extension of the BR-DUE problem is the so-called variable tolerance dynamic user equilibrium (VT-BR-DUE) wherein endogenously determined tolerances may depend not only on paths, but also on the established path departure rates. This paper presents a unified approach for modeling both BR-DUE and VT-BR-DUE, which makes significant contributions to the model formulation, analysis of existence, solution characterization, and numerical computation of such problems. The VT-BR-DUE problem, together with the BR-DUE problem as a special case, is formulated as a variational inequality. We provide a very general existence result for VT-BR-DUE and BR-DUE that relies on assumptions weaker than those required for normal DUE models. Moreover, a characterization of the solution set is provided based on rigorous topological analysis. Finally, three computational algorithms with convergence results are proposed based on the VI and DVI formulations. Numerical studies are conducted to assess the proposed algorithms in terms of solution quality, convergence, and computational efficiency
Varieties whose tolerances are homomorphic images of their congruences
The homomorphic image of a congruence is always a tolerance (relation) but,
within a given variety, a tolerance is not necessarily obtained this way. By a
Maltsev-like condition, we characterize varieties whose tolerances are
homomorphic images of their congruences (TImC). As corollaries, we prove that
the variety of semilattices, all varieties of lattices, and all varieties of
unary algebras have TImC. We show that a congruence n-permutable variety has
TImC if and only if it is congruence permutable, and construct an idempotent
variety with a majority term that fails TImC
Semiclassical States in Quantum Cosmology: Bianchi I Coherent States
We study coherent states for Bianchi type I cosmological models, as examples
of semiclassical states for time-reparametrization invariant systems. This
simple model allows us to study explicitly the relationship between exact
semiclassical states in the kinematical Hilbert space and corresponding ones in
the physical Hilbert space, which we construct here using the group averaging
technique. We find that it is possible to construct good semiclassical physical
states by such a procedure in this model; we also discuss the sense in which
the original kinematical states may be a good approximation to the physical
ones, and the situations in which this is the case. In addition, these models
can be deparametrized in a natural way, and we study the effect of time
evolution on an "intrinsic" coherent state in the reduced phase space, in order
to estimate the time for this state to spread significantly.Comment: 21 pages, 1 figure; Version to be published in CQG; The discussion
has been slightly reorganized, two references added, and some typos correcte
The devil is in the detail: hints for practical optimisation
Finding the minimum of an objective function, such as a least squares or negative log-likelihood function, with respect to the unknown model parameters is a problem often encountered in econometrics. Consequently, students of econometrics and applied econometricians are usually well-grounded in the broad differences between the numerical procedures employed to solve these problems. Often, however, relatively little time is given to understanding the practical subtleties of implementing these schemes when faced with illbehaved problems. This paper addresses some of the details involved in practical optimisation, such as dealing with constraints on the parameters, specifying starting values, termination criteria and analytical gradients, and illustrates some of the general ideas with several instructive examples
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