61,970 research outputs found
Computing the demagnetizing tensor for finite difference micromagnetic simulations via numerical integration
In the finite difference method which is commonly used in computational
micromagnetics, the demagnetizing field is usually computed as a convolution of
the magnetization vector field with the demagnetizing tensor that describes the
magnetostatic field of a cuboidal cell with constant magnetization. An
analytical expression for the demagnetizing tensor is available, however at
distances far from the cuboidal cell, the numerical evaluation of the
analytical expression can be very inaccurate.
Due to this large-distance inaccuracy numerical packages such as OOMMF
compute the demagnetizing tensor using the explicit formula at distances close
to the originating cell, but at distances far from the originating cell a
formula based on an asymptotic expansion has to be used. In this work, we
describe a method to calculate the demagnetizing field by numerical evaluation
of the multidimensional integral in the demagnetization tensor terms using a
sparse grid integration scheme. This method improves the accuracy of
computation at intermediate distances from the origin.
We compute and report the accuracy of (i) the numerical evaluation of the
exact tensor expression which is best for short distances, (ii) the asymptotic
expansion best suited for large distances, and (iii) the new method based on
numerical integration, which is superior to methods (i) and (ii) for
intermediate distances. For all three methods, we show the measurements of
accuracy and execution time as a function of distance, for calculations using
single precision (4-byte) and double precision (8-byte) floating point
arithmetic. We make recommendations for the choice of scheme order and
integrating coefficients for the numerical integration method (iii)
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Predicativity and parametric polymorphism of Brouwerian implication
A common objection to the definition of intuitionistic implication in the
Proof Interpretation is that it is impredicative. I discuss the history of that
objection, argue that in Brouwer's writings predicativity of implication is
ensured through parametric polymorphism of functions on species, and compare
this construal with the alternative approaches to predicative implication of
Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten
Adaptive control in rollforward recovery for extreme scale multigrid
With the increasing number of compute components, failures in future
exa-scale computer systems are expected to become more frequent. This motivates
the study of novel resilience techniques. Here, we extend a recently proposed
algorithm-based recovery method for multigrid iterations by introducing an
adaptive control. After a fault, the healthy part of the system continues the
iterative solution process, while the solution in the faulty domain is
re-constructed by an asynchronous on-line recovery. The computations in both
the faulty and healthy subdomains must be coordinated in a sensitive way, in
particular, both under and over-solving must be avoided. Both of these waste
computational resources and will therefore increase the overall
time-to-solution. To control the local recovery and guarantee an optimal
re-coupling, we introduce a stopping criterion based on a mathematical error
estimator. It involves hierarchical weighted sums of residuals within the
context of uniformly refined meshes and is well-suited in the context of
parallel high-performance computing. The re-coupling process is steered by
local contributions of the error estimator. We propose and compare two criteria
which differ in their weights. Failure scenarios when solving up to
unknowns on more than 245\,766 parallel processes will be
reported on a state-of-the-art peta-scale supercomputer demonstrating the
robustness of the method
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