12,882 research outputs found
Computation of sum of squares polynomials from data points
We propose an iterative algorithm for the numerical computation of sums of
squares of polynomials approximating given data at prescribed interpolation
points. The method is based on the definition of a convex functional
arising from the dualization of a quadratic regression over the Cholesky
factors of the sum of squares decomposition. In order to justify the
construction, the domain of , the boundary of the domain and the behavior at
infinity are analyzed in details. When the data interpolate a positive
univariate polynomial, we show that in the context of the Lukacs sum of squares
representation, is coercive and strictly convex which yields a unique
critical point and a corresponding decomposition in sum of squares. For
multivariate polynomials which admit a decomposition in sum of squares and up
to a small perturbation of size , is always
coercive and so it minimum yields an approximate decomposition in sum of
squares. Various unconstrained descent algorithms are proposed to minimize .
Numerical examples are provided, for univariate and bivariate polynomials
A variational approach to Ising spin glasses in finite dimensions
We introduce a hierarchical class of approximations of the random Ising spin
glass in dimensions. The attention is focused on finite clusters of spins
where the action of the rest of the system is properly taken into account. At
the lower level (cluster of a single spin) our approximation coincides with the
SK model while at the highest level it coincides with the true -dimensional
system. The method is variational and it uses the replica approach to spin
glasses and the Parisi ansatz for the order parameter. As a result we have
rigorous bounds for the quenched free energy which become more and more precise
when larger and larger clusters are considered.Comment: 16 pages, Plain TeX, uses Harvmac.tex, 4 ps figures, submitted to J.
Phys. A: Math. Ge
An efficient surrogate model for emulation and physics extraction of large eddy simulations
In the quest for advanced propulsion and power-generation systems,
high-fidelity simulations are too computationally expensive to survey the
desired design space, and a new design methodology is needed that combines
engineering physics, computer simulations and statistical modeling. In this
paper, we propose a new surrogate model that provides efficient prediction and
uncertainty quantification of turbulent flows in swirl injectors with varying
geometries, devices commonly used in many engineering applications. The novelty
of the proposed method lies in the incorporation of known physical properties
of the fluid flow as {simplifying assumptions} for the statistical model. In
view of the massive simulation data at hand, which is on the order of hundreds
of gigabytes, these assumptions allow for accurate flow predictions in around
an hour of computation time. To contrast, existing flow emulators which forgo
such simplications may require more computation time for training and
prediction than is needed for conducting the simulation itself. Moreover, by
accounting for coupling mechanisms between flow variables, the proposed model
can jointly reduce prediction uncertainty and extract useful flow physics,
which can then be used to guide further investigations.Comment: Submitted to JASA A&C
Spin-stiffness and topological defects in two-dimensional frustrated spin systems
Using a {\it collective} Monte Carlo algorithm we study the low-temperature
and long-distance properties of two systems of two-dimensional classical tops.
Both systems have the same spin-wave dynamics (low-temperature behavior) as a
large class of Heisenberg frustrated spin systems. They are constructed so that
to differ only by their topological properties. The spin-stiffnesses for the
two systems of tops are calculated for different temperatures and different
sizes of the sample. This allows to investigate the role of topological defects
in frustrated spin systems. Comparisons with Renormalization Group results
based on a Non Linear Sigma model approach and with the predictions of some
simple phenomenological model taking into account the topological excitations
are done.Comment: RevTex, 25 pages, 14 figures, Minor changes, final version. To appear
in Phys.Rev.
Stable, Robust and Super Fast Reconstruction of Tensors Using Multi-Way Projections
In the framework of multidimensional Compressed Sensing (CS), we introduce an
analytical reconstruction formula that allows one to recover an th-order
data tensor
from a reduced set of multi-way compressive measurements by exploiting its low
multilinear-rank structure. Moreover, we show that, an interesting property of
multi-way measurements allows us to build the reconstruction based on
compressive linear measurements taken only in two selected modes, independently
of the tensor order . In addition, it is proved that, in the matrix case and
in a particular case with rd-order tensors where the same 2D sensor operator
is applied to all mode-3 slices, the proposed reconstruction
is stable in the sense that the approximation
error is comparable to the one provided by the best low-multilinear-rank
approximation, where is a threshold parameter that controls the
approximation error. Through the analysis of the upper bound of the
approximation error we show that, in the 2D case, an optimal value for the
threshold parameter exists, which is confirmed by our
simulation results. On the other hand, our experiments on 3D datasets show that
very good reconstructions are obtained using , which means that this
parameter does not need to be tuned. Our extensive simulation results
demonstrate the stability and robustness of the method when it is applied to
real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity
based CS methods specialized for multidimensional signals is also included. A
very attractive characteristic of the proposed method is that it provides a
direct computation, i.e. it is non-iterative in contrast to all existing
sparsity based CS algorithms, thus providing super fast computations, even for
large datasets.Comment: Submitted to IEEE Transactions on Signal Processin
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