1,792 research outputs found
Minimizing Communication in Linear Algebra
In 1981 Hong and Kung proved a lower bound on the amount of communication
needed to perform dense, matrix-multiplication using the conventional
algorithm, where the input matrices were too large to fit in the small, fast
memory. In 2004 Irony, Toledo and Tiskin gave a new proof of this result and
extended it to the parallel case. In both cases the lower bound may be
expressed as (#arithmetic operations / ), where M is the size
of the fast memory (or local memory in the parallel case). Here we generalize
these results to a much wider variety of algorithms, including LU
factorization, Cholesky factorization, factorization, QR factorization,
algorithms for eigenvalues and singular values, i.e., essentially all direct
methods of linear algebra. The proof works for dense or sparse matrices, and
for sequential or parallel algorithms. In addition to lower bounds on the
amount of data moved (bandwidth) we get lower bounds on the number of messages
required to move it (latency). We illustrate how to extend our lower bound
technique to compositions of linear algebra operations (like computing powers
of a matrix), to decide whether it is enough to call a sequence of simpler
optimal algorithms (like matrix multiplication) to minimize communication, or
if we can do better. We give examples of both. We also show how to extend our
lower bounds to certain graph theoretic problems.
We point out recently designed algorithms for dense LU, Cholesky, QR,
eigenvalue and the SVD problems that attain these lower bounds; implementations
of LU and QR show large speedups over conventional linear algebra algorithms in
standard libraries like LAPACK and ScaLAPACK. Many open problems remain.Comment: 27 pages, 2 table
Existence Theory for the Radically Symmetric Contact Lens Equation
In this paper we present a variational formulation of the problem of determining the elastic stresses in a contact lens on an eye and the induced suction pressure distribution in the tear film between the eye and the lens. This complements the force-balance derivation that we used in earlier work [K. L. Maki and D. S. Ross, J. Bio. Sys., 22 (2014), pp. 235–248]. We investigate the existence of solutions of the relevant boundary value problem for the singular, second-order Euler–Lagrange equation. We prove that, for lenses of constant thickness, solutions exist. We present an example to show that in some cases in which the lens thickness increases with distance from the lens center no solution exists
Experiments reveal enrichment of 11B in granitic melt resulting from tourmaline crystallisation
Tourmaline is the most common boron-rich mineral in magmatic systems. In this study, we determined experimentally the fractionation of boron isotopes between granitic melt and tourmaline for the first time. Our crystallisation experiments were performed using a boron-rich granitic glass (B2O3 ≈ 8 wt. %) at 660−800 °C, 300 MPa, and aH2O = 1, in which tourmaline occurs as the only boron-hosting mineral. Our experimental results at four different temperatures show a small and temperature-dependent boron isotope fractionation between granitic melt and tourmaline (Δ11Bmelt–Tur = þ0.90 ± 0.05 ‰ at 660 °C and þ0.23 ± 0.12 ‰ at 800 °C), and the temperature dependence can be defined as Δ11Bmelt–Tur = 4.51 × (1000/T [K]) − 3.94 (R2 = 0.96). Using these boron isotope fractionation factors, tourmaline can serve as a tracer to quantitatively interpret boron isotopic ratios in evolved magmatic systems. Our observation that 11B is enriched in granitic melt relative to tourmaline suggests that the δ11B of late-magmatic tourmaline should be higher than tourmaline that crystallised at an early stage, if B isotope fractionation is not affected by other processes, such as fluid loss. © 2022 The Authors
Structured matrices, continued fractions, and root localization of polynomials
We give a detailed account of various connections between several classes of
objects: Hankel, Hurwitz, Toeplitz, Vandermonde and other structured matrices,
Stietjes and Jacobi-type continued fractions, Cauchy indices, moment problems,
total positivity, and root localization of univariate polynomials. Along with a
survey of many classical facts, we provide a number of new results.Comment: 79 pages; new material added to the Introductio
Local asymptotic normality for qubit states
We consider n identically prepared qubits and study the asymptotic properties
of the joint state \rho^{\otimes n}. We show that for all individual states
\rho situated in a local neighborhood of size 1/\sqrt{n} of a fixed state
\rho^0, the joint state converges to a displaced thermal equilibrium state of a
quantum harmonic oscillator. The precise meaning of the convergence is that
there exist physical transformations T_{n} (trace preserving quantum channels)
which map the qubits states asymptotically close to their corresponding
oscillator state, uniformly over all states in the local neighborhood.
A few consequences of the main result are derived. We show that the optimal
joint measurement in the Bayesian set-up is also optimal within the pointwise
approach. Moreover, this measurement converges to the heterodyne measurement
which is the optimal joint measurement of position and momentum for the quantum
oscillator. A problem of local state discrimination is solved using local
asymptotic normality.Comment: 16 pages, 3 figures, published versio
Enhancement of the Binding Energy of Charged Excitons in Disordered Quantum Wires
Negatively and positively charged excitons are identified in the
spatially-resolved photoluminescence spectra of quantum wires. We demonstrate
that charged excitons are weakly localized in disordered quantum wires. As a
consequence, the enhancement of the "binding energy" of a charged exciton is
caused, for a significant part, by the recoil energy transferred to the
remaining charged carrier during its radiative recombination. We discover that
the Coulomb correlation energy is not the sole origin of the "binding energy",
in contrast to charged excitons confined in quantum dots.Comment: 4 Fig
Coupled Numerical Analysis of Variations in the Capacity of Driven Energy Piles in Clay
Energy piles are an emerging alternative for the reduction of energy consumption to heat and cool buildings. Most of the research to date has focused on thermodynamic properties or axial and radial stress and strain of piles. This paper focuses on the effects of temperature fluctuation on the capacity of driven energy piles in clayey soils. Consolidation of clay surrounding driven piles affects the pile capacity (i.e., set up in clay). The heating and cooling periods of energy piles can create the excess pore-water pressure (EPWP, ue) or relax the existing one (e.g., due to pile driving or previous thermal loads) in clayey soils (due to the contraction and expansion of water) affecting the pile capacity. In the meantime, the thermal expansion and contraction of the pile also generate or relax the EPWP in the soil, which can be computed using the cavity-expansion theory. This paper studies the resulting changes in the pile capacity due to the daily and seasonal thermal cycles. The results show that thermal cycles in an energy pile can cause a decrease in the pile capacity leading to a delay in reaching the capacity after a complete clay set up
Tensor completion in hierarchical tensor representations
Compressed sensing extends from the recovery of sparse vectors from
undersampled measurements via efficient algorithms to the recovery of matrices
of low rank from incomplete information. Here we consider a further extension
to the reconstruction of tensors of low multi-linear rank in recently
introduced hierarchical tensor formats from a small number of measurements.
Hierarchical tensors are a flexible generalization of the well-known Tucker
representation, which have the advantage that the number of degrees of freedom
of a low rank tensor does not scale exponentially with the order of the tensor.
While corresponding tensor decompositions can be computed efficiently via
successive applications of (matrix) singular value decompositions, some
important properties of the singular value decomposition do not extend from the
matrix to the tensor case. This results in major computational and theoretical
difficulties in designing and analyzing algorithms for low rank tensor
recovery. For instance, a canonical analogue of the tensor nuclear norm is
NP-hard to compute in general, which is in stark contrast to the matrix case.
In this book chapter we consider versions of iterative hard thresholding
schemes adapted to hierarchical tensor formats. A variant builds on methods
from Riemannian optimization and uses a retraction mapping from the tangent
space of the manifold of low rank tensors back to this manifold. We provide
first partial convergence results based on a tensor version of the restricted
isometry property (TRIP) of the measurement map. Moreover, an estimate of the
number of measurements is provided that ensures the TRIP of a given tensor rank
with high probability for Gaussian measurement maps.Comment: revised version, to be published in Compressed Sensing and Its
Applications (edited by H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral
Electron Accumulation and Emergent Magnetism in LaMnO3/SrTiO3 Heterostructures
Emergent phenomena at polar-nonpolar oxide interfaces have been studied
intensely in pursuit of next-generation oxide electronics and spintronics. Here
we report the disentanglement of critical thicknesses for electron
reconstruction and the emergence of ferromagnetism in polar-mismatched
LaMnO3/SrTiO3 (001) heterostructures. Using a combination of element-specific
X-ray absorption spectroscopy and dichroism, and first-principles calculations,
interfacial electron accumulation and ferromagnetism have been observed within
the polar, antiferromagnetic insulator LaMnO3. Our results show that the
critical thickness for the onset of electron accumulation is as thin as 2 unit
cells (UC), significantly thinner than the observed critical thickness for
ferromagnetism of 5 UC. The absence of ferromagnetism below 5 UC is likely
induced by electron over-accumulation. In turn, by controlling the doping of
the LaMnO3, we are able to neutralize the excessive electrons from the polar
mismatch in ultrathin LaMnO3 films and thus enable ferromagnetism in films as
thin as 3 UC, extending the limits of our ability to synthesize and tailor
emergent phenomena at interfaces and demonstrating manipulation of the
electronic and magnetic structures of materials at the shortest length scales.Comment: Accepted by Phys. Rev. Let
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