Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

Upcoming fifth-generation (5G) networks need to support novel ultrareliable and low-latency (URLLC) traffic that utilizes short packets. This requires a paradigm shift as traditional communication systems are designed to transmit only long data packets based on Shannon’s capacity formula, which poses a challenge for system designers. To address this challenge, this article relies on an unmanned aerial vehicle (UAV) and a reconfigurable intelligent surface (RIS) to deliver short URLLC instruction packets between ground Internet-of-Things (IoT) devices. In this context, we perform passive beamforming of RIS antenna elements as well as nonlinear and nonconvex optimization to minimize the total decoding error rate and find the UAV’s optimal position and blocklength. In this article, a novel, polytope-based method from the class of direct search methods (DSMs) named Nelder–Mead simplex (NMS) is used to solve the optimization problem based on its computational efficiency; in terms of lesser number of required iterations to evaluate objective function. The proposed approach yields better convergence performance than the traditional gradient-descent optimization algorithm and a lower computation time and equivalent performance for the blocklength variable as the exhaustive search. Moreover, the proposed approach allows ultrahigh reliability, which can be attained by increasing the number of antenna elements in RIS as well as increasing the allocated blocklengths. Simulations demonstrate the RIS’s performance gain and conclusively show that the UAV’s position is crucial for achieving ultrahigh reliability in short packet transmission.

URLLC Facilitated by Mobile UAV Relay and RIS: A Joint Design of Passive Beamforming, Blocklength, and UAV Positioning

The existence of the curse of dimensionality is well known, and its general effects are well acknowledged. However, and perhaps due to this colloquial understanding, specific measurements on the curse of dimensionality and its effects are not as extensive. In continuous domains, the volume of the search space grows exponentially with dimensionality. Conversely, the number of function evaluations budgeted to explore this search space usually grows only linearly. The divergence of these growth rates has important effects on the parameters used in particle swarm optimization and differential evolution as dimensionality increases. New experiments focus on the effects of population size and key changes to the search characteristics of these popular metaheuristics when population size is less than the dimensionality of the search space. Results show how design guidelines developed for low-dimensional implementations can become unsuitable for high-dimensional search spaces.

Measuring the curse of dimensionality and its effects on particle swarm optimization and differential evolution

We estimate the momentum diffusion coefficient of a heavy quark within a pure SU(3) plasma at a temperature of about 1.5Tc. Large-scale Monte Carlo simulations on a series of lattices extending up to 1923×48 permit us to carry out a continuum extrapolation of the so-called color-electric imaginary-time correlator. The extrapolated correlator is analyzed with the help of theoretically motivated models for the corresponding spectral function. Evidence for a nonzero transport coefficient is found and, incorporating systematic uncertainties reflecting model assumptions, we obtain κ=(1.8–3.4)T3. This implies that the “drag coefficient,” characterizing the time scale at which heavy quarks adjust to hydrodynamic flow, is η−1D=(1.8–3.4)(Tc/T)2(M/1.5  GeV)  fm/c, where M is the heavy quark kinetic mass. The results apply to bottom and, with somewhat larger systematic uncertainties, to charm quarks.

/pdf/nonperturbative-estimate-of-the-heavy-quark-momentum-3a7vhxgoo5.pdf

Nonperturbative estimate of the heavy quark momentum diffusion coefficient

Gliomas are the most common type of tumor in the brain. Although the definite diagnosis is routinely made ex vivo by histopathologic and molecular examination, diagnostic work-up of patients with suspected glioma is mainly done using MRI. Nevertheless, l-S-methyl-11C-methionine (11C-MET) PET holds great potential in the characterization of gliomas. The aim of this study was to establish machine-learning-driven survival models for glioma built on in vivo 11C-MET PET characteristics, ex vivo characteristics, and patient characteristics. Methods: The study included 70 patients with a treatment-naive glioma that was 11C-MET-positive and had histopathology-derived ex vivo feature extraction, such as World Health Organization 2007 tumor grade, histology, and isocitrate dehydrogenase 1 R132H mutational status. The 11C-MET-positive primary tumors were delineated semiautomatically on PET images, followed by the extraction of tumor-to-background-based general and higher-order textural features by applying 5 different binning approaches. In vivo and ex vivo features, as well as patient characteristics (age, weight, height, body mass index, Karnofsky score), were merged to characterize the tumors. Machine-learning approaches were used to identify relevant in vivo, ex vivo, and patient features and their relative weights for predicting 36-mo survival. The resulting feature weights were used to establish 3 predictive models per binning configuration: one model based on a combination of in vivo, ex vivo, and clinical patient information (M36IEP); another based on in vivo and patient information only (M36IP); and a third based on in vivo information only (M36I). In addition, a binning-independent model based on ex vivo and patient information only (M36EP) was created. The established models were validated in a Monte Carlo cross-validation scheme. Results: The most prominent machine-learning-selected and -weighted features were patient-based and ex vivo-based, followed by in vivo-based. The highest areas under the curve for our models as revealed by the Monte Carlo cross-validation were 0.9 for M36IEP, 0.87 for M36EP, 0.77 for M36IP, and 0.72 for M36IConclusion: Prediction of survival in amino acid PET-positive glioma patients was highly accurate using computer-supported predictive models based on in vivo, ex vivo, and patient features.

https://jnm.snmjournals.org/content/jnumed/59/6/892.full.pdf

Glioma Survival Prediction with Combined Analysis of In Vivo 11C-MET PET Features, Ex Vivo Features, and Patient Features by Supervised Machine Learning

The Nelder–Mead or simplex search algorithm is one of the best known algorithms for unconstrained optimization of non–smooth functions. Even though the basic algorithm is quite simple, it is implemented in many different ways. Apart from some minor computational details, the main difference between various implementations lies in the selection of convergence (or termination) tests, which are used to break the iteration process. A fairly simple efficiency analysis of each iteration step reveals a potential computational bottleneck in the domain convergence test. To be efficient, such a test has to be sublinear in the number of vertices of the working simplex. We have tested some of the most common implementations of the Nelder–Mead algorithm, and none of them is efficient in this sense.

Therefore, we propose a simple and efficient domain convergence test and discuss some of its properties. This test is based on tracking the volume of the working simplex throughout the iterations. Similar termination tests can also be applied in some other simplex–based direct search methods. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Efficient Implementation of the Nelder–Mead Search Algorithm

A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm, using a massively parallel graphics processing unit (GPU), is developed. The algorithm also serves as the final stage of solving a symmetric indefinite eigenvalue problem. Numerical testing demonstrates the gains in speed and accuracy over sequential and MPI-parallelized variants of similar Jacobi-type HSVD algorithms. Finally, possibilities of hybrid CPU–GPU parallelism are discussed.

/pdf/a-gpu-based-hyperbolic-svd-algorithm-1qwj0l7kjb.pdf

A GPU-based hyperbolic SVD algorithm

A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm, using a massively parallel graphics processing unit (GPU), is developed The algorithm also serves as the final stage of solving a symmetric indefinite eigenvalue problem Numerical testing demonstrates the gains in speed and accuracy over sequential and MPI-parallelized variants of similar Jacobi-type HSVD algorithms Finally, possibilities of hybrid CPU--GPU parallelism are discussed

/pdf/a-gpu-based-hyperbolic-svd-algorithm-2ql6vxbwp8.pdf

https://web.math.pmf.unizg.hr/~hari/fbj.pdf

Full block J-Jacobi method for Hermitian matrices

New parallel Jacobi-type algorithm for the generalized singular value problem.The algorithm is 15 times faster than DGGSVD from LAPACK.It is also more accurate. The paper describes how to modify the two-sided Hari-Zimmermann algorithm for computation of the generalized eigenvalues of a matrix pair (A, B), where B is positive definite, to an implicit algorithm that computes the generalized singular values of a pair (F, G). In addition, we present blocking and parallelization techniques for speedup of the computation.For triangular matrix pairs of a moderate size, numerical tests show that the double precision sequential pointwise algorithm is several times faster than the Lapack DTGSJA algorithm, while the accuracy is slightly better, especially for small generalized singular values.Cache-aware algorithms, implemented either as the block-oriented, or as the full block algorithm, are several times faster than the pointwise algorithm. The algorithm is almost perfectly parallelizable, so parallel shared memory versions of the algorithm are perfectly scalable, and their speedup almost solely depends on the number of cores used. A hybrid shared/distributed memory algorithm is intended for huge matrices that do not fit into the shared memory.

Sanja Singer

Papers

Efficient Implementation of the Nelder–Mead Search Algorithm

A GPU-based hyperbolic SVD algorithm

A GPU-based hyperbolic SVD algorithm

Full block J-Jacobi method for Hermitian matrices

Blocking and parallelization of the Hari-Zimmermann variant of the Falk-Langemeyer algorithm for the generalized SVD