Parallel methods for absolute irreducibility testing

Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polynomial of a certain degree is absolutely irreducible, if and only if all the corresponding irreducibility forms vanish when evaluated at the coefficients of the specific polynomial. Our forms have much smaller degrees and coefficients than the forms derived originally by Emmy Noether. We can also apply our estimates to derive more effective versions of irreducibility theorems by Ostrowski and Deuring and of the Hilbert irreducibility theorem. We also give an effective estimate on the diameter of the neighborhood of an absolutely irreducible polynomial with respect to the coefficient space in which absolute irreducibility is preserved. Furthermore, we can apply the effective estimates to derive several factorization results in parallel computational complexity theory: we show how to compute arbitrary high precision approximations of the complex factors of a multivariate integral polynomial and how to count the number of absolutely irreducible factors of a multivariate polynomial with coefficients in a rational function field, both in the complexity class NC. The factorization results also extend to the case where the coefficient field is a function field.

We study the polynomial reconstruction problem for low-degree multivariate polynomials over finite field $\mathbb{F}[2]$. In this problem, we are given a set of points $\mathbf{x}\in\{0,1\}^n$ and target values $f(\mathbf{x})\in\{0,1\}$ for each of these points, with the promise that there is a polynomial over $\mathbb{F}[2]$ of degree at most d that agrees with f at $1-\varepsilon$ fraction of the points. Our goal is to find a degree d polynomial that has good agreement with f. We show that it is NP-hard to find a polynomial that agrees with f on more than $1-2^{-d}+\delta$ fraction of the points for any $\epsilon,\delta>0$. This holds even with the stronger promise that the polynomial that fits the data is in fact linear, whereas the algorithm is allowed to find a polynomial of degree d. Previously the only known hardness of approximation (or even NP-completeness) was for the case when $d =1$, which follows from a celebrated result of Håstad [J. ACM, 48 (2001), pp. 798–859]. In the setting of Computational Learning, our result shows the hardness of nonproper agnostic learning of parities, where the learner is allowed a low-degree polynomial over $\mathbb{F}[2]$ as a hypothesis. This is the first nonproper hardness result for this central problem in computational learning. Our results can be extended to multivariate polynomial reconstruction over any finite field.

Randomizing polynomials represent a function f(x) by a low-degree randomized mapping p(x, r) over a finite field F such that, for any input x, the output distribution of p(x, r) depends only on the value of f(x). We study the class of functions f which admit an efficient representation by constant-degree randomizing polynomials. It is known that this class contains NC1 as well as log-space classes contained in NC2. Whether it contains all polynomial-time computable functions is a wide open question. A positive answer would have major and unexpected consequences, including the existence of efficient constant-round multiparty protocols with unconditional security, and the equivalence of (polynomial-time) cryptography and cryptography in NC0.We obtain evidence for the limited power of randomizing polynomials by showing that a useful subclass of constant-degree randomizing polynomials cannot efficiently capture functions beyond NC. Concretely, we consider randomizing polynomials over fields F of a small characteristic in which each monomial has degree (at most) 2 in the random inputs r and constant degree in x. This subclass captures most constructions of randomizing polynomials from the literature. Our main result is that all functions f which can be efficiently represented by such randomizing polynomials over fields of a small characteristic are in non-uniform NC. (The same holds over arbitrary fields given a quadratic residuosity oracle.) This result is obtained in two steps: (1) we observe that computing f as above reduces to counting roots of degree-2 multivariate polynomials; (2) we design parallel algorithms for the latter problem. These parallel root counting algorithms may be of independent interest.On the flip side, our main result provides an avenue for obtaining new parallel algorithms via the construction of randomizing polynomials. This gives an unexpected application of cryptography to algorithm design. We provide several examples for the potential usefulness of this approach.

Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. In this dissertation thesis, we analyze the growth properties of real multivariate polynomials in terms of their so-called Newton polytopes at infinity. We show some applications of our results, relate them to the existing literature and illustrate them with several examples.

Interpolation methods on Chebyshev nodes give rather good approximations resp. error bounds. We show how certain improvements can be obtained with little effort: Main principle FFE (Few Function Evaluations). The remainder estimate based on one degree of approximation will be refined by introducing several such degrees (in combination with estimates for Chebyshev coefficients). Next we investigate the question whether the interpolation polynomial can be abridged (modified to a polynomial of lower degree). An error bound is obtained by considering the original nodes and the new coefficients. On the other hand we refine the interpolation process by introducing two auxiliary nodes. Finally, we mention modifications, linear programming methods and the multivariate case.

In polynomial interpolation, the choice of the polynomial basis and the location of the interpolation points play an important role numerically, even more so in the multivariate case. We explore the concept of spherical orthogonality for multivariate polynomials in more detail on the disk. We focus on two items: on the one hand the construction of a fully orthogonal cartesian basis for the space of multivariate polynomials starting from this sequence of spherical orthogonal polynomials, and on the other hand the connection between these orthogonal polynomials and the Lebesgue constant in multivariate polynomial interpolation on the disk. We point out the many links of the two topics under discussion with the existing literature. The new results are illustrated with an example of polynomial interpolation and approximation on the unit disk. The numerical example is also compared with the popular radial basis function interpolation.

This paper addresses scheduling problems in hybrid flow shop-like systems with a migration parallel genetic algorithm (PGA_MIG). This parallel genetic algorithm model allows genetic diversity by the application of selection and reproduction mechanisms nearer to nature. The space structure of the population is modified by dividing it into disjoined subpopulations. From time to time, individuals are exchanged between the different subpopulations (migration). Influence of parameters and dedicated strategies are studied. These parameters are the number of independent subpopulations, the interconnection topology between subpopulations, the choice/replacement strategy of the migrant individuals, and the migration frequency. A comparison between the sequential and parallel version of genetic algorithm (GA) is provided. This comparison relates to the quality of the solution and the execution time of the two versions. The efficiency of the parallel model highly depends on the parameters and especially on the migration frequency. In the same way this parallel model gives a significant improvement of computational time if it is implemented on a parallel architecture which offers an acceptable number of processors (as many processors as subpopulations).

With the increase in computational power in the last decade and the forthcoming Exascale supercomputers, a new horizon in computational modelling and simulation is envisioned in combustion science. Considering the multiscale and multiphysics characteristics of turbulent reacting flows, combustion simulations are considered as one of the most computationally demanding applications running on cutting-edge supercomputers. Exascale computing opens new frontiers for the simulation of combustion systems as more realistic conditions can be achieved with high-fidelity methods. However, an efficient use of these computing architectures requires methodologies that can exploit all levels of parallelism. The efficient utilization of the next generation of supercomputers needs to be considered from a global perspective, that is, involving physical modelling and numerical methods with methodologies based on High-Performance Computing (HPC) and hardware architectures. This review introduces recent developments in numerical methods for large-eddy simulations (LES) and direct-numerical simulations (DNS) to simulate combustion systems, with focus on the computational performance and algorithmic capabilities. Due to the broad scope, a first section is devoted to describe the fundamentals of turbulent combustion, which is followed by a general description of state-of-the-art computational strategies for solving these problems. These applications require advanced HPC approaches to exploit modern supercomputers, which is addressed in the third section. The increasing complexity of new computing architectures, with tightly coupled CPUs and GPUs, as well as high levels of parallelism, requires new parallel models and algorithms exposing the required level of concurrency. Advances in terms of dynamic load balancing, vectorization, GPU acceleration and mesh adaptation have permitted to achieve highly-efficient combustion simulations with data-driven methods in HPC environments. Therefore, dedicated sections covering the use of high-order methods for reacting flows, integration of detailed chemistry and two-phase flows are addressed. Final remarks and directions of future work are given at the end.

Parallel partitioning algorithm for numerical simulation of gas-liquid two-phase flow during the mold filling process

This paper is devoted to studying two main problems: 1) computing the apparent distance of an Abelian code and 2) giving a notion of Bose, Ray-Chaudhuri, Hocquenghem (BCH) multivariate code. To do this, we first strengthen the notion of an apparent distance by introducing the notion of a strong apparent distance; then, we present an algorithm to compute the strong apparent distance of an Abelian code, based on some manipulations of hypermatrices associated with its generating idempotent. Our method uses less computations than those given by Camion and Sabin; furthermore, in the bivariate case, the order of computation complexity is reduced from exponential to linear. Then, we use our techniques to develop a notion of a BCH code in the multivariate case, and we extend most of the classical results on cyclic BCH codes. Finally, we apply our method to the design of Abelian codes with maximum dimension with respect to a fixed apparent distance and a fixed length.

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $$K$$ -theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the $$K$$ -theoretic stable envelopes and our elliptic stable envelopes. We show that the $$K$$ -theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the $${\mathfrak {gl}}_2$$ case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the $$K$$ -theoretic stable envelopes.

Parallel programming and data-parallel algorithms have been the main techniques supporting high-performance computing for many decades. A major conceptual step was taken by L. Valiant who introduced the Bulk-Synchronous Parallel (BSP) model. Parallel algorithms on BSP can be designed and measured by taking into account not only the classical balance between time and parallel space but also communication and synchronization. Inspired by BSP, the SGL bridging model was proposed in order to improve the simplicity of parallel program development, the portability of parallel program code, and the precision of parallel algorithm performance prediction on both classical parallel machines and novel hierarchical machines. The programming model of SGL replaces the BSML (BSP-OCaml) programming primitives with scatter, gather and pardo. However, SGL does not express "horizontal" communication patterns. In this paper we introduce the GPS theorem which can be implemented later in a compiler to optimize the SGL's "horizontal" all-to-all communication. We then propose a simplified version of BSML's put based on GPS and implement Tiskin-McColl parallel sample-sort with it. The comparison of BSML's put and SGL's GPS shows that GPS has a better code readability and lower execution time.

Starting from two known continuous univariate distributions, a bivariate distribution is constructed depending on a parameter which measures the degree of stochastic dependence between the two random variables. From the foregoing construction we then pass to a multivariate-type distribution, constructed using only univariate distributions and an association matrix. Some properties of the multivariate and bivariate case are studied.

The zero set of one general multivariate exponential polynomial with interval coefficients is enclosed by unions and intersections of closed half-spaces. Tighter enclosures are derived in the bivariate case. Common zeros of polynomial systems can be located by an appropriate intersection of these enclosure sets in an appropriate space. The resulting domains are directly brought into polynomial equation solvers.

Statistical causal discovery (SCD) has the potential to advance the development and evaluation of Adverse Outcome Pathways (AOPs) by inferring causal relationships directly from data. However, ecotoxicology data often has challenges for SCD applications, such as violations of SCD algorithm assumptions and small sample sizes. As a proof-of-concept, we applied DirectLiNGAM, a direct method for estimating linear non-Gaussian acyclic model (LiNGAM), a representative SCD algorithm, to three types of ecotoxicology datasets: (1) bivariate dose-response relationships, (2) bivariate response-response relationships, and (3) a multivariate dataset with a known causal structure involving thyroid hormone disruption in zebrafish. DirectLiNGAM identified correct causal directions with high statistical reliabilities in three of four bivariate dose-response cases, even when assumptions such as linearity and non-Gaussianity were partially violated. In contrast, response-response cases did not yield a single dominant direction in two of three cases, likely due to the limited sample size. In the multivariate case, some inferred graphs resembled the expert-curated causal graph but exhibited low statistical stability. By increasing effective sample size through pooling datasets with a shared causal ordering, the inference became more stable and more closely matched the expert-curated graph. These results demonstrate the utility of SCD in identifying relevant key events and the causal relationships under realistic ecotoxicological constraints.