Maximum likelihood estimation is an important statistical technique for estimating missing data, for example in climate and environmental applications, which are usually large and feature data points that are irregula...
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Maximum likelihood estimation is an important statistical technique for estimating missing data, for example in climate and environmental applications, which are usually large and feature data points that are irregularly spaced. In particular, the Gaussian log-likelihood function is the de facto model, which operates on the resulting sizable dense covariance matrix. The advent of high performance systems with advanced computing power and memory capacity have enabled full simulations only for rather small dimensional climate problems, solved at the machine precision accuracy. The challenge for high dimensional problems lies in the computation requirements of the log-likelihood function, which necessitates O(n 2 ) storage and O(n 3 ) operations, where n represents the number of given spatial locations. This prohibitive computational cost may be reduced by using approximation techniques that not only enable large-scale simulations otherwise intractable, but also maintain the accuracy and the fidelity of the spatial statistics model. In this paper, we extend the Exascale GeoStatistics software framework (i.e., ExaGeoStat 1 ) to support the Tile Low-Rank (TLR) approximation technique, which exploits the data sparsity of the dense covariance matrix by compressing the off-diagonal tiles up to a user-defined accuracy threshold. The underlying linear algebra operations may then be carried out on this data compression format, which may ultimately reduce the arithmetic complexity of the maximum likelihood estimation and the corresponding memory footprint. Performance results of TLR-based computations on shared and distributed-memory systems attain up to 13X and 5X speedups, respectively, compared to full accuracy simulations using synthetic and real datasets (up to 2M), while ensuring adequate prediction accuracy.
The article introduces a new class of polynomials that first appeared in the probability distribution density function of the hyperbolic cosine type. With an integer change in one of the parameters of this distributio...
The article introduces a new class of polynomials that first appeared in the probability distribution density function of the hyperbolic cosine type. With an integer change in one of the parameters of this distribution, polynomials in the form of a product of positive factors are written out with an increasing degree. Earlier, the author found a connection between the distribution of the hyperbolic cosine type and numerical sets, in particular, in the simplest cases with the triangle of coefficients of Bessel polynomials, the triangle of Stirling numbers, sequences of coefficients in the expansion of various functions, etc. Also from the distribution formed numerous numerical sequences, both new and widely known. Consideration of polynomials separately from the density function made it possible to reconstruct numerical sets of coefficients, ordered in the form of numerical triangles and numerical sequences. The connections between the elements of the sets are established. Among the sequences obtained, in the simplest cases, there are those known from others, for example, physical problems. However, the overwhelming majority of the found number sets have not been encountered earlier in the literature. The obvious applications of this research are numbertheory and algebra. And the interdisciplinarity of the results indicates the possibility of applications and enhances their practical significance in other areas of knowledge.
The amount of information people use is steadily increasing. At the same time, the information flow associated with our daily life is growing. It permeates our everyday life, from school grades to financial transactio...
The amount of information people use is steadily increasing. At the same time, the information flow associated with our daily life is growing. It permeates our everyday life, from school grades to financial transactions in the securities markets. All human activities, both daily and professional are closely connected with various information technologies. IT allow workers to do their jobs in comfort, fast, and as efficient as possible. But, dependence on digital and computer technologies implies the need to transfer information securely and confidentially manner using cryptography methods. Cryptography is the science of writing secret messages. Its role has grown dramatically these days. As a subject of great historical and modern significance, cryptology makes it possible to assess the need for many applications of mathematics to be used. Since many cryptography algorithms require intensive computations, it is advisable to start getting acquainted with this subject using the Maple computeralgebra system as the main tool. To solve the research problems, theoretical methods were used (analysis, synthesis, comparison and generalization were used while studying the literature on the given problem; interpretation method; Kasiski’s method, methods of numbertheory, mathematical modeling), as well as exact analytical solutions to problems of elementary numbertheory and methods of the Maple computeralgebra system (CAS). The developed Maplets make it possible to demonstrate the order of performing the basic functions of elementary numbertheory as well as the encryption and decryption stages of various ciphers, and to assess their cryptographic strength. In addition, the Maplets allow students to easily perform encryption and decryption algorithms with very little programming language knowledge required.
As the most powerful tool in discrepancy theory, the partial coloring method has wide applications in many problems including the Beck-Fiala problem [BF81] and Spencer’s celebrated result [Spe85, Ban10, LM12, Rot17]....
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As the most powerful tool in discrepancy theory, the partial coloring method has wide applications in many problems including the Beck-Fiala problem [BF81] and Spencer’s celebrated result [Spe85, Ban10, LM12, Rot17]. Currently, there are two major algorithmic approaches for the partial coloring method: the first approach uses linear algebraic tools to update the partial coloring for many rounds [Ban10, LM12, BG17, LRR17, BDGL19];and the second one, called Gaussian measure algorithm [Rot17, RR23], projects a random Gaussian vector to the feasible region that satisfies all discrepancy constraints in [−1, 1]n. In this work, we explore the advantages of these two approaches and show the following results for them separately. 1. Spencer [Spe86] conjectured that the prefix discrepancy of any A ∈ {0, 1}m×n is O(√m), i.e., ∃x ∈ {±1}n such that maxt≤n k Pi≤t A(·, i) · x(i)k∞ = O(√m) where A(·, i) denotes column i of A. Combining small deviations bounds of the Gaussian processes and the Gaussian measure algorithm [Rot17], we show how to find a partial coloring with prefix discrepancy O(√m) and Ω(n) entries in {±1} efficiently. To the best of our knowledge, this provides the first partial coloring whose prefix discrepancy is almost optimal [Spe85, Spe86] (up to constants). However, unlike the classical discrepancy problem [Bec81, Spe85], there is no reduction on the number of variables n for the prefix problem. By recursively applying partial coloring, we obtain a full coloring with prefix discrepancy O(√m · log Om(n) ). Prior to this work, the best bounds of the prefix Spencer conjecture for arbitrarily large n were 2m [BG81] and O(√m log n) [Ban12, BG17]. 2. Our second result extends the first linear algebraic approach to a sampling algorithm in Spencer’s classical setting. On the first hand, besides the six deviation bound [Spe85], Spencer also proved that there are 1.99m good colorings with discrepancy O(√m) for any A ∈ {0, 1}m×m. Hence a natural question is to design effi
Availability of computeralgebra systems (CAS) lead to the resurrection of the resultant method for eliminating one or more variables from the polynomials system. The resultant matrix method has advantages over the Gr...
Availability of computeralgebra systems (CAS) lead to the resurrection of the resultant method for eliminating one or more variables from the polynomials system. The resultant matrix method has advantages over the Groebner basis and Ritt-Wu method due to their high complexity and storage requirement. This paper focuses on the current resultant matrix formulations and investigates their ability or otherwise towards producing optimal resultant matrices. A determinantal formula that gives exact resultant or a formulation that can minimize the presence of extraneous factors in the resultant formulation is often sought for when certain conditions that it exists can be determined. We present some applications of elimination theory via resultant formulations and examples are given to explain each of the presented settings.
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