scholarly journals A sieve algorithm based on overlattices

2014 ◽  
Vol 17 (A) ◽  
pp. 49-70 ◽  
Author(s):  
Anja Becker ◽  
Nicolas Gama ◽  
Antoine Joux
Keyword(s):  
Heuristic Algorithm ◽  
Orthonormal Basis ◽  
Complexity Analysis ◽  
Computer Architectures ◽  
High Dimensions ◽  
Average Case ◽  
Solution Vector ◽  
New Approach ◽  
Bounded Norm

AbstractIn this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case.This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$in time$2^{0.3774\, n}$using memory$2^{0.2925\, n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.

scholarly journals Understanding and Predicting Nonlinear Turbulent Dynamical Systems with Information Theory

Atmosphere ◽  
2019 ◽  
Vol 10 (5) ◽  
pp. 248
Author(s):  
Nan Chen ◽  
Xiao Hou ◽  
Qin Li ◽  
Yingda Li
Keyword(s):  
Information Theory ◽  
Dynamical Systems ◽  
Information Criterion ◽  
Gaussian Mixture ◽  
Model Error ◽  
High Dimensions ◽  
New Approach ◽  
Fluctuation Dissipation ◽  
Spectrum Density ◽  
Non Gaussian

Complex nonlinear turbulent dynamical systems are ubiquitous in many areas. Quantifying the model error and model uncertainty plays an important role in understanding and predicting complex dynamical systems. In the first part of this article, a simple information criterion is developed to assess the model error in imperfect models. This effective information criterion takes into account the information in both the equilibrium statistics and the temporal autocorrelation function, where the latter is written in the form of the spectrum density that permits the quantification via information theory. This information criterion facilitates the study of model reduction, stochastic parameterizations, and intermittent events. In the second part of this article, a new efficient method is developed to improve the computation of the linear response via the Fluctuation Dissipation Theorem (FDT). This new approach makes use of a Gaussian Mixture (GM) to describe the unperturbed probability density function in high dimensions and avoids utilizing Gaussian approximations in computing the statistical response, as is widely used in the quasi-Gaussian (qG) FDT. Testing examples show that this GM FDT outperforms qG FDT in various strong non-Gaussian regimes.

Initial attempt to optimize the operation of a hydrometric network using branch and bound algorithm

1988 ◽  
Vol 15 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Patrice M. Pelletier ◽  
Slobodan P. Simonovic
Keyword(s):  
Heuristic Algorithm ◽  
Optimal Solution ◽  
Optimization Techniques ◽  
Traveling Salesman ◽  
Future Research ◽  
Initial Attempt ◽  
Research Directions ◽  
New Approach ◽  
Future Research Directions ◽  
Field Area

The paper presents a new approach developed to formulate and solve the problem of field operation of a hydrometric network. In present practice, optimization techniques are not used for solving this problem. In the initial attempt, documented in the paper, the traveling salesman algorithm is used to find the optimal solution. Assumptions used and experience gained during the research are addressed in the paper. For the Dauphin hydrometric field area in Manitoba, an "optimal" solution has been obtained and presented. Based on the analysis of the results, conclusions were drawn about future research directions incorporating development of an effective heuristic algorithm for solving the traveling hydrometric technician problem. Key words: operation, hydrometric network, optimization, heuristic algorithm, traveling salesman algorithm.

A new approach to assessing model risk in high dimensions

2015 ◽  
Vol 58 ◽  
pp. 166-178 ◽  
Author(s):  
Carole Bernard ◽  
Steven Vanduffel
Keyword(s):  
High Dimensions ◽  
Model Risk ◽  
New Approach

scholarly journals Time Complexity Analysis of the Binary Tree Roll Algorithm

2017 ◽  
Vol 12 (2) ◽  
Author(s):  
Adrijan Božinovski ◽  
George Tanev ◽  
Biljana Stojčevska ◽  
Veno Pačovski ◽  
Nevena Ackovska
Keyword(s):  
Theoretical Analysis ◽  
Empirical Analysis ◽  
Binary Tree ◽  
Time Complexity ◽  
Recurrence Relations ◽  
Complexity Analysis ◽  
Worst Case ◽  
Average Case ◽  
The Empirical Analysis

This paper presents the time complexity analysis of the Binary Tree Roll algorithm. The time complexity is analyzed theoretically and the results are then confirmed empirically. The theoretical analysis consists of finding recurrence relations for the time complexity, and solving them using various methods. The empirical analysis consists of exhaustively testing all trees with given numbers of nodes  and counting the minimum and maximum steps necessary to complete the roll algorithm. The time complexity is shown, both theoretically and empirically, to be linear in the best case and quadratic in the worst case, whereas its average case is shown to be dominantly linear for trees with a relatively small number of nodes and dominantly quadratic otherwise.

scholarly journals Space Complexity Analysis of the Binary Tree Roll Algorithm

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Adrijan Božinovski ◽  
George Tanev ◽  
Biljana Stojčevska ◽  
Veno Pačovski ◽  
Nevena Ackovska
Keyword(s):  
Theoretical Analysis ◽  
Empirical Analysis ◽  
Binary Tree ◽  
Complexity Analysis ◽  
Space Complexity ◽  
Worst Case ◽  
Average Case ◽  
Tree Topologies ◽  
The Given ◽  
The Empirical Analysis

This paper presents the space complexity analysis of the Binary Tree Roll algorithm. The space complexity is analyzed theoretically and the results are then confirmed empirically. The theoretical analysis consists of determining the amount of memory occupied during the execution of the algorithm and deriving functions of it, in terms of the number of nodes of the tree n, for the worst - and best-case scenarios. The empirical analysis of the space complexity consists of measuring the maximum and minimum amounts of memory occupied during the execution of the algorithm, for all binary tree topologies with the given number of nodes. The space complexity is shown, both theoretically and empirically, to be logarithmic in the best case and linear in the worst case, whereas its average case is shown to be dominantly logarithmic.

scholarly journals Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems

2011 ◽  
Vol 11 (3) ◽  
pp. 363-381 ◽  
Author(s):  
Daniel Kressner ◽  
Christine Tobler
Keyword(s):  
Degrees Of Freedom ◽  
Quantum Physics ◽  
Eigenvalue Problems ◽  
Low Rank ◽  
Elliptic Pde ◽  
Tucker Decomposition ◽  
High Dimensions ◽  
Solution Vector ◽  
Matrix Eigenvalue ◽  
Matrix Eigenvalue Problems

AbstractWe consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that the resulting matrix eigenvalue problem Ax=λx exhibits Kronecker product structure. In particular, we are concerned with the case of high dimensions, where standard approaches to the solution of matrix eigenvalue problems fail due to the exponentially growing degrees of freedom. Recent work shows that this curse of dimensionality can in many cases be addressed by approximating the desired solution vector x in a low-rank tensor format. In this paper, we use the hierarchical Tucker decomposition to develop a low-rank variant of LOBPCG, a classical preconditioned eigenvalue solver. We also show how the ALS and MALS (DMRG) methods known from computational quantum physics can be adapted to the hierarchical Tucker decomposition. Finally, a combination of ALS and MALS with LOBPCG and with our low-rank variant is proposed. A number of numerical experiments indicate that such combinations represent the methods of choice.

A new approach in the analysis of linear systems with periodic coefficients for applications in rotorcraft dynamics

1994 ◽  
Vol 98 (971) ◽  
pp. 9-16 ◽  
Author(s):  
D.-H. Wu ◽  
S.C. Sinha
Keyword(s):  
Numerical Technique ◽  
Numerical Accuracy ◽  
Solution Vector ◽  
Suggested Approach ◽  
New Approach ◽  
Stiffness Matrices ◽  
Bladed Rotor ◽  
The Stability ◽  
Varying Mass ◽  
Second Order Equations

Abstract A numerical technique for the stability analysis of linear mechanical dynamic systems with periodically varying parameters is proposed. The technique is based on representation of the solution vector in terms of Chebyshev polynomials defined over the principal period. Two formulations have been presented. The first formulation is suitable for systems described by state space equations, while the second can be applied directly to a set of second order equations with periodically varying mass, damping and stiffness matrices. As an illustrative example, the flap-lag stability of a multi-bladed rotor is examined. The numerical accuracy and efficiency of the proposed technique is compared with standard numerical codes based on Runge-Kutta, Adams-Moulton and Gear algorithms. The results indicate that the suggested approach is by far the most efficient one, particularly for systems with larger dimensions.

scholarly journals Statistical Properties of Lambda Terms

10.37236/8491 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Maciej Bendkowski ◽  
Olivier Bodini ◽  
Sergey Dovgal
Keyword(s):  
Generating Functions ◽  
Complexity Analysis ◽  
Parameter Distribution ◽  
Random Generation ◽  
Average Case ◽  
Case Complexity ◽  
Average Case Complexity ◽  
Novel Method ◽  
De Bruijn ◽  
Index Value

We present a quantitative, statistical analysis of random lambda terms in the De Bruijn notation. Following an analytic approach using multivariate generating functions, we investigate the distribution of various combinatorial parameters of random open and closed lambda terms, including the number of redexes, head abstractions, free variables or the De Bruijn index value profile. Moreover, we conduct an average-case complexity analysis of finding the leftmost-outermost redex in random lambda terms showing that it is on average constant. The main technical ingredient of our analysis is a novel method of dealing with combinatorial parameters inside certain infinite, algebraic systems of multivariate generating functions. Finally, we briefly discuss the random generation of lambda terms following a given skewed parameter distribution and provide empirical results regarding a series of more involved combinatorial parameters such as the number of open subterms and binding abstractions in closed lambda terms.

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