14,455 research outputs found
Fast Quantum Algorithm for Solving Multivariate Quadratic Equations
In August 2015 the cryptographic world was shaken by a sudden and surprising
announcement by the US National Security Agency NSA concerning plans to
transition to post-quantum algorithms. Since this announcement post-quantum
cryptography has become a topic of primary interest for several standardization
bodies. The transition from the currently deployed public-key algorithms to
post-quantum algorithms has been found to be challenging in many aspects. In
particular the problem of evaluating the quantum-bit security of such
post-quantum cryptosystems remains vastly open. Of course this question is of
primarily concern in the process of standardizing the post-quantum
cryptosystems. In this paper we consider the quantum security of the problem of
solving a system of {\it Boolean multivariate quadratic equations in
variables} (\MQb); a central problem in post-quantum cryptography. When ,
under a natural algebraic assumption, we present a Las-Vegas quantum algorithm
solving \MQb{} that requires the evaluation of, on average,
quantum gates. To our knowledge this is the fastest algorithm for solving
\MQb{}
An efficient mixed variational reduced order model formulation for non-linear analyses of elastic shells
The Koiter-Newton method had recently demonstrated a superior performance for non-linear analyses of structures, compared to traditional path-following strategies. The method follows a predictor-corrector scheme to trace the entire equilibrium path. During a predictor step a reduced order model is constructed based on Koiter's asymptotic post-buckling theory which is followed by a Newton iteration in the corrector phase to regain the equilibrium of forces.
In this manuscript, we introduce a robust mixed solid-shell formulation to further enhance the efficiency of stability analyses in various aspects. We show that a Hellinger-Reissner variational formulation facilitates the reduced order model construction omitting an expensive evaluation of the inherent fourth order derivatives of the strain energy. We demonstrate that extremely large step sizes with a reasonable out-of-balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path. More importantly, the numerical effort of the corrector phase involving a Newton iteration of the full order model is drastically reduced thus revealing the true strength of the proposed formulation. We study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems
Practical Sparse Matrices in C++ with Hybrid Storage and Template-Based Expression Optimisation
Despite the importance of sparse matrices in numerous fields of science,
software implementations remain difficult to use for non-expert users,
generally requiring the understanding of underlying details of the chosen
sparse matrix storage format. In addition, to achieve good performance, several
formats may need to be used in one program, requiring explicit selection and
conversion between the formats. This can be both tedious and error-prone,
especially for non-expert users. Motivated by these issues, we present a
user-friendly and open-source sparse matrix class for the C++ language, with a
high-level application programming interface deliberately similar to the widely
used MATLAB language. This facilitates prototyping directly in C++ and aids the
conversion of research code into production environments. The class internally
uses two main approaches to achieve efficient execution: (i) a hybrid storage
framework, which automatically and seamlessly switches between three underlying
storage formats (compressed sparse column, Red-Black tree, coordinate list)
depending on which format is best suited and/or available for specific
operations, and (ii) a template-based meta-programming framework to
automatically detect and optimise execution of common expression patterns.
Empirical evaluations on large sparse matrices with various densities of
non-zero elements demonstrate the advantages of the hybrid storage framework
and the expression optimisation mechanism.Comment: extended and revised version of an earlier conference paper
arXiv:1805.0338
Practical Sparse Matrices in C++ with Hybrid Storage and Template-Based Expression Optimisation
Despite the importance of sparse matrices in numerous fields of science,
software implementations remain difficult to use for non-expert users,
generally requiring the understanding of underlying details of the chosen
sparse matrix storage format. In addition, to achieve good performance, several
formats may need to be used in one program, requiring explicit selection and
conversion between the formats. This can be both tedious and error-prone,
especially for non-expert users. Motivated by these issues, we present a
user-friendly and open-source sparse matrix class for the C++ language, with a
high-level application programming interface deliberately similar to the widely
used MATLAB language. This facilitates prototyping directly in C++ and aids the
conversion of research code into production environments. The class internally
uses two main approaches to achieve efficient execution: (i) a hybrid storage
framework, which automatically and seamlessly switches between three underlying
storage formats (compressed sparse column, Red-Black tree, coordinate list)
depending on which format is best suited and/or available for specific
operations, and (ii) a template-based meta-programming framework to
automatically detect and optimise execution of common expression patterns.
Empirical evaluations on large sparse matrices with various densities of
non-zero elements demonstrate the advantages of the hybrid storage framework
and the expression optimisation mechanism.Comment: extended and revised version of an earlier conference paper
arXiv:1805.0338
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