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The detailed implementation and analysis of a finite element multigrid scheme for the solution of elliptic optimal control problems is presented. A particular focus is in the definition of smoothing strategies for the case of constrained control problems. For this setting, convergence of the multigrid scheme is discussed based on the BPX framework. Results of numerical experiments are reported to...
Sparse grids, combined with gradient penalties provide an attractive tool for regularised least squares fitting. It has earlier been found that the combination technique, which builds a sparse grid function using a linear combination of approximations on partial grids, is here not as effective as it is in the case of elliptic partial differential equations. We argue that this is due to the irregular...
The present article deals with fictitious domain methods for numerical realization of scalar variational inequalities with the Signorini type conditions on the boundary. Two variants are introduced and analyzed. A discretization is done by finite elements. It leads to a system of non-smooth, piecewise linear equations. This system is solved by the semismooth Newton method. Numerical experiments confirm...
In this paper, we give an analysis and a general procedure for 4D variational data assimilation (4D-Var). In functional partial differential equation setting, the adjoint equation method, sensitivity analysis, and multicomponent operator splitting are discussed. Nonlinear optimization methods and convergence analysis are also investigated for 4D-Var.
In the paper we deal with lower bounds constructed for the asymptotic competitive ratio of semi-online bin packing and batched bin packing algorithms.We determine the bounds as the solutions of a related nonlinear optimization problem using theoretical analysis and a reliable numerical global optimization method. Our results improve the lower bounds given in Gutin et al. (Discrete Optim 2:71–82, 2005)...
Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic...
A matching $${E_\mathcal{M}}$$ of graph G = (V, E) is a subset of the edges E, such that no vertex in V is incident to more than one edge in $${E_\mathcal{M}}$$ . The matching $${E_\mathcal{M}}$$ is maximum if there is no matching in G with size strictly larger than the size of $${E_\mathcal{M}}$$ . In this paper, we present a distributed stabilizing algorithm for finding maximum...
We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular...
In this paper, the maximal abelian dimension is computationally obtained for an arbitrary finite-dimensional Lie algebra, defined by its nonzero brackets. More concretely, we describe and implement an algorithm which computes such a dimension by running it in the symbolic computation package MAPLE. Finally, we also show a computational study related to this implementation, regarding both the computing...
In this paper, we propose a numerical scheme which is almost second-order spatial accurate for a one-dimensional singularly perturbed parabolic convection-diffusion problem exhibiting a regular boundary layer. The proposed numerical scheme consists of classical backward-Euler method for the time discretization and a hybrid finite difference scheme for the spatial discretization. We analyze the scheme...
In this paper, spectral properties and computational performance of a generalized block triangular preconditioner for symmetric saddle point problems are discussed in detail. We will provide estimates for the region containing both the nonreal and the real eigenvalues and generalize the results of Simoncini (Appl Numer Math 49:63–80, 2004) and Cao (Appl Numer Math 57:899–910, 2007). Finally, numerical...
In this paper we consider the secret sharing problem on special access structures with minimal qualified subsets of size two, i.e. secret sharing on graphs. This means that the participants are the vertices of the graph and the qualified subsets are the subsets of V(G) spanning at least one edge. The information ratio of a graph G is denoted by R(G) and is defined as the ratio of the greatest size...
This survey provides a comparative overview of lattice-based signature schemes with respect to security and performance. Furthermore, we explicitly show how to construct a competitive and provably secure Merkle tree signature scheme solely based on worst-case lattice problems.
We consider a key distribution scheme for wireless sensor networks which uses deployment knowledge. Deployment is modeled as a grid of hexagonal clusters, into centers of which the sensor nodes are dropped according to a given probability distribution (e.g. a Gaussian one). We consider sensor connectivity in a random intersection graph model, instead of the more commonly used in literature G(n, p)...
Let $${{\mathcal S}}$$ be one of the two multiplicative semigroups: M × M Boolean matrices, or the semigroup of M × M matrices over the field GF(2). Then for any matrix $${A\in {\mathcal S}}$$ there exist two unique smallest numbers, namely the index and period k, d, such that Ak = Ak+d. This fact allows us to form a new statistical test for randomness which we call the Semigroup Matrix...
Group signature schemes enable to create digital signatures such that the signers are hidden in a group of potential signers. However, in a case of need it is possible to reveal the actual signer either by a group administrator or collectively by the group members. We design a new kind of signatures that we call step-out group signature where the situation is reversed: any member of the group except...
We present group-theoretic and cryptographic properties of a generalization of the traditional discrete logarithm problem from cyclic to arbitrary finite groups. Questions related to properties which contribute to cryptographic security are investigated, such as distributional, coverage and complexity properties. We show that the distribution of elements in a certain multiset tends to uniform. In...
This paper proposes and analyzes an approach for design of stream ciphers based on joint computing over random and secret data. Feasibility of encryption/ decryption computation when the ciphertext involve pure random data is shown. The core element of the proposed approach for stream ciphering is a pseudo-random embedding of the random bits into the ciphertext and this embedding plays role of a homophonic...
Wiener’s attack is a well-known polynomial-time attack on a RSA cryptosystem with small secret decryption exponent d, which works if d < n0.25, where n = pq is the modulus of the cryptosystem. Namely, in that case, d is the denominator of some convergent pm/qm of the continued fraction expansion of e/n, and therefore d can be computed efficiently from the public key (n, e). There are several...
In this paper we study the security of the Advanced Encryption Standard (AES) and AES-like block ciphers against differential cryptanalysis. Differential cryptanalysis is one of the most powerful methods for analyzing the security of block ciphers. Even though no formal proofs for the security of AES against differential cryptanalysis have been provided to date, some attempts to compute the maximum...
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