Let(b1, . . .,bn) be a lattice basis with Gram-Schmidt orthogonalization(b1, . . . ,b∗n), the quantities‖b1‖/‖bi ‖for i = 1, . . . , n play important roles in analyzing lattice reduction algorithms and lattice enumeration algorithms.In this paper, we study the problem of minimizing the quantity‖b1‖/‖bn‖ over all bases(b1, . . .,bn) of a givenn...

We improve the zeroizing attack on the multilinear map of Garg, Gentry and Halevi (GGH). Our algorithm can solve the Graded Decisional Diffie-Hellman (GDDH) problem on the GGH scheme when the dimension n of the ideal lattice Z[X]/(X+1) is O(κλ) as suggested for the κ-linear GGH scheme. The zeroizing attack is to recover a basis of an ideal generated by a secret element g ∈ Z[X]/(X + 1) from the...

In this thesis, different aspects of lattice-based precoding and decoding for the transmission of digital and analog data over MIMO fading channels are investigated: 1) Lattice-based precoding in MIMO broadcast systems: A new viewpoint for adopting the lattice reduction in communication over MIMO broadcast channels is introduced. Lattice basis reduction helps us to reduce the average transmitte...

A component-based approach is introduced for fast and flexible solution of parameter-dependent eigenproblems. We consider a shifted version of the eigenproblem where the left hand side operator corresponds to an equilibrium between the stiffness operator and a weighted mass operator. This permits to apply the Static Condensation Reduced Basis Element method, a domain synthesis approach with red...

As numerical simulations find more and more use in real-world scenarios and industrial applications, demands concerning efficiency and reliability increase as well. Especially scenarios that call for real-time simulations or multi-query evaluations of partial differential equations (PDEs) often require means of model order reduction. Examples for such scenarios are optimal control and optimizat...

Ballistic transport simulation in nanodevices, which involves self-consistently solving a coupled Schrödinger-Poisson system of equations, is usually computationally intensive. Here, we propose coupling the reduced basis method with the subband decomposition method to improve the overall efficiency of the simulation. By exploiting a posteriori error estimation procedure and greedy sampling algo...

We introduce a reduced basis method for the efficient numerical solution of partial integro-differential equations which arise in option pricing theory. Our method uses a basis of functions constructed from a sequence of Black-Scholes solutions with different volatilities. We show that this choice of basis leads to a sparse representation of option pricing functions, yielding an approximation w...

We analyze reduced basis acceleration of recently proposed deterministic Bayesian inversion algorithms for partial differential equations with uncertain distributed parameter, for observation data subject to additive, Gaussian observation noise. Specifically, Bayesian inversion of affine-parametric, linear operator families on possibly high-dimensional parameter spaces. We consider “high-fideli...

The main theme of this volume is the efficient solution of families of stochastic or parametric partial differential equations. This article focuses on the theoretical underpinnings of such methods. It shows how concepts from approximation theory, such as entropy or widths, can help quantify a priori how well such methods can perform. This theory is then used to analyze one of the primary numer...

We describe the application of the reduced-basis method in rapid and accurate determination of band energies in band structure calculations. The method is well suited for problems requiring repetitive evaluations of the band energies, especially in the many-query limit. We demonstrate the efficacy of the method in the determination of the spectral properties of crystalline silicon.