Threshold functions and Poisson convergence for systems of equations in random sets

We present a unified framework to study threshold functions for the existence of solutions to linear systems of equations in random sets which includes arithmetic progressions, sum-free sets, $B_{h}[g]$-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property "$\mathcal{A}$ contains a non-trivial solution of $M\cdot\textbf{x}=\textbf{0}$", where $\mathcal{A}$ is a random set and each of its elements is chosen independently with the same probability from the interval of integers $\{1,\dots,n\}$. Our study contains a formal definition of trivial solutions for any combinatorial structure, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the behaviour of the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page

Sequential Compressed Sensing

Compressed sensing allows perfect recovery of sparse signals (or signals sparse in some basis) using only a small number of random measurements. Existing results in compressed sensing literature have focused on characterizing the achievable performance by bounding the number of samples required for a given level of signal sparsity. However, using these bounds to minimize the number of samples requires a-priori knowledge of the sparsity of the unknown signal, or the decay structure for near-sparse signals. Furthermore, there are some popular recovery methods for which no such bounds are known. In this paper, we investigate an alternative scenario where observations are available in sequence. For any recovery method, this means that there is now a sequence of candidate reconstructions. We propose a method to estimate the reconstruction error directly from the samples themselves, for every candidate in this sequence. This estimate is universal in the sense that it is based only on the measurement ensemble, and not on the recovery method or any assumed level of sparsity of the unknown signal. With these estimates, one can now stop observations as soon as there is reasonable certainty of either exact or sufficiently accurate reconstruction. They also provide a way to obtain "run-time" guarantees for recovery methods that otherwise lack a-priori performance bounds. We investigate both continuous (e.g. Gaussian) and discrete (e.g. Bernoulli) random measurement ensembles, both for exactly sparse and general near-sparse signals, and with both noisy and noiseless measurements.Comment: to appear in IEEE transactions on Special Topics in Signal Processin

A nonlinear vehicle-structure interaction methodology with wheel-rail detachment and reattachment

. A vehicle-structure interaction methodology with a nonlinear contact formulation based on contact and target elements has been developed. To solve the dynamic equations of motion, an incremental formulation has been used due to the nonlinear nature of the contact mechanics, while a procedure based on the Lagrange multiplier method imposes the contact constraint equations when contact occurs. The system of nonlinear equations is solved by an efficient block factorization solver that reorders the system matrix and isolates the nonlinear terms that belong to the contact elements or to other nonlinear elements that may be incorporated in the model. Such procedure avoids multiple unnecessary factorizations of the linear terms during each Newton iteration, making the formulation efficient and computationally attractive. A numerical example has been carried out to validate the accuracy and efficiency of the present methodology. The obtained results have shown a good agreement with the results obtained with the commercial finite element software ANSY