Regularization matrices determined by matrix nearness problems

This paper is concerned with the solution of large-scale linear discrete ill-posed problems with error-contaminated data. Tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. This matrix should be chosen to promote the recovery of known important features of the desired solution, such as smoothness and monotonicity. We describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem. The constructed regularization matrix is the closest matrix in the Frobenius norm with a prescribed null space to a given matrix. Numerical examples illustrate the performance of the regularization matrices so obtained

Transients in sheared granular matter

As dense granular materials are sheared, a shear band and an anisotropic force network form. The approach to steady state behavior depends on the history of the packing and the existing force and contact network. We present experiments on shearing of dense granular matter in a 2D Couette geometry in which we probe the history and evolution of shear bands by measuring particle trajectories and stresses during transients. We find that when shearing is stopped and restarted in the same direction, steady state behavior is immediately reached, in agreement with the typical assumption that the system is quasistatic. Although some relaxation of the force network is observed when shearing is stopped, quasistatic behavior is maintained because the contact network remains essentially unchanged. When the direction of shear is reversed, a transient occurs in which stresses initially decrease, changes in the force network reach further into the bulk, and particles far from the wheel become more mobile. This occurs because the force network is fragile to changes transverse to the force network established under previous shear; particles must rearrange before becoming jammed again, thereby providing resistance to shear in the reversed direction. The strong force network is reestablished after displacing the shearing surface $\approx 3d$, where $d$ is the mean grain diameter. Steady state velocity profiles are reached after a shear of $\leq 30d$. Particles immediately outside of the shear band move on average less than 1 diameter before becoming jammed again. We also examine particle rotation during this transient and find that mean particle spin decreases during the transient, which is related to the fact that grains are not interlocked as strongly.Comment: 7 pages, 11 figures, accepted to Eur. Phys. J. E, revised version based on referee suggestion

Brain Activity Mapping from MEG Data via a Hierarchical Bayesian Algorithm with Automatic Depth Weighting

A recently proposed iterated alternating sequential (IAS) MEG inverse solver algorithm, based on the coupling of a hierarchical Bayesian model with computationally efficient Krylov subspace linear solver, has been shown to perform well for both superficial and deep brain sources. However, a systematic study of its ability to correctly identify active brain regions is still missing. We propose novel statistical protocols to quantify the performance of MEG inverse solvers, focusing in particular on how their accuracy and precision at identifying active brain regions. We use these protocols for a systematic study of the performance of the IAS MEG inverse solver, comparing it with three standard inversion methods, wMNE, dSPM, and sLORETA. To avoid the bias of anecdotal tests towards a particular algorithm, the proposed protocols are Monte Carlo sampling based, generating an ensemble of activity patches in each brain region identified in a given atlas. The performance in correctly identifying the active areas is measured by how much, on average, the reconstructed activity is concentrated in the brain region of the simulated active patch. The analysis is based on Bayes factors, interpreting the estimated current activity as data for testing the hypothesis that the active brain region is correctly identified, versus the hypothesis of any erroneous attribution. The methodology allows the presence of a single or several simultaneous activity regions, without assuming that the number of active regions is known. The testing protocols suggest that the IAS solver performs well with both with cortical and subcortical activity estimation

Iterative Updating of Model Error for Bayesian Inversion

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when optimization algorithms are used to find a single estimate, or to speed up Markov chain Monte Carlo (MCMC) calculations in the Bayesian framework. The use of an approximate model introduces a discrepancy, or modeling error, that may have a detrimental effect on the solution of the ill-posed inverse problem, or it may severely distort the estimate of the posterior distribution. In the Bayesian paradigm, the modeling error can be considered as a random variable, and by using an estimate of the probability distribution of the unknown, one may estimate the probability distribution of the modeling error and incorporate it into the inversion. We introduce an algorithm which iterates this idea to update the distribution of the model error, leading to a sequence of posterior distributions that are demonstrated empirically to capture the underlying truth with increasing accuracy. Since the algorithm is not based on rejections, it requires only limited full model evaluations. We show analytically that, in the linear Gaussian case, the algorithm converges geometrically fast with respect to the number of iterations. For more general models, we introduce particle approximations of the iteratively generated sequence of distributions; we also prove that each element of the sequence converges in the large particle limit. We show numerically that, as in the linear case, rapid convergence occurs with respect to the number of iterations. Additionally, we show through computed examples that point estimates obtained from this iterative algorithm are superior to those obtained by neglecting the model error.Comment: 39 pages, 9 figure

Electromagnetic penguin operators and direct CP violation in K --> pi l^+ l^-

Supersymmetric extensions of the Standard Model predict a large enhancement of the Wilson coefficients of the dimension-five electromagnetic penguin operators affecting the direct CP violation in K_L --> pi^0 e^+ e^- and the charge asymmetry in K^\pm --> pi^\pm l^+ l^-. Here we compute the relevant matrix elements in the chiral quark model and compare these with the ones given by lattice calculationsComment: 12 pages, JHEP style, gluonic corrections to B_T adde

A Model for Granular Texture with Steric Exclusion

We propose a new method to characterize the geometrical texture of a granular packing at the particle scale including the steric hindrance effect. This method is based on the assumption of a maximum disorder (entropy) compatible both with strain-induced anisotropy of the contact network and steric exclusions. We show that the predicted statistics for the local configurations is in a fairly agreement with our numerical data.Comment: 9 pages, 5 figure

"Studio preliminare per il progetto del Parco di Banditella, Livorno"

Oggetto della presente tesi è lo studio preliminare di progetto per la vasta superficie denominata Parco di Banditella in prossimità dell’omonimo quartiere, destinata a verde pubblico ma attualmente in stato di degrado e priva di opere di urbanizzazione. Lo studio risponde alla volontà del Comune di Livorno di realizzare all’interno del Parco strutture flessibili in grado di ospitare, durante l’anno, manifestazioni, iniziative e meeting, in linea con la destinazione urbanistica dell’area che è “verde pubblico e servizi”, e sviluppa ulteriormente il programma di progetto prevedendo la realizzazione di strutture ricettive, attività commerciali e ricreative. La proposta parte dalla constatazione che attraverso la sovrapposizione di diverse occasioni di aggregazione è possibile ottenere un interscambio efficace e produttivo tra diverse esperienze sociali, valorizzando maggiormente gli spazi pubblici come i parchi urbani, troppo spesso banalizzati nella progettazione e dedicati ad un’utenza che vive il verde in momenti esclusivamente di disimpegno. In primo luogo è stata svolta un’analisi storica della città di Livorno, e in particolare dello sviluppo del lungomare labronico e del quartiere di Banditella ad esso connesso. In seguito al rilievo dello stato attuale del sito e delle previsioni urbanistiche correlate, si sono individuati ed esplicitati i criteri decisionali utilizzati, gli obiettivi da perpetrare e il sistema funzionale di progetto. Lo studio affrontato si è posto il fine di non compromettere l’identificabilità dell’area verde, a favore delle nuove strutture previste; con tale scopo si è dunque condotta l’analisi di molteplici progetti di architettura ipogea, rappresentanti una possibile soluzione organica al problema. Sulla base dell’indagine svolta si è passati infine alla fase creativa vera e propria, con la definizione di un progetto preliminare in scala urbanistica dell’intera area, e un approfondimento in scala architettonica 1:200 delle strutture di principale interesse