Adaptive Regularization for Nonconvex Optimization Using Inexact Function Values and Randomly Perturbed Derivatives

A regularization algorithm allowing random noise in derivatives and inexact function values is proposed for computing approximate local critical points of any order for smooth unconstrained optimization problems. For an objective function with Lipschitz continuous $p$-th derivative and given an arbitrary optimality order $q \leq p$, it is shown that this algorithm will, in expectation, compute such a point in at most $O\left(\left(\min_{j\in\{1,\ldots,q\}}\epsilon_j\right)^{-\frac{p+1}{p-q+1}}\right)$ inexact evaluations of $f$ and its derivatives whenever $q\in\{1,2\}$, where $\epsilon_j$ is the tolerance for $j$th order accuracy. This bound becomes at most $O\left(\left(\min_{j\in\{1,\ldots,q\}}\epsilon_j\right)^{-\frac{q(p+1)}{p}}\right)$ inexact evaluations if $q>2$ and all derivatives are Lipschitz continuous. Moreover these bounds are sharp in the order of the accuracy tolerances. An extension to convexly constrained problems is also outlined.Comment: 22 page

Adaptive Regularization Algorithms with Inexact Evaluations for Nonconvex Optimization

A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that is constraints whose evaluation and enforcement has negligible cost) under the assumption that the derivative of highest degree is $\beta$-H\"{o}lder continuous. It features a very flexible adaptive mechanism for determining the inexactness which is allowed, at each iteration, when computing objective function values and derivatives. The complexity analysis covers arbitrary optimality order and arbitrary degree of available approximate derivatives. It extends results of Cartis, Gould and Toint (2018) on the evaluation complexity to the inexact case: if a $q$th order minimizer is sought using approximations to the first $p$ derivatives, it is proved that a suitable approximate minimizer within $\epsilon$ is computed by the proposed algorithm in at most $O(\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ iterations and at most $O(|\log(\epsilon)|\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ approximate evaluations. An algorithmic variant, although more rigid in practice, can be proved to find such an approximate minimizer in $O(|\log(\epsilon)|+\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ evaluations.While the proposed framework remains so far conceptual for high degrees and orders, it is shown to yield simple and computationally realistic inexact methods when specialized to the unconstrained and bound-constrained first- and second-order cases. The deterministic complexity results are finally extended to the stochastic context, yielding adaptive sample-size rules for subsampling methods typical of machine learning.Comment: 32 page

Influences of urban fabric on pyroclastic density currents at Pompeii (Italy): 1. Flow direction and deposition

To assess ways in which the products of explosive eruptions interact with human settlements, we performed volcanological and rock magnetic analyses on the deposits of the A.D. 79 eruption at the Pompeii excavations (Italy). During this eruption the Roman town of Pompeii was covered by 2.5 m of fallout pumice and then partially destroyed by pyroclastic density currents (PDCs). Anisotropy of magnetic susceptibility measurements performed on the fine matrix of the deposits allowed the quantification of the variations in flow direction and emplacement mechanisms of the parental PDCs that entered the town. These results, integrated with volcanological field investigations, revealed that the presence of buildings, still protruding through the fallout deposits, strongly affected the distribution and accumulation of the erupted products. All of the PDCs that entered the town, even the most dilute ones, were density stratified currents in which interaction with the urban fabric occurred in the lower part of the current. The degree of interaction varied mainly as a function of obstacle height and density stratification within the current. For examples, the lower part of the EU4pf current left deposits up to 3 m thick and was able to interact with 2- to 4-m-high obstacles. However, a decrease in thickness and grain size of the deposits across the town indicates that even though the upper portion of the current was able to decouple from the lower portion, enabling it to flow over the town, it was not able to fully restore the sediment supply to the lower portion in order to maintain the deposition observed upon entry into the town.PublishedB052134.3. TTC - Scenari di pericolosità vulcanica3.6. Fisica del vulcanismoJCR Journalreserve

Large K-exciton dynamics in GaN epilayers: the non-thermal and thermal regime

We present a detailed investigation concerning the exciton dynamics in GaN epilayers grown on c-plane sapphire substrates, focussing on the exciton formation and the transition from the nonthermal to the thermal regime. The time-resolved kinetics of LO-phonon replicas is used to address the energy relaxation in the excitonic band. From ps time-resolved spectra we bring evidence for a long lasting non-thermal excitonic distribution which accounts for the rst 50 ps. Such a behavior is con rmed in di erent experimental conditions, both when non-resonant and resonant excitation are used. At low excitation power density the exciton formation and their subsequent thermalization is dominated by impurity scattering rather than by acoustic phonon scattering. The estimate of the average energy of the excitons as a function of delay after the excitation pulse provides information on the relaxation time, which describes the evolution of the exciton population to the thermal regime.Comment: 9 pages,8 figure

Cutaneous sarcoid-like reaction in a patient treated with target therapy for metastatic melanoma: the hue is the clue

Cutaneous sarcoid-like reaction in a patient treated with target therapy for metastatic melanoma: the hue is the clu

Endometrioid Cancer Associated With Endometriosis: From the Seed and Soil Theory to Clinical Practice

Endometriosis is a benign condition characterized by the presence of ectopic endometrial tissue. It is still debated whether endometriosis is a disease that can predispose to the pathogenesis of endometrial cancer outside the uterus. Deficiencies in mismatch repair (MMR) genes are a known risk factor for developing endometrioid cancer. Starting from two cases of patients with abnormal MMR endometrioid carcinoma of the uterus and synchronous endometrioid carcinoma in non-ovarian and ovarian endometriosis, we performed a somatic mutation profile and phylogenetic analysis of the lesions in order to identify if they were metastasis or primary de novo tumors. In the first case, we identified de novo activating mutations in PIK3CA and KRAS in endometrioid cancer lesions but not in endometriosis. Although the acquisition of a de novo mutation in ESR1 and a decrease in mutant allele fraction (MAF) from the endometrial tumor to the localizations in the endometriosis lesions, the clonal relationship was confirmed by the limited number of heteroplasmic mutations in D-loop mitochondrial DNA region. In the other case, the clonal behavior was demonstrated by the overlap of MAF at each site. Our data support the hypothesis of a retrograde dissemination of tumor cells, moving from the primary carcinoma in the endometrium to ectopic sites of endometriosis where localizations of tumor arise

On the exciton binding energy in a quantum well

We consider a model describing the one-dimensional confinement of an exciton in a symmetrical, rectangular quantum-well structure and derive upper and lower bounds for the binding energy $E_b$ of the exciton. Based on these bounds, we study the dependence of $E_b$ on the width of the confining potential with a higher accuracy than previous reports. For an infinitely deep potential the binding energy varies as expected from $1 Ry$ at large widths to $4 Ry$ at small widths. For a finite potential, but without consideration of a mass mismatch or a dielectric mismatch, we substantiate earlier results that the binding energy approaches the value $1 Ry$ for both small and large widths, having a characteristic peak for some intermediate size of the slab. Taking the mismatch into account, this result will in general no longer be true. For the specific case of a $Ga_{1-x}Al_{x}As/GaAs/Ga_{1-x}Al_{x}As$ quantum-well structure, however, and in contrast to previous findings, the peak structure is shown to survive.Comment: 32 pages, ReVTeX, including 9 figure