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

Experimental and Numerical Analysis of Single Phase Flow in a micro T-junction

This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.In this work the fluid-dynamic behaviour of a micro T-junction has been investigated both numerically and experimentally for low Reynolds numbers (Re<14) with water as working fluid. The velocity profiles within the T-junction has been experimentally determined by using the micro Particle Image Velocimetry (μPIV). The experimental data have been compared with the numerical results obtained by means of a 3D model implemented in Comsol Multiphysics® environment for incompressible, isothermal, laminar flows with constant properties. The comparison between the experimental and the numerical data puts in evidence a perfect agreement among the results. In the central region of the T-junction where the velocity profiles of the inlet branches interact, the maximum difference is less than 5.8% for different flow rates imposed at the inlet (with the ratio 1:2) and less than 4.4% in the case of the same flow rate at the inlets (1:1). Since the estimated uncertainty of the experimental velocity is about 3%, the obtained result can be considered very good and it demonstrates that no significant scaling effects influences the liquid mixing for low Reynolds numbers (Re<14) and the behaviour of the micro T-junction can be considered as conventional. The detailed analysis of the velocity profile evolution within the central region of the mixer has allowed to determine where the fully developed laminar profile is reached (for instance 260 mm far from the centre of the T-junction when a maximum water flow rate of 8 ml/h is considered)

A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation

We address in this paper the study of a geometric evolution, corresponding to a curvature which is non-local and singular at the origin. The curvature represents the first variation of the energy recently proposed as a variant of the standard perimeter penalization for the denoising of nonsmooth curves. To deal with such degeneracies, we first give an abstract existence and uniqueness result for viscosity solutions of non-local degenerate Hamiltonians, satisfying suitable continuity assumption with respect to Kuratowsky convergence of the level sets. This abstract setting applies to an approximated flow. Then, by the method of minimizing movements, we also build an "exact" curvature flow, and we illustrate some examples, comparing the results with the standard mean curvature flow

A second order minimality condition for the Mumford-Shah functional

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\Gamma)$, $\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.Comment: 30 page

BeppoSAX Observations of the Radio Galaxy Centaurus A

We present preliminary results from two observations of the radio galaxy Centaurus A performed by the BeppoSAX satellite. We did not detect any spectral variation of the nuclear continuum in spite of the long-term flux change (by a factor 1.3) between the two observations. At both epochs, the nuclear point-like emission was well fitted with a strongly absorbed power law with an exponential cutoff at high energies (E_cutoff>200 keV). We also observed a significant flux variation of the iron line between the two observations. The flux of the line and of the continuum changed in the opposite sense. The line is more intense at the first epoch, when the nuclear source was at the lower intensity level. The implied delay between the continuum and line variations strongly suggests that the cold material responsible for the iron line production is not located very near to the primary X-ray source. There is also evidence that the line profile changed between the two epochs, being broader and slightly blueshifted when the source was fainter. It is possible that the emission feature is a blend of cold and ionized iron lines produced in separate regions surrounding the nuclear source.Comment: 4 pages, 4 figures, accepted for publication in Advances in Space Research, proceedings of 32nd COSPAR Symposium (1998

Effect of Electron Correlation on the Bragg Reflection

We study the effect of correlation on the Bragg reflection in the 3D electron gas, the 1D Luttinger liquid, and the 1D Hubbard model in an alternating periodic potential at half-filling. In the last system, we suggest a Luttinger-liquid-type quasi-metallic state in the crossover region from the band insulator to the Mott insulator. We explain the appearance of this state in terms of the incompatibility of the Bragg reflection with the concept of Luttinger liquids.Comment: 4 pages, 3 figure

Estudo do complexo valvar atrioventricular cardíaco esquerdo em búfalos (Bubalus bubalis) da raça Jafarabadi

Atrioventricular valve complex of 30 Jafarabadi water buffaloes, adult males were studied in this research with no heart diseases. The animals were obtained from a slaughterhouse in Brazilian State of Parana. The hearts were opened at the third portion affording access to the valve complex. The complexes had its area, number and type of tendinous cords submitted to analysis. The results showed that the complex is composed by two cusps and four accessory cusps, two or three papillary muscles in which 10-25 tendinous cords fix on the cusps that face the ventricle wall. The total area of the complex was on average 38.56cm², with a minimum of 24.96cm² and a maximum of 55.54cm². Statistically, no relation between the number of cords and the cusps' area where they are inserted or with the number of papillary muscle where they originated from was observed.Foram estudados os complexos valvares atrioventricular esquerdo de 30 búfalos da raça Jafarabadi, machos e adultos, sem alterações cardíacas, provenientes de abatedouros do Estado do Paraná. Os corações foram examinados em seu terço médio para acesso ao complexo valvar, que foi submetido a estudos de área, número e tipificação de cordas tendíneas. Os resultados demonstram que este complexo é formado por duas cúspides principais e quatro cúspides acessórias, apresentam em sua formação de 2-3 músculos papilares, nos quais se inserem de 10-25 cordas tendíneas, que se fixam em cúspides voltadas para a parede do ventrículo. A área total deste complexo apresenta uma média de 38,56cm² com um mínimo de 24,96cm² e um máximo de 55,54cm². Estatisticamente não há relação entre número de cordas e a área da cúspide onde estas estão inseridas, nem com o número de músculos papilares dos quais elas provem

Taste Perception And Food Choices

OBJECTIVES: The extent to which variation in taste perception influences food preferences is, to date, controversial. Bitterness in food triggers an innate aversion that is responsible for dietary restriction in children. We investigated the association among genetic variations in bitter receptor TAS2R38 and food choices in healthy children in the Mediterranean area, to develop appropriate tools to evaluate the relation among genetic predisposition, dietary habits, and feeding disorders. The aims of the study were to get a first baseline picture of taste sensitivity in healthy adults and their children and to explore taste sensitivity in a preliminary sample of obese children and in samples affected by functional gastrointestinal diseases. METHODS: Individuals (98 children, 87 parents, 120 adults) were recruited from the general population in southern Italy. Bitterness sensitivity was assessed by means of a suprathreshold method with 6-propyl-2-thiouracil. Genomic DNA from saliva was used to genotype individuals for 3 polymorphisms of TAS2R38 receptor, A49P, A262 V, and V296I. Food intake was assessed by a food frequency questionnaire. RESULTS: Children's taste sensation differed from that of adults: we observed a higher frequency of supertasters among children even in the mother-child dyads with the same diplotypes. Among adults, supertaster status was related with proline-alanine-valine (taster allele) homozygous haplotype, whereas supertaster children were mainly heterozygous. Regarding the food choices, we found that a higher percentage of taster children avoided bitter vegetables or greens altogether compared with taster adults. Taster status was also associated with body mass index in boys. CONCLUSIONS: Greater sensitivity to 6-propyl-2-thiouracil predicts lower preferences for vegetables in children, showing an appreciable effect of the genetic predisposition on food choices. None of the obese boys was a supertaster