Stochastic optimization applied to scheduling mining problems

Tesis para optar al grado de Doctor en Ciencias de la Ingeniería, Mención Modelación MatemáticaEn esta tesis, abordaremos el Problema de la Plani ficación de la Producción en una Mina Subterránea para una mina explotada mediante el método de hundimientos de bloques y/o paneles, a través de la formulación de un programa de optimización lineal estocástica multietapa. El objetivo es acrecentar la información considerada por los modelos existentes incorporando el riesgo sísmico, además de los tradicionalmente considerados como la incertidumbre nanciera y geológica. La formulación de un modelo que considere múltiples fuentes de incertidumbre implica la consideración de muchísimas variables que tornan imposible la resolución directa de este mediante métodos convencionales. Aún recurriendo a la clusterización de bloques, como se hace en este trabajo, el problema de optimización formulado continúa intratable. Para abordar la resolución del modelo propuesto utilizamos el bien conocido principio 'Divide y vencerás', más específicamente se echa mano a la Relajación Lagrangiana de restricciones que complican el problema. En este trabajo se relajan las restricciones de no anticipatividad, que relacionan los escenarios, lo que nos permite dividir el problema maestro en tantos subproblemas como escenarios, y se propone una metodología para obtener soluciones factibles para el problema maestro empleando las soluciones obtenidas para los subproblemas. Se estudia la relación de la producción y la actividad sísmica, estableciendo un modelo empírico simple, que nos ayuda a entender y a estimar la intensidad de la micro sismicidad inducida por la minería. Este análisis nos posibilita estudiar y cuanti ficar los benefi cios del fracturamiento de rocas, mediante explosivos y preacondicionamiento hidráulico. Se relaciona el preacondicionamiento y las tasas de extracción y el impacto en el Valor Presente Neto. Para la realización de esta tesis, se consideraron datos reales proporcionados por la mina Codelco Chile División El Teniente, la mina de cobre más grande del mundo. Esta mina es explotada mediante hundimiento de bloques y paneles.Beca Doctorado Nacional, Conicyt 2014, N° 2114106

Estimativas de magnitudes da dinâmica quântica envolvendo dimensões de Hausdorff e de empacotamento

In this work we approach the nonrelativistic quantum mechanics from the point of view of mathematical physics; we present estimates of some dynamical quantities that are used to describe the return probability to the initial state. Such estimates are related to Hausdorff and packing dimensions of the relevant spectral measures. After defining such dimensions we discuss some tools to characterize the quantum dynamics of a particle in one dimension. Then we summarize some standard facts from the literature and the quantities we are interested in. Finally, as an application, estimates of some dynamical quantities based on concepts from statistical mechanics are presented.Financiadora de Estudos e ProjetosNa abordagem da mecânica quântica não-relativística e do ponto de vista da física matemática, neste trabalho apresentamos estimativas de quantidades que descrevem o crescimento de magnitudes físicas que caracterizam a evolução temporal de um sistema quântico, tal como o momento associado a probabilidade de retorno. Estas estimativas envolvem as dimensões de Hausdorff e de empacotamento da medida espectral associada ao grupo unitário que descreve a evolução temporal de um sistema. Em primeiro lugar, introduzimos as medidas de Hausdorff e empacotamento de uma medida. A continuacão, definimos as ferramentas que nos permitirão caracterizar matematicamente a dinâmica de uma partícula em uma dimensão. Em seguida apresentamos um resumo que inclui resultados da teoria de operadores auto-adjuntos e grupos unitários de evolução, tais como o teorema espectral e o teorema de Stone, os quais serão relevantes em nosso estudo. Após isso, apresentamos as quantidades que desejamos estimar e suas estimativas. Finalmente, o leitor encontrará uma aplicação destas estimativas, que consiste na estimativa de quantidades definidas a partir de conceitos da mecânica estatística

On

Based on the superstatistical formalism, we propose a new probability distribution. By means of a transformation of the q-exponential, we construct the q-Pareto distribution by Beck formalism. Some properties of this distribution are studied which show that the q-Pareto is a proper generalization of the Pareto distribution. By applying this new q-distribution to larger earthquakes, we show its goodness-of-fit to model great earthquakes recorded in global earthquake catalogs. It is noted that a small change in the entropic $$q_w$$ index associated with the moment magnitude implies an exponential change in the corresponding $$q_0$$ index associated with the seismic moments. Our results suggest that q-Pareto distribution is a better candidate than Pareto distribution when modeling extreme values

Statistical Modeling of the Seismic Moments via Mathai Distribution

Mathai’s pathway model is playing an increasingly prominent role in statistical distributions. As a generalization of a great variety of distributions, the pathway model allows the studying of several non-linear dynamics of complex systems. Here, we construct a model, called the Pareto–Mathai distribution, using the fact that the earthquakes’ magnitudes of full catalogues are well-modeled by a Mathai distribution. The Pareto–Mathai distribution is used to study artificially induced microseisms in the mining industry. The fitting of a distribution for entire range of magnitudes allow us to calculate the completeness magnitude (Mc). Mathematical properties of the new distribution are studied. In addition, applying this model to data recorded at a Chilean mine, the magnitude Mc is estimated for several mine sectors and also the entire mine

Full-waveform inversion based on generalized Rényi entropy using patched Green's function techniques.

The estimation of physical parameters from data analyses is a crucial process for the description and modeling of many complex systems. Based on Rényi α-Gaussian distribution and patched Green's function (PGF) techniques, we propose a robust framework for data inversion using a wave-equation based methodology named full-waveform inversion (FWI). From the assumption that the residual seismic data (the difference between the modeled and observed data) obeys the Rényi α-Gaussian probability distribution, we introduce an outlier-resistant criterion to deal with erratic measures in the FWI context, in which the classical FWI based on l2-norm is a particular case. The new misfit function arises from the probabilistic maximum-likelihood method associated with the α-Gaussian distribution. The PGF technique works on the forward modeling process by dividing the computational domain into outside target area and target area, where the wave equation is solved only once on the outside target (before FWI). During the FWI processing, Green's functions related only to the target area are computed instead of the entire computational domain, saving computational efforts. We show the effectiveness of our proposed approach by considering two distinct realistic P-wave velocity models, in which the first one is inspired in the Kwanza Basin in Angola and the second in a region of great economic interest in the Brazilian pre-salt field. We call our proposal by the abbreviation α-PGF-FWI. The results reveal that the α-PGF-FWI is robust against additive Gaussian noise and non-Gaussian noise with outliers in the limit α → 2/3, being α the Rényi entropic index

Hydraulic fracturing assessment on seismic hazard by Tsallis statistics

Nowadays, because of the intensive use of hydraulic fracturing (HF) in mining, the study of its impact on mining extraction becomes crucial. Here we use the Sotolongo-Costa–Posadas (SCP) model, which is based on Tsallis formalism, to assess the impact of HF on the microearthquakes’ magnitudes and interevent times in a Chilean underground mine. In this regard, we analyse the seismic hazard at regions with HF and HF-free, which is an important issue for workers’ safety and continuity of the mining operations. The results reveal that the HF diminishes the value of the q entropic index associated with the SCP model, which implies that the magnitude and the autocorrelation of the microearthquakes decrease