Numerical reconstruction of inverse problems for partial differential equations

Abstract

This thesis is devoted to the study of inverse problems in a numerical approach. We will focus on problems applied to engineering and science, specifically we will study two problems that we will state below: In the first part of the thesis, we will study the problem of ore extraction in mines. For underground mine, the current usual technique for ore extraction is block caving, which generates and induces seismic activity in the mine. To understand the block caving method is one of the most challenging problems in underground mining. This method relies on gravity to break and transport enormous amounts of ore and waste. The state of the art in damage models is not able to represent the real effect of the mining in the rock mass since for example the damage appears in the bottom of the domain under consideration and with this is not possible to recover the subsidence seen in the mine. In this work we study the effects that occur on rock mass as mining activity develops. In this thesis we present an extended and improved damage model that recovers the effect the block caving based on the gradient damage model proposed by Pham and Marigo in 2010 we will consider a variational formulation where gradient damage models appear as an elliptical approach to the problem of variational fracture mechanics and the damage criterion depends on the spherical and deviatoric part of the stress tensor. We show interesting numerical tests and simulations producing realistic damage behaviors in the domain, improving all previous approaches to this issue. In the second part of this thesis, we will study the numerical reconstruction of the CGO solutions for a conductivity system. We study numerical methods for computing numerically the CGO solutions to the conductivity system $div\left(\sigma \cdot \nabla U\right)=0$ in $\mathbb{R}^2$ for symmetric, positive definite matrix functions $\sigma$. The way to compute those solutions is to use solutions to the Beltrami system. In this work, we first prove the existence of CGO solution and then use a numerical strategy based on the method introduced by Huhntanem and Perämäki in 2012 for the Beltrami equation. Numerical experiments are considered to show the influence of coupled equations.