In this thesis, we focus on the numerical study on some inverse problems and optimal control problems. In the first part, we consider some linear inverse problems with discontinuous or piecewise constant solutions. We use the total variation to regularize these inverse problems and then the finite element technique to discretize the regularized problems. These discretized problems are treated from the saddle-point perspective; and some primal-dual numerical schemes are proposed. We intensively investigate the convergence of these primal-dual type schemes, establishing the global convergence and estimating their worst-case convergence rates measured by the iteration complexity. We test these schemes by some experiments and verify their efficiency numerically. In the second part, we consider the finite difference and finite element discretization for an optimal control problem which is governed by time fractional diffusion equation. The prior error estimate of the discretized model is analyzed, and a projection gradient method is applied for iteratively solving the fully discretized surrogate. Some numerical experiments are conducted to verify the efficiency of the proposed method. Overall speaking, the thesis has been mainly inspired by some most recent advances developed in optimization community, especially in the area of operator splitting methods for convex programming; and it can be regarded as a combination of some contemporary optimization techniques with some relatively mature inverse and control problems. Keywords: Total variation minimization, linear inverse problem, saddle-point problem, finite element method, primal-dual method, convergence rate, optimal control problem, time fractional diffusion equation, projection gradient method.

[pt] Nesta dissertação, discutiremos o método de equações diferenciais de Wormald, que possui muitas aplicações recentes em Combinatória. Esse método explora a interação entre a matemática discreta e contínua e pode ser usado para provar concentração em uma grande quantidade de processos aleatórios discretos. Em particular, estudaremos o processo livre de H e o algoritmo guloso aleatório para gerar conjuntos independentes em hipergrafos. Esses processos tem sido amplamente estudados nos últimos anos, culminando com o recente grande avanço de Tom Bohman e Patrick Bennett em 2016, que obtiveram uma cota inferior para hipergrafos com certas condições de densidade. Nós não só reproduzimos sua demonstração mas também obtemos um resultado mais forte (expandindo seu resultado para hipergrafos mais esparsos) e analisamos o caso de hipergrafos lineares, com o intuito de progredir rumo a uma conjectura de Johnson e Pinto sobre o processo livre de Q2 no hipercubo Qd. / [en] In this dissertation, we will discuss Wormald s differential equations method, which has recently had many intriguing applications in Combinatorics. This method explores the interplay between discrete and continuous mathematics and it can be used to prove concentration in a number of discrete random processes. In particular, we will discuss the H-free process and the random greedy algorithm to obtain independent sets in hypergraphs. These processes had been extensively studied through the past few years, culminating in the recent breakthrough of Tom Bohman and Patrick Bennett in 2016, who obtained a lower bound for hypergraphs with certain density conditions. We not only reproduce the proof given by them but also obtain a stronger result (expanding their result to sparser hypergraphs) and we analyze the case of linear hypergraphs, in order to make progress towards a conjecture by Johnson and Pinto concerning the Q2-free process in the hypercube Qd.