Optimization and control theory are well developed techniques to quantize, model, understand and optimize real world systems and they have been widely used in engineering, economics, and science. In this thesis, we focus on applications in biomedical systems ranging from cells to microbial communities, and to something as complex as the human body. The first problem we consider is that of medication dosage control for drugs delivered intravenously to the patient. We focus specifically on a blood thinner (called bivalirudin) used in the post cardiac surgery Intensive Care Unit (ICU). We develop two approaches (a model-free and a model-based one) that predict the effect of bivalirudin. After obtaining the model and its best fit parameters by solving a non-linear optimization problem, we develop automatic dosage controllers that adaptively regulate its effect to desired levels. Our algorithms are validated using actual data from a large hospital in the Boston area. In the second problem, we introduce a cellular objective function inference mechanism in metabolic networks. We develop an inverse optimization method, called InvFBA (Inverse Flux Balance Analysis), to infer the objective functions of growing cells by using their reaction fluxes. InvFBA can be seen as an inverse version of FBA (Flux Balance Analysis) which predicts the distribution of the cell's reaction fluxes by using a hypothetical objective function. The objective functions can be linear, quadratic and non-parametric. The efficiency of the InvFBA approach matches the structure of the FBA and ensures scalability to large networks and optimality of the solution. After testing our algorithm on simulated E. coli data and time-dependent S. oneidensis fluxes inferred from gene expression data, we apply our inverse approach to flux measurements in long-term evolved E. coli strains, revealing objective functions that provide insight into metabolic adaptation trajectories. In the final problem in this thesis, we formulate a novel resource allocation problem in microbial ecosystems. We consider a given number of microbial species living symbiotically in a community and a list of all metabolic reactions present in the community, expressed in terms of the metabolite proportions involved in each reaction. We are interested in allocating reactions to organisms so that each organism maintains a minimal level of growth and the community optimizes certain objectives, such as maximizing growth and/or the uptake of specific compounds from the common environment. We leverage tools from Flux Balance Analysis (FBA) and formulate the problem as a mixed integer linear programming problem. We test our method in a toy model involving two organisms that can only survive through cross-feeding, demonstrating that the method can recover this interaction. We also test the method in a community of two simplified bacteria described in terms of their core, simplified metabolic network. We demonstrate that the method can obtain syntrophic cross-feeding species that would be very difficult to design manually.

Advances in wireless technology and the growing popularity of multimedia applications have brought about a need for energy efficient and cost effective portable supercomputers capable of delivering performance beyond the capabilities of current microprocessors and DSP chips. The SIMPil architecture currently being developed at Georgia Institute of Technology is a promising candidate for this task. In order to develop applications for SIMPil, a high level language and an optimizing compiler for the language are essential. However, with the recent trend of interconnect latency becoming a major bottleneck on computer systems, optimizations focusing on reducing latency are becoming more important, especially with SIMPil, as it is highly scalable. The compiler tracks the path of data through the network and buffers data in each processor to eliminate redundant communication. With a buffer size of 5, the compiler was able to eliminate 96 percent of the redundant communication for a 9x9 convolution and 8x8 DCT algorithms. With 5x5 convolution, only 89 percent elimination was observed. In terms of performance, 106 percent speedup was observed with 9x9 convolution at buffer size of 5 while 5x5 convolution and 8x8 DCT which have a much lower number of communication showed only 101 percent speedup.

This thesis is concerned with the development of novel numerical methods for solving nondifferentiable convex optimization problems in real Hilbert spaces and with the investigation of their asymptotic behavior. To this end, we are also making use of monotone operator theory as some of the provided algorithms are originally designed to solve monotone inclusion problems. After introducing basic notations and preliminary results in convex analysis, we derive two numerical methods based on different smoothing strategies for solving nondifferentiable convex optimization problems. The first approach, known as the double smoothing technique, solves the optimization problem with some given a priori accuracy by applying two regularizations to its conjugate dual problem. A special fast gradient method then solves the regularized dual problem such that an approximate primal solution can be reconstructed from it. The second approach affects the primal optimization problem directly by applying a single regularization to it and is capable of using variable smoothing parameters which lead to a more accurate approximation of the original problem as the iteration counter increases. We then derive and investigate different primal-dual methods in real Hilbert spaces. In general, one considerable advantage of primal-dual algorithms is that they are providing a complete splitting philosophy in that the resolvents, which arise in the iterative process, are only taken separately from each maximally monotone operator occurring in the problem description. We firstly analyze the forward-backward-forward algorithm of Combettes and Pesquet in terms of its convergence rate for the objective of a nondifferentiable convex optimization problem. Additionally, we propose accelerations of this method under the additional assumption that certain monotone operators occurring in the problem formulation are strongly monotone. Subsequently, we derive two Douglas–Rachford type primal-dual methods for solving monotone inclusion problems involving finite sums of linearly composed parallel sum type monotone operators. To prove their asymptotic convergence, we use a common product Hilbert space strategy by reformulating the corresponding inclusion problem reasonably such that the Douglas–Rachford algorithm can be applied to it. Finally, we propose two primal-dual algorithms relying on forward-backward and forward-backward-forward approaches for solving monotone inclusion problems involving parallel sums of linearly composed monotone operators. The last part of this thesis deals with different numerical experiments where we intend to compare our methods against algorithms from the literature. The problems which arise in this part are manifold and they reflect the importance of this field of research as convex optimization problems appear in lots of applications of interest.

This thesis is concerned with the development of novel numerical methods for solving nondifferentiable convex optimization problems in real Hilbert spaces and with the investigation of their asymptotic behavior. To this end, we are also making use of monotone operator theory as some of the provided algorithms are originally designed to solve monotone inclusion problems. After introducing basic notations and preliminary results in convex analysis, we derive two numerical methods based on different smoothing strategies for solving nondifferentiable convex optimization problems. The first approach, known as the double smoothing technique, solves the optimization problem with some given a priori accuracy by applying two regularizations to its conjugate dual problem. A special fast gradient method then solves the regularized dual problem such that an approximate primal solution can be reconstructed from it. The second approach affects the primal optimization problem directly by applying a single regularization to it and is capable of using variable smoothing parameters which lead to a more accurate approximation of the original problem as the iteration counter increases. We then derive and investigate different primal-dual methods in real Hilbert spaces. In general, one considerable advantage of primal-dual algorithms is that they are providing a complete splitting philosophy in that the resolvents, which arise in the iterative process, are only taken separately from each maximally monotone operator occurring in the problem description. We firstly analyze the forward-backward-forward algorithm of Combettes and Pesquet in terms of its convergence rate for the objective of a nondifferentiable convex optimization problem. Additionally, we propose accelerations of this method under the additional assumption that certain monotone operators occurring in the problem formulation are strongly monotone. Subsequently, we derive two Douglas–Rachford type primal-dual methods for solving monotone inclusion problems involving finite sums of linearly composed parallel sum type monotone operators. To prove their asymptotic convergence, we use a common product Hilbert space strategy by reformulating the corresponding inclusion problem reasonably such that the Douglas–Rachford algorithm can be applied to it. Finally, we propose two primal-dual algorithms relying on forward-backward and forward-backward-forward approaches for solving monotone inclusion problems involving parallel sums of linearly composed monotone operators. The last part of this thesis deals with different numerical experiments where we intend to compare our methods against algorithms from the literature. The problems which arise in this part are manifold and they reflect the importance of this field of research as convex optimization problems appear in lots of applications of interest.