An Efficient Algorithm for Video Super-Resolution Based On a Sequential Model

In this work, we propose a novel procedure for video super-resolution, that is the recovery of a sequence of high-resolution images from its low-resolution counterpart. Our approach is based on a "sequential" model (i.e., each high-resolution frame is supposed to be a displaced version of the preceding one) and considers the use of sparsity-enforcing priors. Both the recovery of the high-resolution images and the motion fields relating them is tackled. This leads to a large-dimensional, non-convex and non-smooth problem. We propose an algorithmic framework to address the latter. Our approach relies on fast gradient evaluation methods and modern optimization techniques for non-differentiable/non-convex problems. Unlike some other previous works, we show that there exists a provably-convergent method with a complexity linear in the problem dimensions. We assess the proposed optimization method on {several video benchmarks and emphasize its good performance with respect to the state of the art.}Comment: 37 pages, SIAM Journal on Imaging Sciences, 201

Phase Retrieval via Matrix Completion

This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we introduce some theory showing that one can design very simple structured illumination patterns such that three diffracted figures uniquely determine the phase of the object we wish to recover

A Stochastic Majorize-Minimize Subspace Algorithm for Online Penalized Least Squares Estimation

Stochastic approximation techniques play an important role in solving many problems encountered in machine learning or adaptive signal processing. In these contexts, the statistics of the data are often unknown a priori or their direct computation is too intensive, and they have thus to be estimated online from the observed signals. For batch optimization of an objective function being the sum of a data fidelity term and a penalization (e.g. a sparsity promoting function), Majorize-Minimize (MM) methods have recently attracted much interest since they are fast, highly flexible, and effective in ensuring convergence. The goal of this paper is to show how these methods can be successfully extended to the case when the data fidelity term corresponds to a least squares criterion and the cost function is replaced by a sequence of stochastic approximations of it. In this context, we propose an online version of an MM subspace algorithm and we study its convergence by using suitable probabilistic tools. Simulation results illustrate the good practical performance of the proposed algorithm associated with a memory gradient subspace, when applied to both non-adaptive and adaptive filter identification problems

Multi-Contact Postures Computation on Manifolds

International audienceWe propose a framework to generate static robot configurations satisfying a set of physical and geometrical constraints. This is done by formulating nonlinear constrained optimization problems over non-Euclidean manifolds and solving them. To do so, we present a new sequential quadratic programming (SQP) solver working natively on general manifolds, and propose an interface to easily formulate the problems, with the tedious and error-prone work automated for the user. We also introduce several new types of constraints for having more complex contacts or working on forces/torques. Our approach allows an elegant mathematical description of the constraints and we exemplify it through formulation and computation examples in complex scenarios with humanoid robots

An elastic primal active-set method for structured QPs

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Distribution network optimisation for an active network management system

The connection of Distributed Generators (DGs) to a distribution network causes technical concerns for Distribution Network Operators (DNOs) which include power flow management, loss increase and voltage management problems. An Active Network Management System can provide monitoring and control of the distribution network as well as providing the infrastructure and technology for full integration of DGs into the distribution network. The Optimal Power Flow (OPF) method is a valuable tool in providing optimal control solutions for active network management system applications. The research presented here has concentrated on the development of a multi-objective OPF to provide power flow management, voltage control solutions and network optimisation strategies. The OPF has been shown to provide accurate solutions for variety of network topologies. It is possible to apply time-series of load and generation data to the OPF in a loop, generating optimal network solutions to maintain the network within thermal and voltage limits. The OPF incorporates not only the DG real power output maximisation, but also network loss minimisation as well as minimising the dispatch of DG reactive power. This investigation uses a direct Interior Point (IP) method as the solution methodology which is speed efficient and converges in polynomial time. Each objective function has been assigned a weighting factor, making it possible to favour one objective function and ignore the others. Contributions to enhance the performance of the IP OPF algorithm include a new generic barrier parameter formulation and a new swing bus formulation to model energy export/import in the main optimisation routine. A Terminal Voltage Regulator Mode (TVRM) and Power Factor Regulation Mode (PFRM) for DG were incorporated in the main optimisation routine. The main motivation is to compare these two decentralised DG control methods in terms of the achieving the maximum DG real power generation. The DG operation methods of TVRM and PFRM are compared with the optimisation results obtained from centralised dispatch in terms of the DG capacity achieved as it produces the optimum overall network solution. A suitable value of the droop and local voltage regulator dead-bands were determined for particular DGs. Furthermore, the effect of these decentralised DG control methods on distribution network losses are considered in a measure to assess the financial implications from a DNO's perspective

Computational methods for geochemical modelling: applications to carbon dioxide sequestration

Geochemical modelling is fundamental for solving many environmental problems, and specially useful for modelling carbon storage into deep saline aquifers. This is because the injected greenhouse gas perturbs the reservoir, causing the subsurface fluid to become acidic, and consequently increasing its reactivity with the formation rock. Assessment of the long term fate of carbon dioxide, therefore, requires accurate calculations of the geochemical processes that occur underground. For this, it is important to take into account the major water-gas-rock effects that play important roles during the gas storage and migration. These reactive processes can in general be formulated in terms of chemical equilibrium or chemical kinetics models. This work proposes novel numerical methods for the solution of multiphase chemical equilibrium and kinetics problems. Instead of adapting or improving traditional algorithms in the geochemical modelling literature, this work adopts an approach of abstracting the underlying mathematics from the chemical problems, and investigating suitable, modern and efficient methods for them in the mathematical literature. This is the case, for example, of the adaptation of an interior-point minimisation algorithm for the calculation of chemical equilibrium, in which the Gibbs energy of the system is minimised. The methods were developed for integration into reactive transport simulators, requiring them to be accurate, robust and efficient. These features are demonstrated in the manuscript. All the methods developed were applied to problems relevant to carbon sequestration in saline aquifers. Their accuracy was assessed by comparing, for example, calculations of pH and CO2 solubility in brines against recent experimental data. Kinetic modelling of carbon dioxide injection into carbonate and sandstone saline aquifers was performed to demonstrate the importance of accounting for the water-gas-rock effects when simulating carbon dioxide sequestration. The results demonstrated that carbonate rocks, for example, increase the potential of the subsurface fluid to dissolve even more mobile CO2.Open Acces

Computational methods for solving optimal industrial process control problems

In this thesis, we develop new computational methods for three classes of dynamic optimization problems: (i) A parameter identification problem for a general nonlinear time-delay system; (ii) an optimal control problem involving systems with both input and output delays, and subject to continuous inequality state constraints; and (iii) a max-min optimal control problem arising in gradient elution chromatography.In the first problem, we consider a parameter identification problem involving a general nonlinear time-delay system, where the unknown time delays and system parameters are to be identified. This problem is posed as a dynamic optimization problem, where its cost function is to measure the discrepancy between predicted output and observed system output. The aim is to find unknown time-delays and system parameters such that the cost function is minimized. We develop a gradient-based computational method for solving this dynamic optimization problem. We show that the gradients of the cost function with respect to these unknown parameters can be obtained via solving a set of auxiliary time-delay differential systems from t = 0 to t = T. On this basis, the parameter identification problem can be solved as a nonlinear optimization problem and existing optimization techniques can be used. Two numerical examples are solved using the proposed computational method. Simulation results show that the proposed computational method is highly effective. In particular, the convergence is very fast even when the initial guess of the parameter values is far away from the optimal values.Unlike the first problem, in the second problem, we consider a time delay identification problem, where the input function for the nonlinear time-delay system is piecewise-constant. We assume that the time-delays—one involving the state variables and the other involving the input variables—are unknown and need to be estimated using experimental data. We also formulate the problem of estimating the unknown delays as a nonlinear optimization problem in which the cost function measures the least-squares error between predicted output and measured system output. This estimation problem can be viewed as a switched system optimal control problem with time-delays. We show that the gradient of the cost function with respect to the unknown state delay can be obtained via solving a auxiliary time-delay differential system. Furthermore, the gradient of the cost function with respect to the unknown input delay can be obtained via solving an auxiliary time-delay differential system with jump conditions at the delayed control switching time points. On this basis, we develop a heuristic computational algorithm for solving this problem using gradient based optimization algorithms. Time-delays in two industrial processes are estimated using the proposed computational method. Simulation results show that the proposed computational method is highly effective.For the third problem, we consider a general optimal control problem governed by a system with input and output delays, and subject to continuous inequality constraints on the state and control. We focus on developing an effective computational method for solving this constrained time delay optimal control problem. For this, the control parameterization technique is used to approximate the time planning horizon [0, T] into N subintervals. Then, the control is approximated by a piecewise constant function with possible discontinuities at the pre-assigned partition points, which are also called the switching time points. The heights of the piecewise constant function are decision variables which are to be chosen such that a given cost function is minimized. For the continuous inequality constraints on the state, we construct approximating smooth functions in integral form. Then, the summation of these approximating smooth functions in integral form, which is called the constraint violation, is appended to the cost function to form a new augmented cost function. In this way, we obtain a sequence of approximate optimization problems subject to only boundedness constraints on the decision variables. Then, the gradient of the augmented cost function is derived. On this basis, we develop an effective computational method for solving the time-delay optimal control problem with continuous inequality constraints on the state and control via solving a sequence of approximate optimization problems, each of which can be solved as a nonlinear optimization problem by using existing gradient-based optimization techniques. This proposed method is then used to solve a practical optimal control problem arising in the study of a real evaporation process. The results obtained are highly satisfactory, showing that the proposed method is highly effective.The fourth problem that we consider is a max-min optimal control problem arising in the study of gradient elution chromatography, where the manipulative variables in the chromatographic process are to be chosen such that the separation efficiency is maximized. This problem has three non-standard characteristics: (i) The objective function is nonsmooth; (ii) each state variable is defined over a different time horizon; and (iii) the order of the final times for the state variable, the so-called retention times, are not fixed. To solve this problem, we first introduce a set of auxiliary decision variables to govern the ordering of the retention times. The integer constraints on these auxiliary decision variables are approximated by continuous boundedness constraints. Then, we approximate the control by a piecewise constant function, and apply a novel time-scaling transformation to map the retention times and control switching times to fixed points in a new time horizon. The retention times and control switching times become decision variables in the new time horizon. In addition, the max-min objective function is approximated by a minimization problem subject to an additional constraint. On this basis, the optimal control problem is reduced to an approximate nonlinear optimization problem subject to smooth constraints, which is then solved using a recently developed exact penalty function method. Numerical results obtained show that this approach is highly effective.Finally, some concluding remarks and suggestions for further study are made in the conclusion chapter