Spike controls for elliptic and parabolic PDEs

We analyze the use of measures of the minimal norm to control elliptic and parabolic equations. We prove the sparsity of the optimal control. In the parabolic case, we prove that the solution of the optimization problem is a Borel measure supported in a set of Lebesgue measure zero. In both cases, the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations. We also analyze the corresponding dual problem

Optimality Conditions for Semilinear Parabolic Equations with Controls in Leading Term

An optimal control problem for semilinear parabolic partial differential equations is considered. The control variable appears in the leading term of the equation. Necessary conditions for optimal controls are established by the method of homogenizing spike variation. Results for problems with state constraints are also stated.Comment: 22 apges, 1 figure

Pontryagin´s principle for state-constrained boundary control problems of semilinear parabolic equations

This paper deals with state-constrained optimal control problems governed by semilinear parabolic equations. We establish a minimum principle of Pontryagin's type. To deal with the state constraints, we introduce a penalty problem by using Ekeland's principle. The key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are de ned. Conditions for normality of optimality conditions are given

Integration of a big data emerging on large sparse simulation and its application on green computing platform

The process of analyzing large data and verifying a big data set are a challenge for understanding the fundamental concept behind it. Many big data analysis techniques suffer from the poor scalability, variation inequality, instability, lower convergence, and weak accuracy of the large-scale numerical algorithms. Due to these limitations, a wider opportunity for numerical analysts to develop the efficiency and novel parallel algorithms has emerged. Big data analytics plays an important role in the field of sciences and engineering for extracting patterns, trends, actionable information from large sets of data and improving strategies for making a decision. A large data set consists of a large-scale data collection via sensor network, transformation from signal to digital images, high resolution of a sensing system, industry forecasts, existing customer records to predict trends and prepare for new demand. This paper proposes three types of big data analytics in accordance to the analytics requirement involving a large-scale numerical simulation and mathematical modeling for solving a complex problem. First is a big data analytics for theory and fundamental of nanotechnology numerical simulation. Second, big data analytics for enhancing the digital images in 3D visualization, performance analysis of embedded system based on the large sparse data sets generated by the device. Lastly, extraction of patterns from the electroencephalogram (EEG) data set for detecting the horizontal-vertical eye movements. Thus, the process of examining a big data analytics is to investigate the behavior of hidden patterns, unknown correlations, identify anomalies, and discover structure inside unstructured data and extracting the essence, trend prediction, multi-dimensional visualization and real-time observation using the mathematical model. Parallel algorithms, mesh generation, domain-function decomposition approaches, inter-node communication design, mapping the subdomain, numerical analysis and parallel performance evaluations (PPE) are the processes of the big data analytics implementation. The superior of parallel numerical methods such as AGE, Brian and IADE were proven for solving a large sparse model on green computing by utilizing the obsolete computers, the old generation servers and outdated hardware, a distributed virtual memory and multi-processors. The integration of low-cost communication of message passing software and green computing platform is capable of increasing the PPE up to 60% when compared to the limited memory of a single processor. As a conclusion, large-scale numerical algorithms with great performance in scalability, equality, stability, convergence, and accuracy are important features in analyzing big data simulation

Integration of a big data emerging on large sparse simulation and its application on green computing platform

The process of analyzing large data and verifying a big data set are a challenge for understanding the fundamental concept behind it. Many big data analysis techniques suffer from the poor scalability, variation inequality, instability, lower convergence, and weak accuracy of the large-scale numerical algorithms. Due to these limitations, a wider opportunity for numerical analysts to develop the efficiency and novel parallel algorithms has emerged. Big data analytics plays an important role in the field of sciences and engineering for extracting patterns, trends, actionable information from large sets of data and improving strategies for making a decision. A large data set consists of a large-scale data collection via sensor network, transformation from signal to digital images, high resolution of a sensing system, industry forecasts, existing customer records to predict trends and prepare for new demand. This paper proposes three types of big data analytics in accordance to the analytics requirement involving a large-scale numerical simulation and mathematical modeling for solving a complex problem. First is a big data analytics for theory and fundamental of nanotechnology numerical simulation. Second, big data analytics for enhancing the digital images in 3D visualization, performance analysis of embedded system based on the large sparse data sets generated by the device. Lastly, extraction of patterns from the electroencephalogram (EEG) data set for detecting the horizontal-vertical eye movements. Thus, the process of examining a big data analytics is to investigate the behavior of hidden patterns, unknown correlations, identify anomalies, and discover structure inside unstructured data and extracting the essence, trend prediction, multi-dimensional visualization and real-time observation using the mathematical model. Parallel algorithms, mesh generation, domain-function decomposition approaches, inter-node communication design, mapping the subdomain, numerical analysis and parallel performance evaluations (PPE) are the processes of the big data analytics implementation. The superior of parallel numerical methods such as AGE, Brian and IADE were proven for solving a large sparse model on green computing by utilizing the obsolete computers, the old generation servers and outdated hardware, a distributed virtual memory and multi-processors. The integration of low-cost communication of message passing software and green computing platform is capable of increasing the PPE up to 60% when compared to the limited memory of a single processor. As a conclusion, large-scale numerical algorithms with great performance in scalability, equality, stability, convergence, and accuracy are important features in analyzing big data simulation

Averaged control

We analyze the problem of controlling parameter-dependent systems. We introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter. First we consider the problem of controllability for linear finite-dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition, in the spirit of the classical rank condition for linear control systems, but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action. We also describe some open problems and directions of possible research, in particular on the average controllability of evolution partial differential equations. In this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system