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Completed
Calculus of Variations, Optimization, and Applications
FunderMinistry of Science, Education and Youth of the Republic of Croatia
Funder Contribution €6,000.00
Duration01 Jan 2016 - 31 Dec 2017
Funded underProgram of scientific-technological cooperation between Republic of Croatia and Republic of Serbia
Project codeRHRS3
Open Access mandatePublications: NoResearch data: No
Identifiers
Project timeline100%
Description
Through our collaboration to date, common scientific interests have been identified, and the planned cooperation on this project would be realized through joint work on the following topics: a) Optimal control, optimal design, generalized solutions, and homogenization. Within this topic, we plan to study the linear-quadratic problem in the framework of Friedrichs systems, with an emphasis on applications to specific initial-boundary value problems. Furthermore, we would investigate a two-phase optimal design problem with multiple states for the stationary diffusion equation, where the goal is to minimize a weighted sum of energy functionals over all mixtures of two isotropic materials. The objective here is to explicitly compute the solution of the relaxed problem, at least in the spherically symmetric case. It would also be interesting to examine whether linearized elasticity can be incorporated into the framework of Friedrichs systems. In addition, research is planned on the characterization of H-distributions, with special attention to the relationship between the support of H-distributions (i.e., H-measures) and the wave front set. b) Optimization problems of special structure. We will consider a class of optimization problems with a particular structure in which the objective function is expressed as a sum of a large number of so-called loss functions. We are especially interested in problems where direct minimization of the objective function, defined over the entire dataset (due to its size), is not feasible. In order to reduce the cost of the iterative optimization procedure, we will consider methods in which the number of local loss functions varies at each iteration. In this context, the objective function is approximated based on sample size, with second-order methods being of particular interest. With the aim of reducing the dimensionality of large datasets, incremental and adaptive methods based on spectral relaxation will be developed to find approximately globally optimal partitions.
Partners (1)
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Through our collaboration to date, common scientific interests have been identified, and the planned cooperation on this project would be realized through joint work on the following topics: a) Optimal control, optimal design, generalized solutions, and homogenization. Within this topic, we plan to study the linear-quadratic problem in the framework of Friedrichs systems, with an emphasis on applications to specific initial-boundary value problems. Furthermore, we would investigate a two-phase optimal design problem with multiple states for the stationary diffusion equation, where the goal is to minimize a weighted sum of energy functionals over all mixtures of two isotropic materials. The objective here is to explicitly compute the solution of the relaxed problem, at least in the spherically symmetric case. It would also be interesting to examine whether linearized elasticity can be incorporated into the framework of Friedrichs systems. In addition, research is planned on the characterization of H-distributions, with special attention to the relationship between the support of H-distributions (i.e., H-measures) and the wave front set. b) Optimization problems of special structure. We will consider a class of optimization problems with a particular structure in which the objective function is expressed as a sum of a large number of so-called loss functions. We are especially interested in problems where direct minimization of the objective function, defined over the entire dataset (due to its size), is not feasible. In order to reduce the cost of the iterative optimization procedure, we will consider methods in which the number of local loss functions varies at each iteration. In this context, the objective function is approximated based on sample size, with second-order methods being of particular interest. With the aim of reducing the dimensionality of large datasets, incremental and adaptive methods based on spectral relaxation will be developed to find approximately globally optimal partitions.
