LCS Tool : A Computational Platform for Lagrangian Coherent Structures

We give an algorithmic introduction to Lagrangian coherent structures (LCSs) using a newly developed computational engine, LCS Tool. LCSs are most repelling, attracting and shearing material lines that form the centerpieces of observed tracer patterns in two-dimensional unsteady dynamical systems. LCS Tool implements the latest geodesic theory of LCSs for two-dimensional flows, uncovering key transport barriers in unsteady flow velocity data as explicit solutions of differential equations. After a review of the underlying theory, we explain the steps and numerical methods used by LCS Tool, and illustrate its capabilities on three unsteady fluid flow examples

Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation

In this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh-Nagumo (FHN) equations. The convective FHN equations consists of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest

Second-order analysis and numerical approximation for bang-bang bilinear control problems

We consider bilinear optimal control problems whose objective functionals do not depend on the controls. Hence, bang-bang solutions will appear. We investigate sufficient secondorder conditions for bang-bang controls, which guarantee local quadratic growth of the objective functional in L1 . In addition, we prove that for controls that are not bang-bang, no such growth can be expected. Finally, we study the finite-element discretization and prove error estimates of bang-bang controls in L1 -norms.The first author was partially supported by the Spanish Ministerio de Economía Industria y Competitividad under research projects MTM2014-57531-P and MTM2017-83185-P. The second author was partially supported by DFG under grant Wa 3626/1-1

Measure control of a semilinear parabolic equation with a nonlocal time delay

We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a nondifferentiable term with the measure norm of the control. Existence, uniqueness, and regularity of the solution of the state equation, as well as differentiability properties of the control-to-state operator are obtained. Next, we provide first order optimality conditions for local solutions. Finally, the control space is suitably discretized and we prove convergence of the solutions of the discrete problems to the solutions of the original problem. Several numerical examples are included to illustrate the theoretical results.The first two authors were partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P. The third author was supported by the collaborative research center SFB 910, TU Berlin, project B6

Numerical Stackelberg--Nash Control for the Heat Equation

This paper deals with a strategy to solve numerically control problems of theStackelberg--Nash kind for heat equations with Dirichlet boundary conditions. We assume thatwe can act on the system through several controls, respecting an order and a hierarchy: a first con-trol (the leader) is assumed to choose the policy; then, a Nash equilibrium pair, determined by thechoice of the leader and corresponding to a noncooperative multiple-objective optimization strat-egy, is found (these are the followers). Our method relies on a formulation inspired by the work ofFursikov and Imanuvilov. More precisely, we minimize over the class of admissible null controls afunctional that involves weighted integrals of the state and the control, with weights that blow upat the final time. The use of the weights is crucial to ensure the existence of the controls and theassociated state in a reasonable space. We present several mixed formulations of the problems and,then, associated mixed finite element approximations that are easy to handle. In a final step, weexhibit some numerical experiments making use of the Freefem++ package

High speed all optical networks

An inherent problem of conventional point-to-point wide area network (WAN) architectures is that they cannot translate optical transmission bandwidth into comparable user available throughput due to the limiting electronic processing speed of the switching nodes. The first solution to wavelength division multiplexing (WDM) based WAN networks that overcomes this limitation is presented. The proposed Lightnet architecture takes into account the idiosyncrasies of WDM switching/transmission leading to an efficient and pragmatic solution. The Lightnet architecture trades the ample WDM bandwidth for a reduction in the number of processing stages and a simplification of each switching stage, leading to drastically increased effective network throughputs. The principle of the Lightnet architecture is the construction and use of virtual topology networks, embedded in the original network in the wavelength domain. For this construction Lightnets utilize the new concept of lightpaths which constitute the links of the virtual topology. Lightpaths are all-optical, multihop, paths in the network that allow data to be switched through intermediate nodes using high throughput passive optical switches. The use of the virtual topologies and the associated switching design introduce a number of new ideas, which are discussed in detail

Introducing Molly: Distributed Memory Parallelization with LLVM

Programming for distributed memory machines has always been a tedious task, but necessary because compilers have not been sufficiently able to optimize for such machines themselves. Molly is an extension to the LLVM compiler toolchain that is able to distribute and reorganize workload and data if the program is organized in statically determined loop control-flows. These are represented as polyhedral integer-point sets that allow program transformations applied on them. Memory distribution and layout can be declared by the programmer as needed and the necessary asynchronous MPI communication is generated automatically. The primary motivation is to run Lattice QCD simulations on IBM Blue Gene/Q supercomputers, but since the implementation is not yet completed, this paper shows the capabilities on Conway's Game of Life

Scalable dimensioning of resilient Lambda Grids

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Critical cones for sufficient second order conditions in PDE constrained optimization

In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed, and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.The authors were partially supported by Spanish Ministerio de Economía y Competitividad under research project MTM2017-83185-P