Approximation of elliptic control problems in measure spaces with sparse solutions

Optimal control problems in measure spaces governed by elliptic equations are considered for distributed and Neumann boundary control, which are known to promote sparse solutions. Optimality conditions are derived and some of the structural properties of their solutions, in particular sparsity, are discussed. A framework for their approximation is proposed which is efficient for numerical computations and for which we prove convergence and provide error estimates.This author was supported by Spanish Ministerio de Ciencia e Innovación under projects MTM2008-04206 and “Ingenio Mathematica (i-MATH)” CSD2006-00032 (Consolider Ingenio 2010)

GC Insights: Lessons from participatory water quality research in the upper Santa River basin, Peru

Here we share four key lessons from an inter-disciplinary project (Nuestro Rio) that gathered community perspectives on local water quality in the Santa River basin (Peru) utilising a digital technological approach where we collected data via a novel photo elicitation app, supported by a field work campaign. The lessons explored in this article provide insights into challenges and opportunities for researchers considering developing technological tools for encouraging participation and engagement in marginalised communities

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included.The first two authors were partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P

Absorbing boundary conditions for the Westervelt equation

The focus of this work is on the construction of a family of nonlinear absorbing boundary conditions for the Westervelt equation in one and two space dimensions. The principal ingredient used in the design of such conditions is pseudo-differential calculus. This approach enables to develop high order boundary conditions in a consistent way which are typically more accurate than their low order analogs. Under the hypothesis of small initial data, we establish local well-posedness for the Westervelt equation with the absorbing boundary conditions. The performed numerical experiments illustrate the efficiency of the proposed boundary conditions for different regimes of wave propagation

Radiological and elemental composition of cryoconite and glacier mice from Vatnajökull, Iceland

Cryoconite has been demonstrated to be an efficient accumulator of some classes of contaminants on glaciers in both mountain and polar environments, however the accumulation of contaminants in cryoconite in Iceland has received very little attention to date. To understand the spatial variability of natural and anthropogenic fallout radionuclides and metals on glaciers in Iceland, we present the first study of this region including both cryoconite from three glaciers: Virkisjökull; Skaftafellsjökull; and Falljökull, together with moss balls (‘glacier mice’) from Falljökull. The cryoconite samples and glacier mice were analysed using XRF spectrometry to assess their elemental composition and gamma spectrometry to identify, and quantify, fallout radionuclides, primarily 7Be, 137Cs, 241Am, excess 210Pb, and 40K. The results revealed that the cryoconite samples had similar compositions, influenced by local geology and natural sources of volcanic ash and dust. Higher concentrations of radionuclides and heavy metals were found in both cryoconite and glacier mice compared to control samples comprising nearby proglacial sediments. In comparison to other glaciers in the Northern Hemisphere, however, cryoconite from Icelandic glaciers contains some of the lowest activity concentrations of key radionuclides. Consequently, cryoconite deposits that are released and diluted during the melt and retreat of Icelandic glaciers are unlikely to be of environmental concern following transport to proglacial areas

A review on sparse solutions in optimal control of partial differential equations

In this paper a review of the results on sparse controls for partial differential equations is presented. There are two different approaches to the sparsity study of control problems. One approach consists of taking functions to control the system, putting in the cost functional a convenient term that promotes the sparsity of the optimal control. A second approach deals with controls that are Borel measures and the norm of the measure is involved in the cost functional. The use of measures as controls allows to obtain optimal controls supported on a zero Lebesgue measure set, which is very interesting for practical implementation. If the state equation is linear, then we can carry out a complete analysis of the control problem with measures. However, if the equation is nonlinear the use of measures to control the system is still an open problem, in general, and the use of functions to control the system seems to be more appropriate.This work was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2014-57531-P