On distributional solutions of local and nonlocal problems of porous medium type

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of $$\partial_tu-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=g(x,t)\quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $\varphi$ is merely continuous and nondecreasing and $\mathfrak{L}^{\sigma,\mu}$ is the generator of a general symmetric L\'evy process. This means that $\mathfrak{L}^{\sigma,\mu}$ can have both local and nonlocal parts like e.g. $\mathfrak{L}^{\sigma,\mu}=\Delta-(-\Delta)^{\frac12}$. New uniqueness results for bounded distributional solutions of this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for $\mathfrak{L}^{\sigma,\mu}$. Existence and a priori estimates are deduced from a numerical approximation, and energy type estimates are also obtained.Comment: 6 pages. Minor revision. Added details to Step 2 of the proof of Theorem 3.

Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $\varphi:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations.Comment: To appear in "Advances in Mathematics

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general symmetric diffusion operator of L ́evy type and φ is merely continuous and non-decreasing. We then use this theory to prove con- vergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion op- σ,μ α are the (fractional) Laplacians ∆ and −(−∆)2 for α ∈ (0,2), erators L discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L ́evy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and conver- gence of the methods under minimal assumptions – including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [28]. We also present some numerical tests, but extensive testing is deferred to the companion paper [31] along with a more detailed discussion of the numerical methods included in our theory

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$\partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $\mathfrak{L}^{\sigma,\mu}$ is a general symmetric diffusion operator of L\'evy type and $\varphi$ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators $\mathfrak{L}^{\sigma,\mu}$ are the (fractional) Laplacians $\Delta$ and $-(-\Delta)^{\frac\alpha2}$ for $\alpha\in(0,2)$, discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L\'evy operators, allows us to give a unified and compact {\em nonlocal} theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions -- including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in \cite{DTEnJa17b}. We also present some numerical tests, but extensive testing is deferred to the companion paper \cite{DTEnJa18b} along with a more detailed discussion of the numerical methods included in our theory.Comment: 34 pages, 3 figures. To appear in SIAM Journal on Numerical Analysi

The Liouville theorem and linear operators satisfying the maximum principle

A result by Courr\`ege says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ where $$\mathcal{L}^{\sigma,b}[u](x)=\text{tr}(\sigma \sigma^{\texttt{T}} D^2u(x))+b\cdot Du(x)$$ and $$\mathcal{L}^\mu[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} \mu(z).$$ This class of operators coincides with the infinitesimal generators of L\'evy processes in probability theory. In this paper we give a complete characterization of the translation invariant operators of this form that satisfy the Liouville theorem: Bounded solutions $u$ of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$ are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$. The proofs combine arguments from PDE and group theories. They are simple and short.Comment: This is an independent and substantial update of arXiv:1807.01843. Here we treat general operators which could be both local and nonlocal, symmetric and nonsymmetric. 13 pages. v3: Update according to the suggestions of the referees. To appear in Journal de Math\'ematiques Pures et Appliqu\'ee

Improving the communication with stakeholders: the infrastructure degradation index and the infrastructure histogram

[EN] Water infrastructures are rapidly ageing without being properly replaced. Communicating the state of the network and the sector¿s needs to stakeholders is key for guaranteeing the sustainability of water and sewage systems. The infrastructure value index (IVI) is becoming a standard in the water industry as a communication tool; however, as a single value metric, it can mask key information. The complementary use of the infrastructure degradation index (IDI) and the infrastructure histogram (Hi) can provide a better understanding of the network's state while maintaining the simplicity of the analysis needed for public dissemination. The IVI is focused on the value of the infrastructure, the IDI on its median remaining life. The HI provides a detailed but simple picture of the network's remaining life, providing a clear idea of the magnitude of the investments needed in the future for maintaining the infrastructureEstruch-Juan, ME.; Cabrera Rochera, E.; Gomez Selles, E.; Del Teso-March, R. (2020). Improving the communication with stakeholders: the infrastructure degradation index and the infrastructure histogram. Water Science & Technology: Water Supply. 20(7):2762-2767. https://doi.org/10.2166/ws.2020.170S2762276720

Herramienta para el seguimiento del aprendizaje a distancia en alumnos de posgrado. El potencial de Office para realizar envíos personalizados

[ES] El número de alumnos en la oferta formativa de docencia online impartida por el ITA ha crecido considerablemente en los últimos años. La autoevaluación y el seguimiento minucioso de los tutores durante el aprendizaje del estudiante son aspectos claves para lograr que adquieran las competencias mínimas requeridas para superar las materias. Con el crecimiento del número de alumnos, el seguimiento personalizado y detallado de los tutores es cada vez más complejo, teniendo que adaptar y modificar las estrategias de seguimiento con el aumento de alumnos. El pasado curso, se desarrolló una herramienta que permite semiautomatizar el seguimiento de los alumnos a partir de los datos extraídos de la Plataforma Cursosagua en la que se desarrollan cada uno de los cursos. Con esta herramienta, los tutores detectan fácilmente las deficiencias y fortalezas de los alumnos. La herramienta muestra advertencias sobre las tareas realizadas, las notas en las evaluaciones, y el tiempo de dedicación, automatizando el envío de mensajes al estudiante en función de las advertencias anteriores. Esto permite que el alumno perciba un seguimiento constante y personalizado, y a su vez facilita la tarea de los tutores, comprobando que con esta estrategia se puede realizar un seguimiento más continuo y personalizado.[EN] The amount of online courses offered by ITA has grown significantly in recent years. As a consecuence, the number of students has rosen accordingly. In order to ensure that students adquire the skills required to pass the subjects, self-assessment and a careful supervision by tutors throughout the student's learning process are key aspects. With the growth of the number of students, it is becoming more complex to realise a personalized and detailed follow-up of the students. For this reason, the strategies had to be adapted. Last year, a tool was developed to allow a semi-automated monitorization of students from the Cursosagua online learning platform, where the courses take place. With this tool, tutors can easily detect students' deficiencies and strengths. The tool shows warnings about the tasks performed, the grades obtained, and the time spent in the course. According to these warnings, students receive personalized and automatic messages concerning their performance. This allows the student to perceive a constant and personalized follow-up. This tool makes it easier for tutors to monitor student learning, proving that with this strategy a more continuous and personalized monitoring can be done.Estruch-Juan, E.; Del Teso, R.; Gómez, E.; Soriano, J. (2021). Herramienta para el seguimiento del aprendizaje a distancia en alumnos de posgrado. El potencial de Office para realizar envíos personalizados. En IN-RED 2020: VI Congreso de Innovación Educativa y Docencia en Red. Editorial Universitat Politècnica de València. 640-654. https://doi.org/10.4995/INRED2020.2020.11938OCS64065

Defining complementary tools to the IVI. The Infrastructure Degradation Index (IDI) and the Infrastructure Histogram (HI)

[EN] The Infrastructure Value Index (IVI) is quickly becoming a standard as a valuable tool to quickly assess the state of urban water infrastructure. However, its simple nature (as a single metric) can mask some valuable information and lead to erroneous conclusions. This paper introduces two complementary tools to IVI: The Infrastructure Degradation Index (IDI) and the Infrastructure Histogram (HI). The IDI is focused on time (compared to the IVI, focused on value), represents an intuitive concept and behaves in a linear way. The joint analysis of IVI and IDI provides results in a more complete understanding of the state of the assets, while maintaining the simplicity of the tools. The Infrastructure Histogram allows for a full evaluation of the infrastructure state and provides a detailed picture of network age compared to its expected life, as well as an order of magnitude of the required investments in the following years.Cabrera Rochera, E.; Estruch-Juan, ME.; Gomez Selles, E.; Del Teso-March, R. (2019). Defining complementary tools to the IVI. The Infrastructure Degradation Index (IDI) and the Infrastructure Histogram (HI). Urban Water Journal. 16(5):343-352. https://doi.org/10.1080/1573062X.2019.1669195S343352165Alegre, H., Vitorino, D., & Coelho, S. (2014). Infrastructure Value Index: A Powerful Modelling Tool for Combined Long-term Planning of Linear and Vertical Assets. Procedia Engineering, 89, 1428-1436. doi:10.1016/j.proeng.2014.11.469Amaral, R., Alegre, H., & Matos, J. S. (2016). A service-oriented approach to assessing the infrastructure value index. Water Science and Technology, 74(2), 542-548. doi:10.2166/wst.2016.250Aware-p.org. 2014. “AWARE-P/Software.” Accessed 25 November 2018. http://www.aware-p.org/np4/software/Baseform. 2018. “Baseform.” Accessed 24 November 2018. https://baseform.com/np4/productCanal de Isabel II Gestión. 2012. Normas Para Redes de Abastecimiento. [Standards for Water Supply Networks.]. https://www.canalgestion.es/es/galeria_ficheros/pie/normativa/normativa/Normas_redes_abastecimiento2012_CYIIG.pdfFrost, and Sullivan. 2011. “Western European Water and Wastewater Utilities Market.” https://store.frost.com/western-european-water-and-wastewater-utilities-market.html#section1Leitão, J. P., Coelho, S. T., Alegre, H., Cardoso, M. A., Silva, M. S., Ramalho, P., … Carriço, N. (2014). Moving urban water infrastructure asset management from science into practice. Urban Water Journal, 13(2), 133-141. doi:10.1080/1573062x.2014.939092Marchionni, V., Cabral, M., Amado, C., & Covas, D. (2016). Estimating Water Supply Infrastructure Cost Using Regression Techniques. Journal of Water Resources Planning and Management, 142(4), 04016003. doi:10.1061/(asce)wr.1943-5452.0000627Marchionni, V., Lopes, N., Mamouros, L., & Covas, D. (2014). Modelling Sewer Systems Costs with Multiple Linear Regression. Water Resources Management, 28(13), 4415-4431. doi:10.1007/s11269-014-0759-zPulido-Velazquez, M., Cabrera Marcet, E., & Garrido Colmenero, A. (2014). Economía del agua y gestión de recursos hídricos. Ingeniería del agua, 18(1), 95. doi:10.4995/ia.2014.3160Rokstad, M. M., Ugarelli, R. M., & Sægrov, S. (2015). Improving data collection strategies and infrastructure asset management tool utilisation through cost benefit considerations. Urban Water Journal, 13(7), 710-726. doi:10.1080/1573062x.2015.102469

A network application approach towards 5G and beyond critical communications use cases

Low latency and high bandwidth heralded with 5G networks will allow transmission of large amounts of Mission-Critical data over a short time period. 5G hence unlocks several capabilities for novel Public Protection and Disaster Relief (PPDR) applications, developed to support first responders in making faster and more accurate decisions during times of crisis. As various research initiatives are giving shape to the Network Application ecosystem as an interaction layer between vertical applications and the network control plane, in this article we explore how this concept can unlock finer network service management capabilities that can be leveraged by PPDR solution developers. In particular, we elaborate on the role of Network Applications as means for developers to assure prioritization of specific emergency flows of data, such as high-definition video transmission from PPDR field users to remote operators. To demonstrate this potential in future PPDR-over-5G services, we delve into the transfer of network-intensive PPDR solutions to the Network Application model. We then explore novelties in Network Application experimentation platforms, aiming to streamline development and deployment of such integrated systems across existing 5G infrastructures, by providing the reliability and multi-cluster environments they require