Uniform in Time Error Estimates for a Finite Element Method Applied to a Downscaling Data Assimilation Algorithm for the Navier Stokes Equations

First Published in SIAM Journal on Numerical Analysis in 2020, Vol. 58, No. 1, published by the Society for Industrial and Applied Mathematics (SIAM)In this paper we analyze a finite element method applied to a continuous downscaling data assimilation algorithm for the numerical approximation of the two- and three-dimensional Navier--Stokes equations corresponding to given measurements on a coarse spatial scale. For representing the coarse mesh measurements we consider different types of interpolation operators including a Lagrange interpolant. We obtain uniform-in-time estimates for the error between a finite element approximation and the reference solution corresponding to the coarse mesh measurements. We consider both the case of a plain Galerkin method and a Galerkin method with grad-div stabilization. For the stabilized method we prove error bounds in which the constants do not depend on inverse powers of the viscosity. Some numerical experiments illustrate the theoretical resultsDepartamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain. Research is supported by Spanish MINECO under grant MTM2015-65608-P ([email protected]). Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain. Research is supported by Spanish MINECO under grant MTM2016-78995-P (AEI/FEDER, UE) and VA024P17 (Junta de Castilla y Leon, ES) co nanced by FEDER funds ([email protected]). Department of Mathematics, Texas A&M University, College Station, TX 77843, USA. De- partment of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. Research is supported in part by the ONR grant N00014-15-1-2333, the Einstein Stiftung/Foundation - Berlin, through the Einstein Visiting Fellow Program, and by the John Simon Guggenheim Memorial Foundation ([email protected], [email protected])

A model to classify users of social networks based on PageRank

In this paper, we present a model to classify users of Social Networks. In particular, we focus on Social Network Sites. The model is based on the PageRank algorithm. We use the personalization vector to bias the PageRank to some users. We give an explicit expression of the personalization vector that allows the introduction of some typical features of the users of SNSs. We describe the model as a seven-step process. We illustrate the applicability of the model with two examples. One example is based on real links of a Facebook network. We also indicate how to take into account real actions of Facebook users to implement the model.This work is supported by Spanish DGI grant MTM2010-18674.Pedroche Sánchez, F. (2012). A model to classify users of social networks based on PageRank. International Journal of Bifurcation and Chaos. 22(7):1-14. https://doi.org/10.1142/S0218127412501623S114227Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., & Zhou, C. (2008). Synchronization in complex networks. Physics Reports, 469(3), 93-153. doi:10.1016/j.physrep.2008.09.002BOCCALETTI, S., LATORA, V., MORENO, Y., CHAVEZ, M., & HWANG, D. (2006). Complex networks: Structure and dynamics. Physics Reports, 424(4-5), 175-308. doi:10.1016/j.physrep.2005.10.009Boldi, P., Santini, M., & Vigna, S. (2009). PageRank. ACM Transactions on Information Systems, 27(4), 1-23. doi:10.1145/1629096.1629097Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-Law Distributions in Empirical Data. SIAM Review, 51(4), 661-703. doi:10.1137/070710111Criado, R., Flores, J., González-Vasco, M. I., & Pello, J. (2007). Choosing a leader on a complex network. Journal of Computational and Applied Mathematics, 204(1), 10-17. doi:10.1016/j.cam.2006.04.024C. De Kerchove and P. Van Dooren, Lectures Notes in Control and Information Sciences 389 (2009) pp. 3–16.Dorogovtsev, S. (2010). Lectures on Complex Networks. doi:10.1093/acprof:oso/9780199548927.001.0001Easley, D., & Kleinberg, J. (2010). Networks, Crowds, and Markets. doi:10.1017/cbo9780511761942Estrada, E., & Higham, D. J. (2010). Network Properties Revealed through Matrix Functions. SIAM Review, 52(4), 696-714. doi:10.1137/090761070Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486(3-5), 75-174. doi:10.1016/j.physrep.2009.11.002Granovetter, M. S. (1973). The Strength of Weak Ties. American Journal of Sociology, 78(6), 1360-1380. doi:10.1086/225469Haveliwala, T. H. (2003). Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search. IEEE Transactions on Knowledge and Data Engineering, 15(4), 784-796. doi:10.1109/tkde.2003.1208999Langville, A. N., & Meyer, C. D. (2006). Google’s PageRank and Beyond. doi:10.1515/9781400830329Lazer, D., Pentland, A., Adamic, L., Aral, S., Barabasi, A.-L., Brewer, D., … Van Alstyne, M. (2009). SOCIAL SCIENCE: Computational Social Science. Science, 323(5915), 721-723. doi:10.1126/science.1167742Lewis, K., Kaufman, J., Gonzalez, M., Wimmer, A., & Christakis, N. (2008). Tastes, ties, and time: A new social network dataset using Facebook.com. Social Networks, 30(4), 330-342. doi:10.1016/j.socnet.2008.07.002Nan Lin, Dayton, P. W., & Greenwald, P. (1978). Analyzing the Instrumental Use of Relations in the Context of Social Structure. Sociological Methods & Research, 7(2), 149-166. doi:10.1177/004912417800700203Mayer, A., & Puller, S. L. (2008). The old boy (and girl) network: Social network formation on university campuses. Journal of Public Economics, 92(1-2), 329-347. doi:10.1016/j.jpubeco.2007.09.001Newman, M. (2010). Networks. doi:10.1093/acprof:oso/9780199206650.001.0001Pedroche Sánchez, F. (2010). Competitivity groups on social network sites. Mathematical and Computer Modelling, 52(7-8), 1052-1057. doi:10.1016/j.mcm.2010.02.031Sabater, J., & Sierra, C. (2005). Review on Computational Trust and Reputation Models. Artificial Intelligence Review, 24(1), 33-60. doi:10.1007/s10462-004-0041-5Serra-Capizzano, S. (2005). Jordan Canonical Form of the Google Matrix: A Potential Contribution to the PageRank Computation. SIAM Journal on Matrix Analysis and Applications, 27(2), 305-312. doi:10.1137/s0895479804441407Vasalou, A., Joinson, A. N., & Courvoisier, D. (2010). Cultural differences, experience with social networks and the nature of «true commitment» in Facebook. International Journal of Human-Computer Studies, 68(10), 719-728. doi:10.1016/j.ijhcs.2010.06.00

An Analysis of Publication Venues for Automatic Differentiation Research

We present the results of our analysis of publication venues for papers on automatic differentiation (AD), covering academic journals and conference proceedings. Our data are collected from the AD publications database maintained by the autodiff.org community website. The database is purpose-built for the AD field and is expanding via submissions by AD researchers. Therefore, it provides a relatively noise-free list of publications relating to the field. However, it does include noise in the form of variant spellings of journal and conference names. We handle this by manually correcting and merging these variants under the official names of corresponding venues. We also share the raw data we get after these corrections.Comment: 6 pages, 3 figure

Research and Education in Computational Science and Engineering

Over the past two decades the field of computational science and engineering (CSE) has penetrated both basic and applied research in academia, industry, and laboratories to advance discovery, optimize systems, support decision-makers, and educate the scientific and engineering workforce. Informed by centuries of theory and experiment, CSE performs computational experiments to answer questions that neither theory nor experiment alone is equipped to answer. CSE provides scientists and engineers of all persuasions with algorithmic inventions and software systems that transcend disciplines and scales. Carried on a wave of digital technology, CSE brings the power of parallelism to bear on troves of data. Mathematics-based advanced computing has become a prevalent means of discovery and innovation in essentially all areas of science, engineering, technology, and society; and the CSE community is at the core of this transformation. However, a combination of disruptive developments---including the architectural complexity of extreme-scale computing, the data revolution that engulfs the planet, and the specialization required to follow the applications to new frontiers---is redefining the scope and reach of the CSE endeavor. This report describes the rapid expansion of CSE and the challenges to sustaining its bold advances. The report also presents strategies and directions for CSE research and education for the next decade.Comment: Major revision, to appear in SIAM Revie