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# 2017 News Archive

 September 07, 2017 Ivanov, Pavlis, and Petrushev publish article in IMI Preprint Series Download Preprint Kamen Ivanov, Nikolaos K. Pavlis, and Pencho Petrushev published their article "Precise and Efficient Evaluation of Gravimetric Quantities at Arbitrarily Scattered Points in Space" in the IMI Preprint Series (2017:01). Please see the link above to the full article and visit Preprints to see other articles in this series. September 07, 2017 Ivanov and Petrushev publish article in IMI Preprint Series Download Original Article Kamen Ivanov and Pencho Petrushev published their article "Highly Localized Kernels on the Sphere Induced by Newtonian Kernals" in the IMI Preprint Series (2017:02). Please see the link above to the full article and visit Preprints to see other articles in this series. August 15, 2017 Pencho Petrushev is awarded a National Science Foundation grant Pencho Petrushev (IMI) is awarded a 3-year NSF grant totaling $162,941 for his project: Nonlinear Approximation in Geometric, Harmonic and Anisotropic Settings with Applications. "Many areas ranging from Physics, Geodesy and Geomagnetism to Cosmology and to Data Analysis require efficient representation and approximation of the underlying functions in the natural topology of the targeted application. The capturing of physical phenomena and data structure occurring at various scales requires approximation from locally supported multiscale systems relative to the application domains. Moreover, these approximation methods should be amenable to fast and accurate computation. The proposed research aims at increasing our fundamental understanding of nonlinear approximation theory and its applications in three main directions. The first objective is to develop nonlinear approximation theory in various geometric and nonclassical settings from multiscale systems that are well adapted to the targeted applications. The second aim is to study the approximation of harmonic functions from shifts of the Newtonian potential with targeted applications to Geodesy, Geomagnetism and Physics. The third goal is to approximate functions that are smooth on domains in space separated by smooth curves or surfaces. Here the idea is to use adaptively anisotropic multiscale dilations of the space, which enable the approximation tool to adjust to curved singularities. More explicitly, a core objective of this project is the development of nonlinear n-term approximation from frames and other systems in various geometric and nonclassical settings such as on the sphere, ball, box and simplex with weights as well as in the context of Lie groups and Riemannian manifolds. All these settings are covered by the general framework of Dirichlet spaces with heat kernel with Gaussian bounds. The key point of the proposed approach is to give us the freedom of dealing with (a) different geometries, (b) compact and noncompact spaces, and (c) spaces with nontrivial weights, and at the same time to allow for the development and frame decomposition of Besov and Triebel-Lizorkin spaces with complete range of indices. The development of the underlying heat kernel theory and nonlinear n-term approximation from localized systems are basics aspects of this theory. Another goal of this project is the development of nonlinear approximation of harmonic functions on the d-dimensional ball from linear combinations of shifts of the Newtonian potential. This includes the complete characterization of the rates of approximation and the development of an effective algorithm that achieves the rates of best approximation. Anisotropic phenomena appear in various contexts in analysis, PDEs and in applications. For instance, functions are frequently very smooth on domains in space separated by smooth curves or manifolds. This project aims at resolving this kind of singularities of functions by utilization of the framework of anisotropic multiscale dilations, which may change rapidly from point to point at any level and in depth. The main strands of this approach are (i) the development of an algorithm for rapid construction of best or near best dilation matrices leading to optimal sparsity, (ii) the construction of highly localized anisotropic frames and their utilization to nonlinear approximation of functions." August 11, 2017 Linyuan Lu and Marco Valtorta are awarded an Office of Naval Research grant Linyuan Lu (IMI) and Marco Valtorta (CSE) are awarded a 1-year ONR grant totaling$100,000 for their project: Hypergraph-based Causal Modeling. "Probabilistic graphical models (PGMs), such as Bayesian networks and chain graphs, are compact representations of independence relations that hold in a domain. PGMs have been very successfully used for representing causal relations and to support the discovery of causal dependence: for example, whether smoking causes cancer, or whether drug A is effective on disease B, etc. However, PGMs are restricted in their ability to represent useful domain information by various restriction on their structure. For example, a Bayesian network structure cannot represent a situation in which two variables, say A and B, are conditionally independent given two other variables, say C and D, which are in turn unconditionally independent. These limitations have led to a plethora of PGMs with additional kinds of edges. We propose a directed acyclic hypergraph framework for a probabilistic graphical model that we call “Bayesian hypergraphs” (a generalization of Bayesian networks and chain graphs). The new framework has the following advantages. First, correlational and causal dependencies are distinguished but co-exist via directed hyperedges: causal dependencies are between heads and tails while correlational dependencies are among head vertices. Second, Bayesian hypergraphs can encode graphically many more independence and functional structures than graph-based PGMs. This generality will help support structure learning better, without the need to resort to additional intermediate representations. Third, methodologies from the area of prediction and causal reasoning can be tested and applied to the area of learning naturally by sharing the same mathematical object. In particular, we can combine learning and intervention. Finally, by leveraging new developments on hypergraph theory, one may reduce the computational cost of structure learning via reduction and approximation at the global level." BACK TO TOP July 28, 2017 Xiaofeng Yang's publications recognized by Web of Science Xiaofeng Yang (IMI) recently had one paper (with Qi Wang) highlighted as "Hot" and two papers highlighted as "Highly Cited Paper" by Web of Science: ENERGY STABLE NUMERICAL SCHEMES FOR A HYDRODYNAMIC MODEL OF NEMATIC LIQUID CRYSTALS is now highlighted as “Hot paper” by web of science. As of Mar/Apr 2017, this paper was cited as 0.1\% in math field. NUMERICAL APPROXIMATIONS OF ALLEN-CAHN AND CAHN-HILLIARD EQUATIONS has been highlighted as "highly cited paper" by web of science. As of Mar/Apr 2017, this paper was cited as 1\% in math field. A PHASE-FIELD MODEL AND ITS NUMERICAL APPROXIMATION FOR TWO-PHASE INCOMPRESSIBLE FLOWS WITH DIFFERENT DENSITIES AND VISCOSITIES has been highlighted as "highly cited paper" by web of science. As of Mar/Apr 2017, this paper was cited as 1\% in math field. BACK TO TOP June 27, 2017 Peter Binev and Wolfgang Dahmen are awarded a National Science Foundation grant Peter Binev and Wolfgang Dahmen (IMI) are awarded a 3-year NSF grant totaling $275,000 for their project: Collaborative Research: Optimal Convergence Rates for hp-Adaptivity. "Despite the ever increasing power of modern digital computing facilities in many areas the enormous challenge remains to guarantee a desired accuracy of large scale simulation tasks at the expense of as few degrees of freedom as possible so as to ultimately further advance the frontiers of computability. An essential component in meeting such goals offered by mathematical sciences is the development of rigorously founded adaptive solution concepts that aim at optimally allocating computational resources - viz. degrees of freedom - in the course of the solution process based on information gathered so far. When dealing with piecewise polynomial or finite element discretizations, which are the by far most common ones, the guiding principle for contriving such allocations is to equilibrate the local errors associated with the underlying mesh partition. The so-called h-adaptivity tries to accomplish this by locally refining a given domain partition based on suitable local error indicators. Such strategies are meanwhile fairly well understood for elliptic boundary value problems while fundamental questions remain open for other problem classes. Minimizing the number of degrees of freedom for a given target accuracy does, however, not only depend on locality but also on the respective polynomial degree p. So called hp-finite element discretizations account for this fact by using high polynomial degrees in regions where the target function is very regular while resorting to low order polynomials near singularities. hp-adaptive methods therefore not only aim at optimally refining the mesh but also a simultaneously assigning the "most appropriate" polynomial degrees to the corresponding piecewise polynomials. While several existing methods demonstrate substantial advantages of such hp-strategies in numerous test cases there seems to exist no rigorous theoretical foundation yet that, in particular, relates the achieved accuracy to the incurred computational cost. The central objective of this proposal is therefore to develop the theoretical foundations of hp-adaptive solution concepts. The challenges have two major sources. On the one hand, the type of the PDE to which such methods are to be applied, of course, matters very much. On the other hand, there are several fundamental problem aspects that are independent of the particular application and primarily of approximation theoretic nature, in particular, regarding multivariate settings and the globalizing effect of high order conforming approximants. To address these aspects and build a solid footing for future specifications to different application areas is the primary goal of this project. The theoretical investigations will be accompanied by implementing the strategies for model problems that shed light on the quantitative behavior of the methods." BACK TO TOP June 23, 2017 Xiaofeng Yang is awarded a National Science Foundation grant Xiaofeng Yang (IMI) is awarded a 3-year NSF grant totaling$159,996 for his project: Collaborative Research: Efficient, Stable and Accurate Numerical Algorithms for a class of Gradient Flow Systems and their Applications. "This research proposal focuses on the development of efficient, stable and accurate numerical algorithms for a class of gradient flow systems with high nonlinearities. It is well known that the dissipative evolution equation that is derived from the gradient flow approach, perform as one of the most effective modeling tools to describe the real-world phenomena. For algorithms design, one main challenging issue is about the time marching problem, i.e., how to develop suitable temporal discretizations for the nonlinear stiffness terms in order to preserve the unconditional energy stability at the discrete level. The proposed research has a four-fold focus: (i) to develop a unified numerical framework of time-marching schemes for solving general gradient flow models with high nonlinearity; (ii) to develop accurate, efficient numerical schemes for a number of challenging gradient flow models of current interests; (iii) to perform numerical simulations to validate the efficiency, stability and accuracy of the proposed numerical schemes; and (iv) to study some physically motivated problems by using the developed predictive numerical tools." BACK TO TOP June 15, 2017 Temlyakov is co-organizer of a semester at the Isaac Newton Institute Vladimir Temlyakov's proposal to co-organize a semester at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, was recently approved. This will be a six month semester (January 3 - June 28, 2019) on the topic of "Approximation, sampling and compression in data science," and is being organized by: Anders Hansen (University of Cambridge); Alexei Shadrin (University of Cambridge); Vladimir Temlyakov (IMI, University of South Carolina, Steklov Mathematical Institute, Russian Academy of Sciences); and Sergey Tikhonov (Scuola Normale Superiore di Pisa). For more information, please visit: http://www.newton.ac.uk/event/asc BACK TO TOP June 15, 2017 Temlyakov's paper named "Classic Paper in Information Theory" by Google Scholar Vladimir Temlyakov's paper (with D. Donoho and M. Elad), "Stable recovery of sparse overcomplete representations in the presence of noise," IEEE Transactions on Information Theory {\bf 52} (2006), 6--18, has been named "Classic Paper in Information Theory" by Google Scholar. According to Google Scholar, "Classic papers are highly-cited papers in their area of research that have stood the test of time. For each area, we list the ten most-cited articles that were published ten years earlier." Temlyakov's paper is number 7 in the list after classical papers on compressed sensing. Please see the list of 2006 Classics in Information Theory. BACK TO TOP May 02, 2017 Ron DeVore elected to the National Academy of Sciences Ron DeVore, current member and former Director of the IMI, the Robert L. Sumwalt Distinguished Professor Emeritus of Mathematics, University of South Carolina, and the Walter E. Koss Professor and Distinguished Professor of Mathematics, Texas A&M University, was elected to the National Academy of Sciences in recognition of his distinguished and continuing achievements in original research (for more information, please visit: http://www.nasonline.org/news-and-multimedia/news/may-2-2017-NAS-Election.html ). BACK TO TOP April 19, 2017 Erik Palmer awarded a 2017 Mathematical Sciences Summer Internship Graduate student Erik Palmer (advised by Paula Vasquez (IMI)) has been offered an internship through the 2017 Mathematical Sciences Summer Internship Program. This program, sponsored by the The National Science Foundation (NSF), Division of Mathematical Sciences (DMS), provides mathematical sciences doctoral students the opportunity to participate in internships at federal national laboratories, industry and other approved facilities, and gain first-hand experience of the use of mathematics in a nonacademic setting to understand the application of advanced mathematical and statistical techniques to "real world" problems. Palmer will spend ten weeks at Lawrence Livermore Labs in Berkeley. BACK TO TOP April 17, 2017 Filaseta awarded 2017 Carolina Trustees Professorship Dr. Michael Filaseta (IMI) has been awarded the 2017 Carolina Trustees Professorship. The Carolina Trustees Professorship awards are presented to tenured full professors who are committed to teaching excellence in any phase of the university’s educational mission (undergraduate classroom, graduate seminar, laboratories, clinical practice, independent study, supervised research, etc.), and demonstrate a record of outstanding performance in research and in public service activities. Three Carolina Trustees Professorship awards are given each year-- two awards are presented to Columbia campus professors and one to a professor at another USC campus. The awards are presented annually in the amount of $2000 each and given by the members of the University of South Carolina Board of Trustees at the spring commencement dinner. BACK TO TOP April 06, 2017 Vasquez's student is awarded a Magellan Scholar Program grant Undergraduate student Erin Kalb (mentored by Paula Vasquez (IMI)) is awarded a one year Magellan Scholar Program grant totaling$3,000 for research on: Mathematical Modeling of Pituitary Organogenesis. "This project focuses on the formulation and solution of mathematical models, closely guided by experiments, that will contribute to our understanding of how the pituitary gland is formed and how different developmental processes affect the final state and function of this gland... Currently, most of the research on the development of the pituitary gland is solely focused in a biology perspective. We are interested in introducing mathematical models since they provide a way to determine the consistency of experimental observations with testable hypotheses. We will compare the results from the new model to experimental data and refine and adjust the model to account for different situations in the development of the pituitary. These new insights will allow us to inquire into processes, length and time scales that, at the moment, are out of experimental resolution." BACK TO TOP March 17, 2017 Francisco Blanco-Silva participates in USC's Out-to-Lunch Program Download USC Times March 2017 Interview Francisco Blanco-Silva was recently interviewed for the March 2017 issue of the USC Times (see online issue here). The article, "Out to Lunch - No Business at Lunch," features, among others, Blanco-Silva, an instructor of mathematics and member of IMI. Blanco-Silva participated in the USC Out-to-Lunch Program with Lyndsey Reynolds, a sophomore psychology major. The Program is designed to promote faculty and undergraduate student interaction outside the classroom. For more information about the program visit: http://www.sc.edu/success/outtolunch.html BACK TO TOP March 14, 2017 Cooper on WACH FOX 57! On Pi day, March 14th, Joshua Cooper represented the University of South Carolina on "Good Day Columbia," a local TV morning show on WACH FOX 57. To see the recorded segment, please visit: http://wach.com/news/local/fun-pi-day-activity-with-a-soccer-ball BACK TO TOP March 09, 2017 Hong Wang is awarded an ARO MURI supplement grant Hong Wang (IMI) is awarded an Army Research Office (ARO) Multidisciplinary University Research Initiatives (MURI) supplement grant totaling \$71,963 (with Brown University as the prime institution) for his research on: Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications. "Fractional PDEs provide significantly improved modeling and simulation capabilities of complex phenomena. However, it is found recently that numerical approximations to FPDEs often exhibit poor accuracy and artifacts near the boundary of the domain, which in turn pollute the numerical solutions in the interior domain. This phenomenon is in sharp contrast to their integer-order analogues and significantly compromises the great potential of FPDE models. Mathematical analysis reveals that the fundamental reason is that the solutions to FPDEs boundary layers for certain fractional orders. The USC Team, led by Dr. Hong Wang, will be responsible for conducting rigorous mathematical analysis on adaptively determining the variable-order of the FPDEs near the boundary, to ensure the solutions to the resulting FPDEs to have the same or compatible regularity near the boundary as in the interior domain. This would minimize or eliminate the boundary layer behavior of the solutions, so the FPDEs and their numerical approximations will behave like their integer-order counterparts and minimize or eliminate the numerical artifacts of the FPDE approximations." BACK TO TOP February 07, 2017 Boyce, DiPrima and Meade Publish Two Books on Elementary Differential Equations As part of its series of books on Differential Equations, Wiley has published the books "Elementary Differential Equations, Enhanced eText, 11th Edition" (ISBN-978-1-119-32063-0) and "Elementary Differential Equations and Boundary Value Problems, Enhanced eText, 11th Edition" (ISBN-978-1-119-38164-8), both by William E. Boyce, Richard C. DiPrima, and Douglas B. Meade (IMI). Both books are "written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between. The authors have sought to combine a sound and accurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 11th edition includes new problems, updated figures and examples to help motivate students. The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two‐ or three‐ semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations." BACK TO TOP January 09, 2017 Binev et al. publish paper in the SIAM/ASA Journal on Uncertainty Quantification Peter Binev (IMI), A. Cohen, W. Dahmen (IMI), R. DeVore (IMI), G. Petrova, and P. Wojtaszczyk published their paper "Data assimilation in reduced modeling" online in the SIAM/ASA Journal on Uncertainty Quantification (Vol. 5, Issue 1, 2017, pp. 1-29, DOI:10.1137/15M1025384). For details, please visit: http://epubs.siam.org/doi/10.1137/15M1025384 BACK TO TOP
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