Relevant bibliographies by topics / Lipschitz estimates

Academic literature on the topic 'Lipschitz estimates'

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Published: 4 June 2021
Last updated: 28 January 2023

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Journal articles on the topic "Lipschitz estimates"

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Coburn, L. A. "Sharp Berezin Lipschitz estimates." Proceedings of the American Mathematical Society 135, no. 04 (April 1, 2007): 1163. http://dx.doi.org/10.1090/s0002-9939-06-08569-8.

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Osȩkowski, Adam. "Sharp Estimates for Lipschitz Class." Journal of Geometric Analysis 26, no. 2 (February 21, 2015): 1346–69. http://dx.doi.org/10.1007/s12220-015-9593-7.

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Varopoulos, N. Th. "Gaussian Estimates in Lipschitz Domains." Canadian Journal of Mathematics 55, no. 2 (April 1, 2003): 401–31. http://dx.doi.org/10.4153/cjm-2003-018-9.

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Bommier-Hato, Hélène. "Lipschitz estimates for the Berezin transform." Journal of Function Spaces and Applications 8, no. 2 (2010): 103–28. http://dx.doi.org/10.1155/2010/461315.

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We consider the generalized Fock spaceA2(μm), whereμmis the measure with weighte−|z|m,m > 0, with respect to the Lebesgue measure on Cn. Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded operatorXonA2(μm), the Berezin transform ofXsatisfies Lipschitz estimates.
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Armstrong, Scott N., and Zhongwei Shen. "Lipschitz Estimates in Almost-Periodic Homogenization." Communications on Pure and Applied Mathematics 69, no. 10 (October 1, 2015): 1882–923. http://dx.doi.org/10.1002/cpa.21616.

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Lu, Shanzhen, Yan Meng, and Qiang Wu. "LIPSCHITZ ESTIMATES FOR MULTILINEAR SINGULAR INTEGRALS, II." Acta Mathematica Scientia 24, no. 2 (April 2004): 291–300. http://dx.doi.org/10.1016/s0252-9602(17)30386-7.

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Ayre, P. J., M. G. Cowling, and F. A. Sukochev. "Operator Lipschitz estimates in the unitary setting." Proceedings of the American Mathematical Society 144, no. 3 (August 5, 2015): 1053–57. http://dx.doi.org/10.1090/proc/12833.

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Kenig, Carlos, and Christophe Prange. "Uniform Lipschitz Estimates in Bumpy Half-Spaces." Archive for Rational Mechanics and Analysis 216, no. 3 (November 28, 2014): 703–65. http://dx.doi.org/10.1007/s00205-014-0818-x.

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Xia, Jingbo. "Diagonalization Modulo Norm Ideals with Lipschitz Estimates." Journal of Functional Analysis 145, no. 2 (April 1997): 491–526. http://dx.doi.org/10.1006/jfan.1996.3036.

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Dong, Hongjie, and Doyoon Kim. "Lq-Estimates for stationary Stokes system with coefficients measurable in one direction." Bulletin of Mathematical Sciences 09, no. 01 (April 2019): 1950004. http://dx.doi.org/10.1142/s1664360719500048.

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We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori [Formula: see text]-estimates for any [Formula: see text] when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a [Formula: see text]-estimate and prove the solvability for any [Formula: see text] when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.
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Dissertations / Theses on the topic "Lipschitz estimates"

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Potapov, Denis, and denis potapov@flinders edu au. "Lipschitz and commutator estimates, a unified approach." Flinders University. School of Informatics and Engineeering, 2007. http://catalogue.flinders.edu.au./local/adt/public/adt-SFU20070723.110059.

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The subject of the thesis is the study of operator functions in the setting of symmetric operator spaces. In this latter setting, it is of great importance to analyze the properties of so-called operator functions A --> f(A), where the variable A is a self-adjoint operator and f is a complex-valued Borel function on the real line. The thesis study the question of differentiability of this type of operator functions. The latter question is intimately related to the study of commutators. Text not only extends existing results to the setting of unbounded self-adjoint linear operators, but it is also shown that this can be obtained via a unified approach utilizing the left regular representation of von Neumann algebras.
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Carroll, T. F. "Growth estimates for conformal mappings and for positive harmonic functions in space." Thesis, Open University, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.382984.

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Sande, Olow. "Boundary Estimates for Solutions to Parabolic Equations." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-281451.

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This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation. Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary. Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary. Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain. Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric.
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Braga, Josà Ederson Melo. "Problemas variacionais de fronteira livre com duas fases e resultados do tipo PhragmÃn-Lindelof regidos por equaÃÃes elÃpticas nÃo lineares singulares/degeneradas." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=15623.

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Neste trabalho de tese discutimos resultados recentes sobre a regularidade e propriedades geomÃtricas de soluÃÃes variacionais de problemas de fronteira livre de duas fases regidos por equaÃÃes elÃpticas nÃo lineares degeneradas/singulares. Discutimos tambÃm resultados do tipo PhragmÃm-Lindelof para tais equaÃÃes classificando essas soluÃÃes em semi-espaÃos.
Neste trabalho de tese discutimos resultados recentes sobre a regularidade e propriedades geomÃtricas de soluÃÃes variacionais de problemas de fronteira livre de duas fases regidos por equaÃÃes elÃpticas nÃo lineares degeneradas/singulares. Discutimos tambÃm resultados do tipo PhragmÃm-Lindelof para tais equaÃÃes classificando essas soluÃÃes em semi-espaÃos.
In this work of thesis we discuss recents results on the regularity and geometric properties of variational solutions of two phase free boundary problems governed by singular/degenerate nonlinear elliptic equations. We also discuss PhragmÃn-Lindelof type results for such equations classifying those solutions in half spaces.
In this work of thesis we discuss recents results on the regularity and geometric properties of variational solutions of two phase free boundary problems governed by singular/degenerate nonlinear elliptic equations. We also discuss PhragmÃn-Lindelof type results for such equations classifying those solutions in half spaces.
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Avelin, Benny. "Boundary Behavior of p-Laplace Type Equations." Doctoral thesis, Uppsala universitet, Analys och tillämpad matematik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-198008.

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This thesis consists of six scientific papers, an introduction and a summary. All six papers concern the boundary behavior of non-negative solutions to partial differential equations. Paper I concerns solutions to certain p-Laplace type operators with variable coefficients. Suppose that u is a non-negative solution that vanishes on a part Γ of an Ahlfors regular NTA-domain. We prove among other things that the gradient Du of u has non-tangential limits almost everywhere on the boundary piece Γ, and that log|Du| is a BMO function on the boundary.  Furthermore, for Ahlfors regular NTA-domains that are uniformly (N,δ,r0)-approximable by Lipschitz graph domains we prove a boundary Harnack inequality provided that δ is small enough.  Paper II concerns solutions to a p-Laplace type operator with lower order terms in δ-Reifenberg flat domains. We prove that the ratio of two non-negative solutions vanishing on a part of the boundary is Hölder continuous provided that δ is small enough. Furthermore we solve the Martin boundary problem provided δ is small enough. In Paper III we prove that the boundary type Riesz measure associated to an A-capacitary function in a Reifenberg flat domain with vanishing constant is asymptotically optimal doubling. Paper IV concerns the boundary behavior of solutions to certain parabolic equations of p-Laplace type in Lipschitz cylinders. Among other things, we prove an intrinsic Carleson type estimate for the degenerate case and a weak intrinsic Carleson type estimate in the singular supercritical case. In Paper V we are concerned with equations of p-Laplace type structured on Hörmander vector fields. We prove that the boundary type Riesz measure associated to a non-negative solution that vanishes on a part Γ of an X-NTA-domain, is doubling on Γ. Paper VI concerns a one-phase free boundary problem for linear elliptic equations of non-divergence type. Assume that we know that the positivity set is an NTA-domain and that the free boundary is a graph. Furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction. We prove that the graph giving the free boundary is Lipschitz continuous.
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Chinot, Geoffrey. "Localization methods with applications to robust learning and interpolation." Electronic Thesis or Diss., Institut polytechnique de Paris, 2020. http://www.theses.fr/2020IPPAG002.

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Cette thèse de doctorat est centrée sur l'apprentissage supervisé. L'objectif principal est l'utilisation de méthodes de localisation pour obtenir des vitesses rapides de convergence, c'est-à-dire, des vitesse de l'ordre O(1/n), où n est le nombre d'observations. Ces vitesses ne sont pas toujours atteignables. Il faut imposer des contraintes sur la variance du problème comme une condition de Bernstein ou de marge. Plus particulièrement, dans cette thèse nous tentons d'établir des vitesses rapides de convergences pour des problèmes de robustesse et d'interpolation.On dit qu'un estimateur est robuste si ce dernier présente certaines garanties théoriques, sous le moins d'hypothèses possibles. Cette problématique de robustesse devient de plus en plus populaire. La raison principale est que dans l'ère actuelle du “big data", les données sont très souvent corrompues. Ainsi, construire des estimateurs fiables dans cette situation est essentiel. Dans cette thèse nous montrons que le fameux minimiseur du risque empirique (regularisé) associé à une fonction de perte Lipschitz est robuste à des bruits à queues lourde ainsi qu'a des outliers dans les labels. En revanche si la classe de prédicteurs est à queue lourde, cet estimateur n'est pas fiable. Dans ce cas, nous construisons des estimateurs appelé estimateur minmax-MOM, optimal lorsque les données sont à queues lourdes et possiblement corrompues.En apprentissage statistique, on dit qu'un estimateur interpole, lorsque ce dernier prédit parfaitement sur un jeu d'entrainement. En grande dimension, certains estimateurs interpolant les données peuvent être bons. En particulier, cette thèse nous étudions le modèle linéaire Gaussien en grande dimension et montrons que l'estimateur interpolant les données de plus petite norme est consistant et atteint même des vitesses rapides
This PhD thesis deals with supervized machine learning and statistics. The main goal is to use localization techniques to derive fast rates of convergence, with a particular focus on robust learning and interpolation problems.Localization methods aim to analyze localized properties of an estimator to obtain fast rates of convergence, that is rates of order O(1/n), where n is the number of observations. Under assumptions, such as the Bernstein condition, such rates are attainable.A robust estimator is an estimator with good theoretical guarantees, under as few assumptions as possible. This question is getting more and more popular in the current era of big data. Large dataset are very likely to be corrupted and one would like to build reliable estimators in such a setting. We show that the well-known regularized empirical risk minimizer (RERM) with Lipschitz-loss function is robust with respect to heavy-tailed noise and outliers in the label. When the class of predictor is heavy-tailed, RERM is not reliable. In this setting, we show that minmax Median of Means estimators can be a solution. By construction minmax-MOM estimators are also robust to an adversarial contamination.Interpolation problems study learning procedure with zero training error. Surprisingly, in large dimension, interpolating the data does not necessarily implies over-fitting. We study a high dimensional Gaussian linear model and show that sometimes the over-fitting may be benign
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Potapov, Denis Sergeevich. "Lipschitz and commutator estimates, a unified approach." 2007. http://catalogue.flinders.edu.au/local/adt/public/adt-SFU20070723.110059/index.html.

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Thesis (Ph.D.)--Flinders University, School of Informatics and Engineering, Dept. of Mathematics.
Typescript bound. Includes bibliographical references: (leaves 135-140) and index. Also available online.
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Bandara, Lashi. "Geometry and the Kato square root problem." Phd thesis, 2013. http://hdl.handle.net/1885/10690.

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The primary focus of this thesis is to consider Kato square root problems for various divergence-form operators on manifolds. This is the study of perturbations of second-order differential operators by bounded, complex, measurable coefficients. In general, such operators are not self-adjoint but uniformly elliptic. The Kato square root problem is then to understand when the square root of such an operator, which exists due to uniform ellipticity, is comparable to its unperturbed counterpart. A remarkably adaptable operator-theoretic framework due to Axelsson, Keith and McIntosh sits in the background of this work. This framework allows us to take a powerful first-order perspective of the problems which we consider in a geometric setting. Through a well established procedure, we reduce these problems to the study of quadratic estimates. Under a set of natural conditions, we prove quadratic estimates for a class of operators on vector bundles over complete measure metric spaces. The first kind of estimates we prove are global, and we establish them on trivial vector bundles when the underlying measure grows at most polynomially. The second kind are local, and there, we allow the vector bundle to be non-trivial but bounded in an appropriate sense. Here, the measure is allowed to grow exponentially. An important consequence of obtaining quadratic estimates on measure metric spaces is that it allows us to consider subelliptic operators on Lie groups. The first-order perspective allows us to reduce the subelliptic problem to a fully elliptic one on a sub-bundle. As a consequence, we are able to solve a homogeneous Kato square root problem for perturbations of subelliptic operators on nilpotent Lie groups. For general Lie groups we solve a similar inhomogeneous problem. In the situation of complete Riemannian manifolds, we consider uniformly elliptic divergence-form operators arising from connections on vector bundles. Under a set of assumptions, we show that the Kato square root problem can be solved for such operators. As a consequence, we solve this problem on functions under the condition that the Ricci curvature and injectivity radius are bounded. Assuming an additional lower bound for the curvature endomorphism on forms, we solve a similar problem for perturbations of inhomogeneous Hodge-Dirac operators. A theorem for tensors is obtained by additionally assuming boundedness of a second-order Riesz transform. Motivated by the study of these Kato problems, where for technical reasons it is useful to know the density of compactly supported functions in the domains of operators, we study connections and their divergence on a vector bundle. Through a first-order formulation, we show that this density property holds for the domains of these operators if the metric and connection are compatible and the underlying manifold is complete. We also show that compactly supported functions are dense in the second-order Sobolev space on complete manifolds under the sole assumption that the Ricci curvature is bounded below, improving a result that previously required an additional lower bound on the injectivity radius.
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Book chapters on the topic "Lipschitz estimates"

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Shen, Zhongwei. "L2 Estimates in Lipschitz Domains." In Periodic Homogenization of Elliptic Systems, 205–81. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91214-1_8.

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Wang, Kelei. "Uniqueness, Stability and Uniform Lipschitz Estimates." In Free Boundary Problems and Asymptotic Behavior of Singularly Perturbed Partial Differential Equations, 17–26. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-33696-6_2.

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Carinci, Gioia, Anna De Masi, Cristian Giardinà, and Errico Presutti. "Lipschitz and $$L^1$$ L 1 Estimates." In SpringerBriefs in Mathematical Physics, 27–29. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-33370-0_5.

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Peller, V. V. "An Elementary Approach to Operator Lipschitz Type Estimates." In 50 Years with Hardy Spaces, 395–416. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-59078-3_20.

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Strongin, Roman, Konstantin Barkalov, and Semen Bevzuk. "Acceleration of Global Search by Implementing Dual Estimates for Lipschitz Constant." In Lecture Notes in Computer Science, 478–86. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-40616-5_46.

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Ding, Shusen, Guannan Shi, and Yuming Xing. "Estimates for Lipschitz and BMO Norms of Operators on Differential Forms." In Nonlinear Analysis, Differential Equations, and Applications, 81–99. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-72563-1_5.

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Brown, Russell M., and Zhongwei Shen. "A Note on L p Estimates for Parabolic Systems in Lipschitz Cylinders." In Partial Differential Equations with Minimal Smoothness and Applications, 105–9. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-2898-1_9.

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Dolcetta, Italo Capuzzo. "Hölder and Lipschitz Estimates for Viscosity Solutions of Some Degenerate Elliptic PDE’s." In Analysis, Partial Differential Equations and Applications, 31–40. Basel: Birkhäuser Basel, 2009. http://dx.doi.org/10.1007/978-3-7643-9898-9_4.

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McIntosh, Alan, and Sylvie Monniaux. "First Order Approach to $$L^p$$ L p Estimates for the Stokes Operator on Lipschitz Domains." In Springer Proceedings in Mathematics & Statistics, 55–75. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-41945-9_3.

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Okumura, Toshiki. "On a Construction of Strong Solutions for Stochastic Differential Equations with Non-Lipschitz Coefficients: A Priori Estimates Approach." In Lecture Notes in Mathematics, 187–219. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-28535-7_10.

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Conference papers on the topic "Lipschitz estimates"

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GULIYEV, VAGIF S., and ZHIJIAN WU. "STRONG-TYPE ESTIMATES AND CARLESON MEASURES FOR WEIGHTED BESOV-LIPSCHITZ SPACES." In Proceedings of the 6th International ISAAC Congress. WORLD SCIENTIFIC, 2009. http://dx.doi.org/10.1142/9789812837332_0007.

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Barucq, H., H. Calandra, M. V. De Hoop, F. Faucher, and J. Shi. "Full waveform inversion for elastic medium using quantitative Lipschitz stability estimates." In 7th EAGE Saint Petersburg International Conference and Exhibition. Netherlands: EAGE Publications BV, 2016. http://dx.doi.org/10.3997/2214-4609.201600240.

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Chatterjee, Anindya, and Joseph P. Cusumano. "Parameter Estimation in a Nonlinear Vibrating System Using an Observer for an Extended System." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/vib-8067.

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Abstract We present a new observer-based method for parameter estimation for nonlinear oscillatory mechanical systems where the unknown parameters appear linearly (they may each be multiplied by bounded and Lipschitz continuous but otherwise arbitrary, possibly nonlinear, functions of the oscillatory state variables and time). The oscillations in the system may be periodic, quasiperiodic or chaotic. The method is also applicable to systems where the parameters appear nonlinearly, provided a good initial estimate of the parameter is available. The observer requires measurements of displacements. It estimates velocities on a fast time scale, and the unknown parameters on a slow time scale. The fast and slow time scales are governed by a single small parameter ϵ. Using asymptotic methods including the method of averaging, it is shown that the observer’s estimates of the unknown parameters converge like e−ϵt where t is time, provided the system response is such that the coefficient-functions of the unknown parameters are not close to being linearly dependent. It is also shown that the method is robust in that small errors in the model cause small errors in the parameter estimates. A numerical example is provided to demonstrate the effectiveness of the method.
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Gimbutas, Albertas, and Antanas Žilinskas. "On global optimization using an estimate of Lipschitz constant and simplicial partition." In NUMERICAL COMPUTATIONS: THEORY AND ALGORITHMS (NUMTA–2016): Proceedings of the 2nd International Conference “Numerical Computations: Theory and Algorithms”. Author(s), 2016. http://dx.doi.org/10.1063/1.4965346.

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Xu, Gen Qi. "State Estimator and Observer Design of One-Sided Lipschitz Systems with Delayed Output." In 2022 41st Chinese Control Conference (CCC). IEEE, 2022. http://dx.doi.org/10.23919/ccc55666.2022.9902334.

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Liao, Fei. "Nonlinear $\mathcal{H}_{\infty}$ observer and estimator transfer for a class of Lipschitz uncertain nonlinear systems." In 2022 41st Chinese Control Conference (CCC). IEEE, 2022. http://dx.doi.org/10.23919/ccc55666.2022.9901919.

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Liu, Zhaoqiang, and Jun Han. "Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priors." In Thirty-First International Joint Conference on Artificial Intelligence {IJCAI-22}. California: International Joint Conferences on Artificial Intelligence Organization, 2022. http://dx.doi.org/10.24963/ijcai.2022/454.

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In this paper, we propose projected gradient descent (PGD) algorithms for signal estimation from noisy nonlinear measurements. We assume that the unknown signal lies near the range of a Lipschitz continuous generative model with bounded inputs. In particular, we consider two cases when the nonlinear link function is either unknown or known. For unknown nonlinearity, we make the assumption of sub-Gaussian observations and propose a linear least-squares estimator. We show that when there is no representation error, the sensing vectors are Gaussian, and the number of samples is sufficiently large, with high probability, a PGD algorithm converges linearly to a point achieving the optimal statistical rate using arbitrary initialization. For known nonlinearity, we assume monotonicity, and make much weaker assumptions on the sensing vectors and allow for representation error. We propose a nonlinear least-squares estimator that is guaranteed to enjoy an optimal statistical rate. A corresponding PGD algorithm is provided and is shown to also converge linearly to the estimator using arbitrary initialization. In addition, we present experimental results on image datasets to demonstrate the performance of our PGD algorithms.
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Gudmundsson, Valthor, Haukur Kristinsson, Soren Petersen, and Agus Hasan. "Robust UAV Attitude Estimation Using a Cascade of Nonlinear Observer and Linearized Kalman Filter." In ASME 2018 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/dscc2018-9123.

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This paper presents a new approach for Unmanned Aerial Vehicle (UAV) attitude estimation using a cascade of nonlinear observer and linearized Kalman filter. The nonlinear observer is globally asymptotically stable and is designed using linear matrix inequalities (LMI). The exogenous signal from the nonlinear observer is used to generate a linearized model for the Kalman filter. The method is implemented for attitude estimation of a quadcopter. The nonlinear model is derived from the Newton-Euler equations. The nonlinear model is locally Lipschitz due to the cross and dot products between the angular and linear velocity vectors. The attitude estimation from the dynamical system presented in this paper can be used as a module for fault detection. Simulations in Gazebo on a PX4 using Software In The Loop (SITL) shows the proposed method is able to estimate the attitude of a quadcopter accurately.
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Tao, Youming, Yulian Wu, Xiuzhen Cheng, and Di Wang. "Private Stochastic Convex Optimization and Sparse Learning with Heavy-tailed Data Revisited." In Thirty-First International Joint Conference on Artificial Intelligence {IJCAI-22}. California: International Joint Conferences on Artificial Intelligence Organization, 2022. http://dx.doi.org/10.24963/ijcai.2022/548.

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In this paper, we revisit the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) with heavy-tailed data, where the gradient of the loss function has bounded moments. Instead of the case where the loss function is Lipschitz or each coordinate of the gradient has bounded second moment studied previously, we consider a relaxed scenario where each coordinate of the gradient only has bounded (1+v)-th moment with some v∈(0, 1]. Firstly, we start from the one dimensional private mean estimation for heavy-tailed distributions. We propose a novel robust and private mean estimator which is optimal. Based on its idea, we then extend to the general d-dimensional space and study DP-SCO with general convex and strongly convex loss functions. We also provide lower bounds for these two classes of loss under our setting and show that our upper bounds are optimal up to a factor of O(Poly(d)). To address the high dimensionality issue, we also study DP-SCO with heavy-tailed gradient under some sparsity constraint (DP sparse learning). We propose a new method and show it is also optimal up to a factor of O(s*), where s* is the underlying sparsity of the constraint.
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Cunha, Sérgio B., and Renan M. Baptista. "Pipeline Leak Detection Using a Moderate Gain Nonlinear Observer." In 2020 13th International Pipeline Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/ipc2020-9333.

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Abstract Most pipeline control systems use some sort of autonomous leak detection system as a safety feature. Among the pipeline leak detection techniques, state observers stand out as the most sophisticated and promising technique. But its use has been inhibited as the dynamic models employed so far are large and estimating the states of nonlinear systems is not trivial. Pipeline pressure and flow dynamics have been modelled in the literature by means of different numerical solutions to a pair of first order partial differential equations that express mass and linear momentum conservation. The numerical solution requires discretizing the pipeline length in a finite number of segments, resulting in a system of equations with size of twice the number of segments. Although there is nothing wrong with this approach, a smaller system is more convenient if one is concerned exclusively with pressure and flow at the pipeline entrance and exit sections. In this paper, energetic modelling principles are employed to obtain a pair of first order ordinary differential equations representing the dynamics of long liquid pipelines. A recently introduced nonlinear observer enables straightforward use of linear, constant-gain observers with Lipschitz nonlinear dynamics. This observer gives the designer freedom to choose the observer eigenvalues and enables mathematically proven asymptotic stability with low gains. In this paper this observer, using a second-order model to represent the pipeline dynamics, is used as a pipeline leak detection algorithm. Initially the observer was employed directly as a leak detection algorithm, the leak being indicated by a non-transient difference between the measured and the estimated flows. Afterwards the leak was modeled as a disturbance flow and a disturbance observer was designed. Both algorithms were verified by means of computer simulations. It was found that the two methodologies are capable of detecting and estimating very small leaks, but the disturbance observer seems capable of indicating small holes further way from the measuring points.
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