@article {2021, title = {A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences}, journal = {Computers and Mathematics with Applications}, volume = {102}, year = {2021}, pages = {261-276}, doi = {10.1016/j.camwa.2021.10.020}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85117948561\&doi=10.1016\%2fj.camwa.2021.10.020\&partnerID=40\&md5=cb57d59a6975a35315b2cf5d0e3a6001}, author = {G. Carere and Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza and R. Stevenson} } @article {2021, title = {Well-Ordered and Non-Well-Ordered Lower and Upper Solutions for Periodic Planar Systems}, journal = {Advanced Nonlinear Studies}, volume = {21}, year = {2021}, month = {2021}, pages = {397 - 419}, url = {https://doi.org/10.1515/ans-2021-2117}, author = {Alessandro Fonda and Giuliano Klun and Andrea Sfecci} } @article {2020, title = {Weak formulation of elastodynamics in domains with growing cracks}, volume = {199}, year = {2020}, month = {2020/08/01}, pages = {1571 - 1595}, abstract = {

In this paper, we formulate and study the system of elastodynamics on domains with arbitrary growing cracks. This includes homogeneous Neumann conditions on the crack sets and mixed general Dirichlet{\textendash}Neumann conditions on the boundary. The only assumptions on the crack sets are to be $$(n-1)$$-rectifiable with finite surface measure, and increasing in the sense of set inclusions. In particular, they might be dense; hence, the weak formulation must fall outside the usual context of Sobolev spaces and Korn{\textquoteright}s inequality. We prove existence of a solution for both the damped and undamped systems, while in the damped case we are also able to prove uniqueness and an energy balance.

}, isbn = {1618-1891}, url = {https://doi.org/10.1007/s10231-019-00932-y}, author = {Emanuele Tasso} } @article {2019, title = {A Weighted POD Method for Elliptic PDEs with Random Inputs}, journal = {Journal of Scientific Computing}, volume = {81}, year = {2019}, pages = {136-153}, abstract = {

In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to assess the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and high dimensional problems.

}, doi = {10.1007/s10915-018-0830-7}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85053798049\&doi=10.1007\%2fs10915-018-0830-7\&partnerID=40\&md5=5cad501b6ef1955da55868807079ee5d}, author = {L.Venturi and F. Ballarin and Gianluigi Rozza} } @article {2019, title = {Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs}, journal = {PoliTO Springer Series}, year = {2019}, pages = {27-40}, abstract = {

In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.

}, doi = {10.1007/978-3-030-04870-9_2}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85084009379\&doi=10.1007\%2f978-3-030-04870-9_2\&partnerID=40\&md5=446bcc1f331167bbba67bc00fb170150}, author = {L. Venturi and D. Torlo and F. Ballarin and Gianluigi Rozza} } @article {Sar18, title = {Weighted Cheeger sets are domains of isoperimetry}, journal = {Manuscripta Math.}, volume = {156}, number = {3{\textendash}4}, year = {2018}, pages = {371{\textendash}381}, doi = {10.1007/s00229-017-0974-z}, author = {Saracco, G.} } @article {Sysoeva:2018xig, title = {Wilson loop and its correlators in the limit of large coupling constant}, journal = {Nucl. Phys. B}, volume = {936}, year = {2018}, pages = {383{\textendash}399}, doi = {10.1016/j.nuclphysb.2018.09.015}, author = {Sysoeva, Ekaterina} } @article {Sysoeva:2017fhr, title = {Wilson loops and its correlators with chiral operators in $\mathcalN=2, 4$ SCFT at large $N$}, journal = {JHEP}, volume = {03}, year = {2018}, pages = {155}, doi = {10.1007/JHEP03(2018)155}, author = {Sysoeva, Ekaterina} } @article {MolaHeltaiDeSimone2017, title = {Wet and Dry Transom Stern Treatment for Unsteady and Nonlinear Potential Flow Model for Naval Hydrodynamics Simulations}, journal = {Journal of Ship Research}, volume = {61}, number = {1}, year = {2017}, pages = {1{\textendash}14}, abstract = {

We present a model for the fast evaluation of the total drag of ship hulls operating in both wet and dry transom stern conditions, in calm or wavy water, based on the combination of an unsteady semi-Lagrangian potential flow formulation with fully nonlinear free-surface treatment, experimental correlations, and simplified viscous drag modeling. The implementation is entirely based on open source libraries. The spatial discretization is solved using a streamline upwind Petrov-Galerkin stabilization of an iso-parametric, collocation based, boundary element method, implemented using the open source library deal.II. The resulting nonlinear differential-algebraic system is integrated in time using implicit backward differentiation formulas, implemented in the open source library SUNDIALS. The Open CASCADE library is used to interface the model directly with computer-aided design data structures. The model accounts automatically for hulls with a transom stern, both in wet and dry regimes, by using a specific treatment of the free-surface nodes on the stern edge that automatically detects when the hull advances at low speeds. In this case, the transom stern is partially immersed, and a pressure patch is applied on the water surface detaching from the transom stern, to recover the gravity effect of the recirculating water on the underlying irrotational flow domain. The parameters of the model used to impose the pressure patch are approximated from experimental relations found in the literature. The test cases considered are those of the U.S. Navy Combatant DTMB-5415 and the National Physical Laboratory hull. Comparisons with experimental data on quasi-steady test cases for both water elevation and total hull drag are presented and discussed. The quality of the results obtained on quasi-steady simulations suggests that this model can represent a promising alternative to current unsteady solvers for simulations with Froude numbers below 0.35.

}, doi = {https://doi.org/10.5957/JOSR.61.1.160016}, author = {Andrea Mola and Luca Heltai and Antonio DeSimone} } @article {2015, title = {The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data}, number = {SISSA;47/2015/MATE}, year = {2015}, abstract = {Given a bounded open set $\Omega \subset \mathbb R^d$ with Lipschitz boundary and an increasing family $\Gamma_t$, $t\in [0,T]$, of closed subsets of $\Omega$, we analyze the scalar wave equation $\ddot{u} - div (A \nabla u) = f$ in the time varying cracked domains $\Omega\setminus\Gamma_t$. Here we assume that the sets $\Gamma_t$ are contained into a prescribed $(d-1)$-manifold of class $C^2$. Our approach relies on a change of variables: recasting the problem on the reference configuration $\Omega\setminus \Gamma_0$, we are led to consider a hyperbolic problem of the form $\ddot{v} - div (B\nabla v) + a \cdot \nabla v - 2 b \cdot \nabla \dot{v} = g$ in $\Omega \setminus \Gamma_0$. Under suitable assumptions on the regularity of the change of variables that transforms $\Omega\setminus \Gamma_t$ into $\Omega\setminus \Gamma_0$, we prove existence and uniqueness of weak solutions for both formulations. Moreover, we provide an energy equality, which gives, as a by-product, the continuous dependence of the solutions with respect to the cracks.}, url = {http://urania.sissa.it/xmlui/handle/1963/34629}, author = {Gianni Dal Maso and Ilaria Lucardesi} } @article {2014, title = {A weighted empirical interpolation method: A priori convergence analysis and applications}, number = {ESAIM: Mathematical Modelling and Numerical Analysis;volume 48; issue 4; pages 943-953}, year = {2014}, publisher = {EDP Sciences}, abstract = {We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Lo{\`e}ve expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.}, doi = {10.1051/m2an/2013128}, url = {http://urania.sissa.it/xmlui/handle/1963/35021}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} } @article {2014, title = {Weighted quantile correlation test for the logistic family}, number = {Acta Scientiarum Mathematicarum;volume 80; issue 1-2; pages 307-326;}, year = {2014}, publisher = {University of Szeged}, abstract = {We summarize the results of investigating the asymptotic behavior of the weighted quantile correlation tests for the location-scale family associated to the logistic distribution. Explicit representations of the limiting distribution are given in terms of integrals of weighted Brownian bridges or alternatively as infinite series of independent Gaussian random variables. The power of this test and the test for the location logistic family against some alternatives are demonstrated by numerical simulations.}, doi = {10.14232/actasm-013-809-8}, url = {http://urania.sissa.it/xmlui/handle/1963/35025}, author = {Ferenc Balogh and {\'E}va Krauczi} } @article {2014, title = {Where best to place a Dirichlet condition in an anisotropic membrane?}, number = {SISSA;61/2014/MATE}, year = {2014}, institution = {SISSA}, abstract = {We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $\Omega$. Dirichlet conditions are imposed along $\partial\Omega$ and, in addition, along a set $\Sigma$ of prescribed length ($1$-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $\Sigma$ in order to maximize the first eigenvalue. We characterize the limit distribution of the optimal sets, as their prescribed length tends to infinity, via $\Gamma$-convergence.}, url = {http://urania.sissa.it/xmlui/handle/1963/7481}, author = {Paolo Tilli and Davide Zucco} } @article {ChenQuarteroniRozza2013a, title = {A weighted reduced basis method for elliptic partial differential equations with random input data}, journal = {SIAM Journal on Numerical Analysis}, volume = {51}, number = {6}, year = {2013}, pages = {3163{\textendash}3185}, abstract = {In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.}, doi = {10.1137/130905253}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} } @article {2012, title = {Weighted barycentric sets and singular Liouville equations on compact surfaces}, journal = {Journal of Functional Analysis 262 (2012) 409-450}, number = {arXiv:1105.2363;}, year = {2012}, publisher = {Elsevier}, abstract = {Given a closed two dimensional manifold, we prove a general existence result\\r\\nfor a class of elliptic PDEs with exponential nonlinearities and negative Dirac\\r\\ndeltas on the right-hand side, extending a theory recently obtained for the\\r\\nregular case. This is done by global methods: since the associated Euler\\r\\nfunctional is in general unbounded from below, we need to define a new model\\r\\nspace, generalizing the so-called space of formal barycenters and\\r\\ncharacterizing (up to homotopy equivalence) its very low sublevels. As a\\r\\nresult, the analytic problem is reduced to a topological one concerning the\\r\\ncontractibility of this model space. To this aim, we prove a new functional\\r\\ninequality in the spirit of [16] and then we employ a min-max scheme based on a cone-style construction, jointly with the blow-up analysis given in [5] (after\\r\\n[6] and [8]). This study is motivated by abelian Chern- Simons theory in\\r\\nself-dual regime, or from the problem of prescribing the Gaussian curvature in\\r\\npresence of conical singularities (hence generalizing a problem raised by\\r\\nKazdan and Warner in [26]).}, doi = {10.1016/j.jfa.2011.09.012}, url = {http://hdl.handle.net/1963/5218}, author = {Alessandro Carlotto and Andrea Malchiodi} } @article {2012, title = {Wild quiver gauge theories}, journal = {JHEP 02(2012)031}, number = {arXiv:1112.1691v1;}, year = {2012}, note = {34 pages}, publisher = {SISSA}, abstract = {

We study $N=2$ supersymmetric $SU(2)$ gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional $A_1 (2,0)$ theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the Omega-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of higher order singularities.

}, doi = {10.1007/JHEP02(2012)031}, url = {http://hdl.handle.net/1963/5184}, author = {Giulio Bonelli and Kazunobu Maruyoshi and Alessandro Tanzini} } @article {DANCHIN2011253, title = {The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces}, journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, volume = {96}, number = {3}, year = {2011}, pages = {253 - 278}, abstract = {

This work is the continuation of the recent paper (Danchin, 2010) [9] devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type B$\infty$,rs embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of H{\"o}lder spaces C1,α and of the endpoint Besov space B$\infty$,11. For such data and under the non-vacuum assumption, we establish the local well-posedness and a continuation criterion in the spirit of that of Beale, Kato and Majda (1984) [2]. In the last part of the paper, we give lower bounds for the lifespan of a solution. In dimension two, we point out that the lifespan tends to infinity when the initial density tends to be a constant. This is, to our knowledge, the first result of this kind for the density-dependent incompressible Euler equations. R{\'e}sum{\'e} Ce travail compl{\`e}te l'article r{\'e}cent (Danchin, 2010) [9] consacr{\'e} au syst{\`e}me d'Euler incompressible {\`a} densit{\'e} variable. Lorsque l'{\'e}tat initial ne comporte pas de vide, on montre ici que le syst{\`e}me est bien pos{\'e} dans tous les espaces de Besov B$\infty$,rs inclus dans l'ensemble des fonctions lipschitziennes. Ce cadre fonctionnel contient en particulier les espaces de H{\"o}lder C1,α et l'espace de Besov limite B$\infty$,11. On {\'e}tablit {\'e}galement un crit{\`e}re de prolongement dans l'esprit de celui de Beale, Kato et Majda (1984) [2] pour le cas homog{\`e}ne. Dans la derni{\`e}re partie de l'article, on donne des minorations pour le temps de vie des solutions du syst{\`e}me. En dimension deux, on montre que ce temps de vie tend vers l'infini lorsque la densit{\'e} tend {\`a} {\^e}tre homog{\`e}ne. {\`A} notre connaissance, il s'agit du premier r{\'e}sultat de ce type pour le syst{\`e}me d'Euler incompressible {\`a} densit{\'e} variable.

}, keywords = {Blow-up criterion, Critical regularity, Incompressible Euler equations, Lifespan, Nonhomogeneous inviscid fluids}, issn = {0021-7824}, doi = {https://doi.org/10.1016/j.matpur.2011.04.005}, url = {http://www.sciencedirect.com/science/article/pii/S0021782411000511}, author = {Rapha{\"e}l Danchin and Francesco Fanelli} } @article {2010, title = {Well-posed infinite horizon variational problems on a compact manifold}, journal = {Proceedings of the Steklov Institute of Mathematics. Volume 268, Issue 1, 2010, Pages 17-31}, number = {arXiv:0906.4433;}, year = {2010}, publisher = {SISSA}, abstract = {We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold M to admit a smooth optimal synthesis, i. e., a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis, we construct an invariant Lagrangian submanifold (well-projected to M) of the flow of extremals in the cotangent bundle T*M. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics}, doi = {10.1134/S0081543810010037}, url = {http://hdl.handle.net/1963/6458}, author = {Andrei A. Agrachev} } @inbook {2006, title = {WDVV equations and Frobenius manifolds}, booktitle = {Encyclopedia of Mathematical Physics. Vol 1 A : A-C. Oxford: Elsevier, 2006, p. 438-447}, year = {2006}, publisher = {SISSA}, organization = {SISSA}, isbn = {0125126611}, url = {http://hdl.handle.net/1963/6473}, author = {Boris Dubrovin} } @article {2005, title = {Wetting of rough surfaces: a homogenization approach}, journal = {Proc. R. Soc. Lon. Ser. A 461 (2005) 79-97}, number = {SISSA;75/2004/M}, year = {2005}, abstract = {The contact angle of a drop in equilibrium on a solid is strongly affected by the roughness of the surface on which it rests. We study the roughness-induced enhancement of the hydrophobic or hydrophilic properties of a solid surface through homogenization theory. By relying on a variational formulation of the problem, we show that the macroscopic contact angle is associated with the solution of two cell problems, giving the minimal energy per unit macroscopic area for a transition layer between the rough solid surface and a liquid or vapor phase. Our results are valid for both chemically heterogeneous and homogeneous surfaces. In the latter case, a very transparent structure emerges from the variational\\napproach: the classical laws of Wenzel and Cassie-Baxter give bounds for the optimal energy, and configurations of minimal energy are those leading to the smallest macroscopic contact angle in the hydrophobic case, to the largest one in the hydrophilic case.}, doi = {10.1098/rspa.2004.1364}, url = {http://hdl.handle.net/1963/2253}, author = {Antonio DeSimone and Giovanni Alberti} } @article {2004, title = {Well-posedness for general 2x2 systems of conservation laws}, journal = {Mem. Amer. Math. Soc. 169 (2004), no. 801, x+170 pp.}, number = {SISSA;27/99/M}, year = {2004}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1241}, author = {Fabio Ancona and Andrea Marson} } @article {Bertola:Warped, title = {Warped products with special Riemannian curvature}, journal = {Bol. Soc. Brasil. Mat. (N.S.)}, volume = {32}, number = {1}, year = {2001}, pages = {45{\textendash}62}, issn = {0100-3569}, author = {Marco Bertola and Gouthier, Daniele} } @book {2000, title = {Well-posedness of the Cauchy problem for n x n systems of conservation laws}, series = {Mem. Amer. Math. Soc. 146 (2000), no. 694, 134 p.}, number = {SISSA;184/96/M}, year = {2000}, note = {Chapter 1 and 2}, publisher = {American Mathematical Society}, organization = {American Mathematical Society}, url = {http://hdl.handle.net/1963/3495}, author = {Alberto Bressan and Graziano Crasta and Benedetto Piccoli} } @article {1993, title = {Workshop on point interactions, Trieste, 21-23 December 1992}, number = {SISSA;8/93/ILAS/MS}, year = {1993}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/71}, author = {Gianfausto Dell{\textquoteright}Antonio} } @article {1985, title = {Weak convergence of measures on spaces of semicontinuous functions.}, journal = {Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 79 (1985), no. 5, 98-106}, number = {SISSA;76/85/MP}, year = {1985}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/463}, author = {Gianni Dal Maso and Ennio De Giorgi and Luciano Modica} }