cvgmt Papershttps://cvgmt.sns.it/papers/en-usWed, 17 Apr 2024 02:56:09 +0000Intermediate domains for scalar conservation lawshttps://cvgmt.sns.it/paper/6544/F. Ancona, A. Bressan, E. Marconi, L. Talamini.<p>For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data $\overline{u}\in \bf{L}^\infty(\mathbb R)$ decays like $t^{-1}$. This paper introduces a class of intermediate domains $\mathcal P_\alpha$, $0<\alpha<1$, such that for $\overline u\in \mathcal P_\alpha$ a faster decay rate is achieved: $\mathrm{Tot.Var.}\bigl\{ u(t,\cdot)\bigr\}\sim t^{\alpha-1}$. A key ingredient of the analysis is a ``Fourier-type" decomposition of $\overline u$ into components which oscillate more and more rapidly. The results aim at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting.</p>https://cvgmt.sns.it/paper/6544/On a degenerate second order traffic model: existence of discrete evolutions, deterministic many-particle limit and first order approximationhttps://cvgmt.sns.it/paper/6543/D. Mazzoleni, E. Radici, F. Riva.<p>We propose and analyse a new microscopic second order Follow-the-Leader type scheme to describe traffic flows. The main novelty of this model consists in multiplying the second order term by a nonlinear function of the global density, with the intent of considering the attentiveness of the drivers in dependence of the amount of congestion. Such term makes the system highly degenerate; indeed, coherently with the modellistic viewpoint, we allow for the nonlinearity to vanish as soon as consecutive vehicles are very close to each other. We first show existence of solutions to the degenerate discrete system. We then perform a rigorous discrete-to-continuum limit, as the number of vehicles grows larger and larger, by making use of suitable piece-wise constant approximations of the relevant macroscopic variables. The resulting continuum system turns out to be described by a degenerate pressure-less Euler-type equation, and we discuss how this could be considered an alternative to the groundbreaking Aw-Rascle-Zhang traffic model. Finally, we study the singular limit to first order dynamics in the spirit of a vanishing-inertia argument. This eventually validates the use of first order macroscopic models with nonlinear mobility to describe a congested traffic stream.</p>https://cvgmt.sns.it/paper/6543/The search for NLS ground states on a hybrid domain: Motivations, methods, and resultshttps://cvgmt.sns.it/paper/6542/R. Adami, F. Boni, R. Carlone, L. Tentarelli.<p>We discuss the problem of establishing the existence of the Ground States forthe subcritical focusing Nonlinear Schr\"odinger energy on a domain made of aline and a plane intersecting at a point. The problem is physically motivatedby the experimental realization of hybrid traps for Bose-Einstein Condensates,that are able to concentrate the system on structures close to the domain weconsider. In fact, such a domain approximates the trap as the temperatureapproaches the absolute zero. The spirit of the paper is mainly pedagogical, sowe focus on the formulation of the problem and on the explanation of theresult, giving references for the technical points and for the proofs.</p>https://cvgmt.sns.it/paper/6542/The p-Laplace "Signature" for Quasilinear Inverse Problems with Large Boundary Datahttps://cvgmt.sns.it/paper/6541/A. Corbo Esposito, L. Faella, G. Piscitelli, R. Prakash, A. Tamburrino.<p> This paper is inspired by an imaging problem encountered in the framework ofElectrical Resistance Tomography involving two different materials, one or bothof which are nonlinear. Tomography with nonlinear materials is in the earlystages of developments, although breakthroughs are expected in thenot-too-distant future. We consider nonlinear constitutive relationships which, at a given point inthe space, present a behaviour for large arguments that is described bymonomials of order p and q. The original contribution this work makes is that the nonlinear problem canbe approximated by a weighted p-Laplace problem. From the perspective oftomography, this is a significant result because it highlights the central roleplayed by the $p-$Laplacian in inverse problems with nonlinear materials.Moreover, when p=2, this provides a powerful bridge to bring all the imagingmethods and algorithms developed for linear materials into the arena ofproblems with nonlinear materials. The main result of this work is that for "large" Dirichlet data in thepresence of two materials of different order (i) one material can be replacedby either a perfect electric conductor or a perfect electric insulator and (ii)the other material can be replaced by a material giving rise to a weightedp-Laplace problem.</p>https://cvgmt.sns.it/paper/6541/Tomography of nonlinear materials via the Monotonicity Principlehttps://cvgmt.sns.it/paper/6539/A. Corbo Esposito, V. Mottola, G. Piscitelli, A. Tamburrino.<p> In this paper we present a first non-iterative imaging method for nonlinearmaterials, based on Monotonicity Principle. Specifically, we deal with theinverse obstacle problem, where the aim is to retrieve a nonlinear anomalyembedded in linear known background. The Monotonicity Principle (MP) is a general property for various class ofPDEs, that has recently generalized to nonlinear elliptic PDEs. Basically, itstates a monotone relation between the point-wise value of the unknown materialproperty and the boundary measurements. It is at the foundation of a class ofnon-iterative imaging methods, characterized by a very low execution time thatmakes them ideal candidates for real-time applications. In this work, we develop an inversion method that overcomes some of thepeculiar difficulties in practical application of MP to imaging of nonlinearmaterials, preserving the feasibility for real-time applications. For the sakeof clarity, we focus on a specific application, i.e. the MagnetostaticPermeability Tomography where the goal is retrieving the unknown (nonlinear)permeability by boundary measurements in DC operations. This choice ismotivated by applications in the inspection of boxes and containers forsecurity. Reconstructions from simulated data prove the effectiveness of the presentedmethod.</p>https://cvgmt.sns.it/paper/6539/The $p_0$-Laplace "Signature" for Quasilinear Inverse Problemshttps://cvgmt.sns.it/paper/6540/A. Corbo Esposito, L. Faella, V. Mottola, G. Piscitelli, R. Prakash, A. Tamburrino.<p> This paper refers to an imaging problem in the presence of nonlinearmaterials. Specifically, the problem we address falls within the framework ofElectrical Resistance Tomography and involves two different materials, one orboth of which are nonlinear. Tomography with nonlinear materials in the earlystages of developments, although breakthroughs are expected in thenot-too-distant future. The original contribution this work makes is that thenonlinear problem can be approximated by a weighted $p_0$-Laplace problem. Fromthe perspective of tomography, this is a significant result because ithighlights the central role played by the $p_0$-Laplacian in inverse problemswith nonlinear materials. Moreover, when $p_0=2$, this result allows all theimaging methods and algorithms developed for linear materials to be broughtinto the arena of problems with nonlinear materials. The main result of thiswork is that for "small" Dirichlet data, (i) one material can be replaced by aperfect electric conductor and (ii) the other material can be replaced by amaterial giving rise to a weighted $p_0$-Laplace problem.</p>https://cvgmt.sns.it/paper/6540/An isoperimetric inequality for the first Robin-Dirichlet eigenvalue of the Laplacianhttps://cvgmt.sns.it/paper/6538/N. Gavitone, G. Piscitelli.<p> In this paper, we study the first eigenvalue of the Laplacian on doublyconnected domains when Robin and Dirichlet conditions are imposed on the outerand the inner part of the boundary, respectively. We provide that the sphericalshell reaches the maximum of the first eigenvalue of this problem among thedomains with fixed measure, outer perimeter and inner $(n-1)^{th}$quermassintegral.</p>https://cvgmt.sns.it/paper/6538/Free boundary regularity for the inhomogeneous one-phase Stefan problemhttps://cvgmt.sns.it/paper/6537/F. Ferrari, N. Forcillo, D. Giovagnoli, D. Jesus.<p> In this paper, we prove that flat solutions to inhomogeneous one-phase Stefanproblem are $C^{1,\alpha}$ in the $x_n$ direction.</p>https://cvgmt.sns.it/paper/6537/Lower semicontinuity and existence results for anisotropic TV functionals with signed measure datahttps://cvgmt.sns.it/paper/6536/E. Ficola, T. Schmidt.<p>We study the minimization of anisotropic total variation functionals with additional measure terms among functions of bounded variation subject to a Dirichlet boundary condition. More specifically, we identify and characterize certain isoperimetric conditions, which prove to be sharp assumptions on the signed measure data in connection with semicontinuity, existence, and relaxation results. Furthermore, we present a variety of examples which elucidate our assumptions and results.</p>https://cvgmt.sns.it/paper/6536/Bifurcation for a sharp interface generation problemhttps://cvgmt.sns.it/paper/6535/E. Acerbi, C. N. Chen, Y. S. Choi.https://cvgmt.sns.it/paper/6535/Stable periodic configurations in nonlocal sharp interface modelshttps://cvgmt.sns.it/paper/6534/E. Acerbi.https://cvgmt.sns.it/paper/6534/Stability of lamellar configurations in a nonlocal sharp interface modelhttps://cvgmt.sns.it/paper/6533/E. Acerbi, C. N. Chen, Y. S. Choi.https://cvgmt.sns.it/paper/6533/Minimal lamellar structures in a periodic FitzHugh-Nagumo systemhttps://cvgmt.sns.it/paper/6532/E. Acerbi, C. N. Chen, Y. S. Choi.https://cvgmt.sns.it/paper/6532/A priori regularity estimates for equations degenerating on nodal setshttps://cvgmt.sns.it/paper/6531/S. Terracini, G. Tortone, S. Vita.<p>We prove $\textit{a priori}$ and $\textit{a posteriori}$ Holder bounds andSchauder $C^{1,\alpha}$ estimates for continuous solutions to singular-degenerate equations with variable coefficients of type \[\mathrm{div}\left(<br>u<br>^a A\nabla w\right)=0\qquad\mathrm{in \}\Omega\subset\mathbb{R}^n,\]where the weight $u$ solves an elliptic equationof type $\mathrm{div}\left(A\nabla u\right)=0$ with a Lipschitz-continuous and uniformly elliptic matrix $A$ and has a nontrivial, possibly singular, nodalset. Such estimates are uniform with respect to $u$ in a class of normalizedsolutions having bounded Almgren's frequency. More precisely, we provide$\textit{a priori}$ Holder bounds in any space dimension, and Schauderestimates when $n=2$. When $a=2$, the results apply to the ratios of two solutions to the same PDE sharing their zero sets. Then, one can infer higherorder boundary Harnack principles on nodal domains by applying the Schauderestimates for solutions to the auxiliary degenerate equation. The results arebased upon a fine blow-up argument, Liouville theorems and quasiconformal maps.</p>https://cvgmt.sns.it/paper/6531/On the Gamma-convergence of some polygonal curvature functionalshttps://cvgmt.sns.it/paper/6530/A. M. Bruckstein, J. A. Iglesias.<p>We study the convergence of polygonal approximations of two variational problems for curves in the plane. These are classical Euler’s elastica and a linear growth model which has connections to minimizing length in a space of positions and orientations. The geometry of these minimizers plays a role in several image-processing tasks, and also in modelling certain processes in visual perception. We prove Gamma-convergence for the linear growth model in a natural topology, and existence of cluster points for sequences of discrete minimizers. Combining the technique for cluster points with a previous Gamma-convergence result for elastica, we also give a proof of convergence of discrete minimizers to continuous minimizers in that case, when a length penalty is present in the functional. Finally, some numerical experiments with these approximations are presented, and a scale invariant modification is proposed for practical applications.</p>https://cvgmt.sns.it/paper/6530/Two-dimensional shape optimisation with nearly conformal transformationshttps://cvgmt.sns.it/paper/6529/J. A. Iglesias, K. Sturm, F. Wechsung.<p>In shape optimisation it is desirable to obtain deformations of a given meshwithout negative impact on the mesh quality. We propose a new algorithm usingleast square formulations of the Cauchy-Riemann equations. Our method allows todeform meshes in a nearly conformal way and thus approximately preserves theangles of triangles during the optimisation process. The performance of ourmethodology is shown by applying our method to some unconstrained shapefunctions and a constrained Stokes shape optimisation problem.</p>https://cvgmt.sns.it/paper/6529/Critical yield numbers of rigid particles settling in Bingham fluids and Cheeger setshttps://cvgmt.sns.it/paper/6528/I. A. Frigaard, J. A. Iglesias, G. Mercier, C. Pöschl, O. Scherzer.<p>We consider the fluid mechanical problem of identifying the critical yieldnumber $Y_c$ of a dense solid inclusion (particle) settling under gravitywithin a bounded domain of Bingham fluid, i.e. the critical ratio of yieldstress to buoyancy stress that is sufficient to prevent motion. We restrictourselves to a two-dimensional planar configuration with a single anti-planecomponent of velocity. Thus, both particle and fluid domains are infinitecylinders of fixed cross-section. We show that such yield numbers arise from aneigenvalue problem for a constrained total variation. We construct particularsolutions to this problem by consecutively solving two Cheeger-type setoptimization problems. We present a number of example geometries in which thesegeometric solutions can be found explicitly and discuss general features of thesolutions. Finally, we consider a computational method for the eigenvalueproblem, which is seen in numerical experiments to produce these geometricsolutions.</p>https://cvgmt.sns.it/paper/6528/Shape Aware Matching of Implicit Surfaces based on Thin Shell Energieshttps://cvgmt.sns.it/paper/6527/J. A. Iglesias, M. Rumpf, O. Scherzer.<p>A shape sensitive, variational approach for the matching of surfacesconsidered as thin elastic shells is investigated. The elasticity functional tobe minimized takes into account two different types of nonlinear energies: amembrane energy measuring the rate of tangential distortion when deforming thereference shell into the template shell, and a bending energy measuring thebending under the deformation in terms of the change of the shape operatorsfrom the undeformed into the deformed configuration. The variational methodapplies to surfaces described as level sets. It is mathematically well-posedand an existence proof of an optimal matching deformation is given. Thevariational model is implemented using a finite element discretization combinedwith a narrow band approach on an efficient hierarchical grid structure. Forthe optimization a regularized nonlinear conjugate gradient scheme and acascadic multilevel strategy are used. The features of the proposed approachare studied for synthetic test cases and a collection of geometry processingexamples.</p>https://cvgmt.sns.it/paper/6527/A note on convergence of solutions of total variation regularized linear inverse problemshttps://cvgmt.sns.it/paper/6526/J. A. Iglesias, G. Mercier, O. Scherzer.<p>In a recent paper by A. Chambolle et al. <a href='Geometric properties of solutionsto the total variation denoising problem. Inverse Problems 33, 2017'>Geometric properties of solutionsto the total variation denoising problem. Inverse Problems 33, 2017</a> it wasproven that if the subgradient of the total variation at the noise free data isnot empty, the level-sets of the total variation denoised solutions converge tothe level-sets of the noise free data with respect to the Hausdorff distance.The condition on the subgradient corresponds to the source condition introducedby Burger and Osher <a href='Convergence rates of convex variational regularization.Inverse Problems 20, 2004'>Convergence rates of convex variational regularization.Inverse Problems 20, 2004</a>, who proved convergence rates results with respectto the Bregman distance under this condition. We generalize the result ofChambolle et al. to total variation regularization of general linear inverseproblems under such a source condition. As particular applications we presentdenoising in bounded and unbounded, convex and non convex domains, deblurringand inversion of the circular Radon transform. In all these examples theconvergence result applies. Moreover, we illustrate the convergence behaviorthrough numerical examples.</p>https://cvgmt.sns.it/paper/6526/Critical yield numbers and limiting yield surfaces of particle arrays settling in a Bingham fluidhttps://cvgmt.sns.it/paper/6525/J. A. Iglesias, G. Mercier, O. Scherzer.<p>We consider the flow of multiple particles in a Bingham fluid in ananti-plane shear flow configuration. The limiting situation in which theinternal and applied forces balance and the fluid and particles stop flowing,that is, when the flow settles, is formulated as finding the optimal ratiobetween the total variation functional and a linear functional. The minimalvalue for this quotient is referred to as the critical yield number or, inanalogy to Rayleigh quotients, generalized eigenvalue. This minimum value canin general only be attained by discontinuous, hence not physical, velocities.However, we prove that these generalized eigenfunctions, whose jumps we referto as limiting yield surfaces, appear as rescaled limits of the physicalvelocities. Then, we show the existence of geometrically simple minimizers.Furthermore, a numerical method for the minimization is then considered. It isbased on a nonlinear finite difference discretization, whose consistency isproven, and a standard primal-dual descent scheme. Finally, numerical examplesshow a variety of geometric solutions exhibiting the properties discussed inthe theoretical sections.</p>https://cvgmt.sns.it/paper/6525/