Samuel Amstutz (University of Avignon)

Around the incompatibility operator: an intrinsic approach to linearized elasto-plasticity

The incompatibility operator is a second order differential operator that represents the classical Saint-Venant compatibility relations in continuum mechanics. It is well-known that the existence of incompatible strain is related to the presence of infinitesimal defects in the medium such as dislocations, and that dislocations are responsible for plastic deformations in crystals. I will present mathematical properties of the incompatibility operator as well as a new model for linearized elasto-plasticity based on the possibility that the total strain be incompatible. The construction of the model and its main features (existence of solution, elastic limit, free energy dissipation) will be discussed.

(Joint work with N. Van Goethem)

 

 

Pedro Antunes (University of Lisboa)

Is it possible to tune a drum?

It is well known that the sound produced by string instruments has a well defined pitch. Essentially, this is due to the fact that all the resonance frequencies of the string have integer ratio with the smallest eigenfrequency. However, it is enough to use Ashbaugh–Benguria bound for the ratio of the smallest two eigenfrequencies to conclude that it is impossible to build a drum with a uniform density membrane satisfying harmonic relations on the eigenfrequencies. On the other hand, it is known since the antiquity, that a drum can produce an almost harmonic sound by using different densities, for example adding a plaster to the membrane. This idea is applied in the construction of some Indian drums like the tabla or the mridangam. In this work we propose a density and shape optimization problem of finding a composite membrane that satisfy approximate harmonic relations of some eigenfrequencies. The problem is solved by a domain decomposition technique applied to the Method of Fundamental Solutions and Hadamard shape derivatives for the optimization of inner and outer boundaries. This method allows to present new configurations of membranes, for example a two- density membrane for which the first 21 eigenfrequencies have approximate five harmonic relations or a three-density membrane for which the first 45 eigenfrequencies have eight harmonic relations, both involving some multiple eigenfrequencies.

 

 

Ana Cristina Barroso (University of Lisboa)

Second-order structured deformations of continua

The theory of structured deformations aims to describe various deformations that may occur in a body, as a response to external loading, at different length scales, capturing geometrical effects at the macroscopic level of both smooth and non-smooth geometrical changes at sub-macroscopic levels.

Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of sub-macroscopic bending and curving.
In this talk an integral representation for a relaxed energy functional in the setting of second-order structured deformations is presented. Our result covers inhomogeneous initial energy densities (i.e., with explicit dependence on the position in the given body). Explicit formulas for bulk relaxed energies in a particular example are also provided.

 

 

Antonin Chambolle (École Polytechnique)

Some properties of SBD displacement fields and the regularity of minimizers of Griffth's energy

In joint works with Sergio Conti (Bonn), Gilles Francfort (Paris), Flaviana Iurlano (Paris) we have studied the properties of "special functions with bounded deformation" which arise in the theory of fractures: roughly speaking, they are deformations with integrable symmetrized gradient, outside of a discontinuity set.

They are needed to express the variational formulation of Griffith's theory (1920) in a sound (weak) mathematical form and show existence of minimizers. I will present some recent results: first, a Poincaré-Korn inequality which holds when the discontinuity set is small, and then applications, in particular to the existence of strong minimisers for Griffith's energy.

 

 

Pierluigi Colli (University of Pavia)

Optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type

This talk will report on recent results obtained in collaboration with Gianni Gilardi and Juergen Sprekels on some optimal control problems for a nonstandard nonlocal phase field system, which constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the mentioned authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation: the latter equation contains both nonlocal and singular terms that render the analysis difficult. Some results are presented on the well-posedness and stability of the system with the aim of discussing the optimal control problem.

 

 

Ana Bela Cruzeiro (University of Lisboa)

An entropic interpolation problem for incompressible viscid fluids

We introduce an analogue of Brenier’s problem for viscous fluids, where generalized flows are no more supported by absolutely continuous paths, but by Brownian sample paths. This stochastic variational problem turns out to correspond to an entropy minimization problem with marginal constraints.

This is a joint work with M. Arnaudon (Bordeaux), C. Léonard (Paris) and J.C. Zambrini (Lisbon).

 

 

Ana Paula Dias (University of Porto)

Networks and coupled cell networks of dynamical systems

This talk concerns coupled dynamical systems, where the coupling structure is described by a graph (network). Some concepts, results and examples of classes of coupled cell networks will be presented aiming to show why a theory of coupled cell networks is important in the study of coupled dynamical systems.

 

 

Isabel N. Figueiredo (University of Coimbra)

Image Processing and Biomathematics: Applications in Gastroenterology

We describe some recent results we have obtained in the fields of medical image processing and of biomathematics, targeting two specific problems in gastroenterology, which are, respectively, the location of the wireless capsule endoscope (WCE) inside the small bowel, and, the modelling and numerical simulation of the morphogenesis of the early stages of colon cancer.

In what concerns the image processing results we explain how an image registration approach can be a valuable auxiliary medical tool for WCE video interpretation. The capsule is a tiny device, that is ingested by the patient and during the time it operates, if films the interior of the gastrointestinal tract and generates thousands of images that are afterwards analysed. The medical doctors search for abnormalities in these images and subsequently, and ideally, their precise location in the small bowel relative to some known physiological landmark - we remark that the intestine moves due to the peristaltic movement, and the exact location of the capsule, that is propelled by peristalsis inside the body, is unknown. The methodology we propose enables an overall, and extremely quick, analysis of the entire video produced by the wireless capsule endoscope and, interestingly, it also provides an indication of the speed of the capsule, which is of course an important information towards capsule location.

The results in biomathematics concern the description of a biomechanical model that intends to represent and simulate the early stages of colon cancer. Aberrant crypt foci are clusters of aberrant (deviant from normal) colonic crypts (small pits, which are compartments of cells, in the colon epithelium) that are thought to be the precursors of colorectal cancer. The top orifices of colonic crypts can be visualised by clinicians, in real time, during a conventional colonoscopy exam. In normal crypts, the orifices are roundish and grouped in a regular pattern. However in aberrant crypts, the shape and pattern of the orifices have different forms, as for example, star-like, elliptical, sulcus-like, branch-like, tubular, or roundish but smaller than the typical size. The biomechanical model we propose couples the cell dynamics occurring inside the colonic crypt with the mechanical behaviour of the material surrounding the crypt. By using numerical simulations, we show that when a modification of the programmed cell mechanism is implemented, the irregular and abnormal patterns of the orifices of the colonic crypts are retrieved.

These are joint works with Carlos Leal (Department of Mathematics, University of Coimbra, Portugal), Luís Pinto (Department of Mathematics, University of Coimbra, Portugal), Pedro Narra Figueiredo (Faculty of Medicine, University of Coimbra, Portugal), Giuseppe Romanazzi (IMECC, State University of Campinas, Brazil), Richard Tsai (ICES, University of Texas at Austin, USA), Björn Engquist (ICES, University of Texas at Austin, USA).

 

 

Mário Figueiredo (University of Lisboa)

Alternating Direction Method of Multipliers with Plug-and-Play Regularizers: Convergence Guarantees and Applications

In this talk, I will address a recent trend in the use of the very popular ADMM (alternating direction method of multipliers) algorithm in imaging inverse problems: the so-called plug-and-play (PnP) approach, wherein a formal regularizer is replaced with a black-box denoiser, aiming at leveraging state-of-the-art denoisers in more general inverse problems. Since these denoisers usually lack an optimization formulation, classical results on the convergence of ADMM cannot be directly invoked. Recently, we have proposed a class of denoisers which, while achieving excellent performance, also allow guaranteeing convergence of the resulting PnP-ADMM algorithm. These denoisers are particularly well suited to certain data fusion problems in imaging, which we will describe and use to illustrate the proposed approach.

This is joint work with Afonso Teodoro and José Bioucas-Dias.

 

 

Jürgen Fuhrmann (Weierstrass Institute)

Robust quality preserving numerical methods for electroosmotic flow

Microscale electroosmotic flows occur in many interesting applications, including pore scale processes in fuel cell membranes and sensing with nanopores. We present a new approach for the numerical solution of coupled fluid flow and ion transport in a self-consistent electric field. Ingredients of the method are

  • Pressure-robust, pointwise divergence free finite element discretization of the Stokes equations describing the barycentric velocity of the ionic mixture
  • Generalization of the Nernst-Planck equations for ion transport to the case of finite ion sizes
  • Thermodynamically consistent, maximum principle observing finite volume method for ion transport including competition for finite available volume
  • Coupling approach between fluid flow and mass transport together with a fixed point iteration to solve the combined system.

The talk introduces the model and the discretization approach. It provides first results of numerical simulations confirming the validity and the advantages of the discretization approach. A number of open problems and challengig directions will be described.

  1. A.Linke. On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Computer methods in applied mechanics and engineering, 268:782--800, 2014.
  2. V. John, A. Linke, Ch. Merdon, M. Neilan, and Leo G. Rebholz. On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Review, 2017, Vol. 59, No. 3, pp. 492-544
  3. W. Dreyer. C. Guhlke, and R. Müller. Overcoming the shortcomings of the Nernst–Planck model. Physical Chemistry Chemical Physics 2013 15(19), 7075-7086
  4. J.Fuhrmann. Comparison and numerical treatment of generalised Nernst-Planck models. Comp.r Phys. Comm., 196 166-178, 2015.
  5. J.Fuhrmann. A numerical strategy for Nernst–Planck systems with solvation effect. Fuel cells, 16(6):704-714, 2016.
  6. Ch. Merdon, J.Fuhrmann, A.Linke, F.Neumann, T.Streckenbach, H.Baltruschat, and M.Khodayari. Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data. Electrochimca Acta, pages 1-10, 2016.

 

 

Holger Heitsch (Weirstrass Institute)

A probabilistic approach to optimization problems in gas transport networks

Stochastic optimization approaches are frequently used to model practical decision processes over time and under uncertainty, e.g., in finance, production, energy and logistics. We present an approach to deal with booked capacity optimization of gas transport networks under uncertainty. The resulting optimization problem represents a new class of models including mixed probabilistic and robust constraints. We establish an approach based on spheric-radial decomposition of Gaussian type random variables to deal with such models algorithmically. For simplicity, our numerical study focuses on stationary gas networks exhibiting tree structure.

 

 

Michael Hintermüller (Humboldt University of Berlin)

(Pre)Dualization, Dense Embeddings of Convex Sets, and Applications in Image Processing

For a class of non-smooth minimization problems in Banach spaces, predualization results and their connection to dense embeddings of convex sets are discussed. Motivating applications are related to non-smooth filters in mathematical image processing. For this problem class also some numerical aspects are highlighted including primal/dual splitting or ADMM-type methods as well as proper (numerical) dissipation reducing discretization.

 

 

Alexander Mielke (Weirstrass Institute)

Mathematical modeling of semiconductors: from quantum mechanics to devices

We discuss the consistent modeling of semiconductor devices from the mathematical point of view.
The task lies in coupling of several physical effects that occur on different temporal or spatial scales, namely optics via the Maxwell equations, charge transport via drift-diffusion models and quantum mechanical processes in embedded quantum dots, wires or layers.

We construct suitable hybrid models that are thermodynamically consistent in the sense that for the isolated system we have energy conservation and positive entropy production. The conservative dynamics is driven by a Hamiltonian structure involving the energy, and the dissipative dynamics is driven by an entropic gradient system.

(This is joint work with Markus Mittnenzweig and Nella Rotundo.)

 

 

Leonard Monsaingeon (University of Lisboa)

Unbalanced optimal transport for reaction-diffusion

The Wasserstein-Fisher-Rao distance on the space of positive measures was recently introduced, and roughly speaking corresponds to optimal transport with mass variations. In this talk I will discuss some applications of this unbalanced optimal transport setting to reaction-diffusion systems, both theoretical and numerical. In particular I will identify a Hele-Shaw model of tumor growth as a gradient flow in the space of measures.

 

 

Robert Patterson (Weirstrass Institute)

Coagulation–Transport Simulations with Stochastic Particles

In a wide range of engineering applications including soot formation and pharmaceutical crystallisation particles are transported in a fluid while occasionally undergoing coagulations with each other.

The integro-diferential equations describing the dynamics of such particle populations are numerically challenging so I will talk about how and why one can use Monte Carlo methods based on stochastic particles in order to generate approximate solutions.

I will sketch some results concerning the convergence and approximation properties of these Monte Carlo methods.

 

 

Tomáš Roubíček (Charles University, Prague, and Czech Academy of Sciences)

Various time discretisations of dynamical damage and phase-field fracture models

Various damage models and their variants as a phase-field fracture will be presented. Two basic variants of their discretisations will be scrutinized: the fully implicit (backward Euler) time discretisation and the splitted (staggered) scheme, both combined with a Crank-Nicolson (mid-point) scheme for the inertial term. A combination with some other phenomena as mass or heat transfer, plasticity, or interaction with fluidic regions will briefly be surveyed. Some 2D FEM computational experiments launched by Jose Reinoso (Univ. Sevilla), Jan Valdman (Czech Acad. Sci.), and Roman Vodička (Tech. Univ. Košice) will be presented, too.

 

 

Lisa Santos (University of Minho)

Lagrange multipliers in problems with nonconstant gradient constraint

We consider a stationary linear variational inequality with coercivity constant δ>0 and nonconstant gradient constraint δ and we prove the existence of solution of an equivalent Lagrange multiplier problem.

Under suitable assumptions on the data, when the constraint satisfies Δg2≤0, it will be shown that the Lagrange multiplier belongs to Lq(Ω), for 1<q<∞. Without the assumption on the sign of Δg2, we are only able to prove that the Lagrange multiplier belongs to L(Ω)'. In both cases, we characterize the limit problem when δ tends to zero.

(Joint work with Assis Azevedo)

 

 

Riccardo Scala (University of Lisboa)

A nonlinear approach to dislocations in 3-dimensional bodies

Dislocations are defects in the atomic lattice of crystals that are responsible for dissipative phenomena and plastic deformations of the material. We study variational problems in single crystals in the setting of nonlinear elasticity. In contrast to the linear approach, we avoid to use a core-radius to regularize the elastic energy, considering a polyconvex bulk energy with p growth, with 1<p<2. Thanks to the presence of high order terms in the energy we prove closeness results for the space of admissible fields by means of geometric measure theory, and specifically using the concept of Cartesian currents.

 

 

Vladimir Spokoiny (Weirstrass Institute)

Quantification of uncertainty in estimation of spectral projectors

Let X1,…,Xn be i.i.d. sample in Rp with zero mean and the covariance matrix Σ. The problem of recovering the projector onto an eigenspace of Σ from these observations naturally arises in many applications. Recent technique from Koltchinskii and Lounici (2015) helps to study the asymptotic distribution of the distance in the Frobenius norm ‖Πr-Π ̂r ‖_2 between the true projector Πr on the subspace of the r th eigenvalue and its empirical counterpart Π ̂r in terms of the effective rank of Σ. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector Πr from the given data. This procedure does not rely on the asymptotic distribution of ‖Πr-Π ̂r ‖_2and its moments. It could be applied for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for finite samples with an explicit error bound for the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples.

 

 

Marita Thomas (Weirstrass Institute)

Delamination processes in solids: GENERIC structure & analytical results

This contribution addresses the modeling of delamination processes in solids using an internal delamination variable in the thermodynamical modeling framework of GENERIC (General Equation of Non-Equilibrium Reversible Irreversible Coupling). This covers models for brittle, Griffith-type delamination, which describe sharp cracks in terms of a non-smooth constraint confining displacement jumps across interfaces to the null set of the delamination variable, as well as adhesive contact models, which regularize this constraint by a finite surface energy contribution. A suitable notion of solution for the resulting non-smooth PDE-system and existence results are adressed. In this context, for a viscoelastic solid with dynamic effects, the limit passage from models for adhesive contact to brittle, Griffith-type delamination is discussed in the sense of evolutionary Gamma-convergence.

 

 

Enrico Valdinoci (University of Milano)

Crystal dislocation, nonlocal equations and fractional dynamical systems

We study heteroclinic and multibump orbits for a system of equations driven by a nonlocal operator. Our motivation comes from the study of the atom dislocation function in a periodic crystal, according to the Peierls-Nabarro model. The evolution of the dislocation function can be studied by analytic techniques of fractional Laplace type. At a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior potential.

Such potential turns out to be either attractive or repulsive, depending on the mutual orientation of the dislocations, and the attractive potentials generate “particle collisions” in finite time. After the collisions, the system relaxes to the equilibrium exponentially fast, and the associated steady states provide a natural setting for the study of dynamics and chaos in a fractional framework.

 

 

Luís Nunes Vicente (University of Coimbra)

A Domain Decomposition Framework for Nonlinear Optimization

In this talk, we will consider continuous nonlinear optimization problems (for simplicity unconstrained) where the direct minimization of the objective function in the full space is out of reach. Our target includes large-scale optimization with derivatives and relatively large-size derivative-free optimization.

With these problems in mind, we propose a space decomposition framework for nonlinear optimization. At each iteration, the space of variables is decomposed into (possibly overlapping) subspaces, and a step is defined in each of the spaces by approximately minimizing a subspace model of the objective function. A synchronization phase is then performed to obtain a full-space step from the subspace ones. The main novelty lies in the synchronization phase which is inspired from successful Overlapping Domain Decomposition techniques for linear PDE systems. In particular, our framework covers the Restricted Additive Schwarz and Additive Schwarz with Harmonic Extension and several other synchronization strategies as special cases. In doing so, the model gradients and the subspace steps may be changed according to the overlaps. Using regularization or globalization schemes, such as Levenberg-Marquardt or trust regions, the new framework is guaranteed to converge globally at appropriated rates.

If time permits, we will discuss related issues such as gradient inexactness or random generation of subspaces. This is joint work with Serge Gratton (Toulouse) and Zaikun Zhang (Hong Kong).

 

 

Barbara Wagner (Weiestrass Institute)

Yield stress for two-phase flow of concentrated suspensions

A two-phase model for concentrated suspensions is derived from underlying microscopic balance laws, that incorporates a constitutive law combining the rheology for non-Brownian suspensions and granular flow. The resulting model exhibits a yield-stress behavior for the solid phase depending on the collision pressure.

This property is investigated for the simple geometry of plane Poiseuille flow, where an unyielded or jammed zone of finite width arises at the center of the channel. The stability properties and their relation to classical Bingham-type flows are discussed.

For a thin-film geometry with a free boundary a new thin-film model for concentrated suspensions with yield-stress properties is derived and investigated regarding existence of solutions.

 

 

Matthias Wolfrum (Weirstrass Institute)

Dynamics of coupled oscillator systems and their continuum limits

The collective dynamics of systems of coupled oscillators play an important role in various applications such as neuronal systems, optoelectronic communication, or power grids. We present some mathematical tools, such as the reduction to phase oscillator systems, their continuum limits, and the so called Ott-Antonsen reduction. Then, we give an overview over recent results on collective dynamical phenomena, including non-universal synchronization transitions and self-organized coherence-incoherence patterns, and discuss the corresponding stability problems in the continuum limit.