ABSTRACT. In this talk, I will present a rigorous theory for topological photonic materials in one dimension. Especially, I will focus on periodic photonic structures with both time-reversal and inversion symmetry, for which the Zak phase is quantized. Given a photonic structure that consists of two semi-infinite systems with distinct topological indices on the two sides of an interface, we show that interface modes exist inside the common gap of two semi-infinite systems. The stability of the interface state mode will also be discussed. For finite topological structures, we investigate the scattering resonances and show that the imaginary parts of the resonances decay exponentially with respect to the size of the structure.

ABSTRACT. Our goal is to advance the development of wave-guiding subwavelength crystals (i.e. high-contrast metamaterials) by developing designs whose properties are stable with respect to geometric imperfections. Using layer-potential formulations and asymptotic techniques, we have studied the topological properties of chains of resonator dimers and seen that stable edge modes exist at the interfaces of topological indices. In this talk, we will examine more recent work which builds on this by proving results which describe how edge mode frequencies cross the subwavelength band gap when dislocations are introduced to the initial periodic array. We study infinite chains of resonators, using the method of fictitious sources, and also conduct a stability analysis through numerical experiments on the corresponding truncated finite structure. This approach gives an intuitive insight into the mechanisms that underpin the existence of robust localized edge modes in topological crystals and reveals an analytic way to fine tune their properties.

ABSTRACT. We study wave propagation inside metamaterials consisting of high-contrast subwavelength resonators subject to amplification and damping. The amplification and damping are modelled through complex material parameters, resulting in a non-Hermitian formulation and the appearance of exceptional points. The usual topological indices may therefore attain fractional values, interpreted as a partial band inversion. Remarkably, purely non-Hermitian edge modes can appear in structures whose Hermitian counterparts do not support edge modes. In this talk, we will use layer potential techniques and asymptotic analysis to explicitly compute the frequency of such non-Hermitian edge modes in an infinite, one-dimensional chain of resonators. Additionally, we will show that the degree of localization increases with the strength of the amplification and damping, also showing that these edge modes vanish in the Hermitian limit. Our analytic results are confirmed through numerical computations on large, finite, systems.

ABSTRACT. We investigate the solution landscape of a reduced Landau–de Gennes model for nematic liquid crystals on a two-dimensional hexagon at a fixed temperature, as a function of λ—the edge length. This is a generic example for reduced approaches on regular polygons. We apply the high-index optimization-based shrinking dimer method to systematically construct the solution landscape consisting of multiple solutions, with different defect configurations, and relationships between them. We report a new stable T state with index-0 that has an interior −1/2 defect; new classes of high-index saddle points with multiple interior defects referred to as H-class and TD-class saddle points; changes in the Morse index of saddle points as λ2 increases and novel pathways mediated by high-index saddle points that can control and steer dynamical pathways on the solution landscape. The range of topological degrees, locations and multiplicity of defects offered by these saddle points can be used to navigate the complex solution landscapes of nematic liquid crystals and other related soft matter systems.

ABSTRACT. Defects in a liquid crystal director field can arise due to external factors such as applied electric or magnetic fields, or the constraining geometry of the cell containing the liquid crystal material. Understanding the formation and dynamics of defects is important in the design and control of liquid crystal devices, and poses significant challenges for numerical modelling. In this talk we consider the numerical solution of a Q-tensor model of a nematic liquid crystal, where defects arise through rapid changes in the Q-tensor over a very small physical region in relation to the dimensions of the liquid crystal device. The efficient solution of the resulting six coupled partial differential equations is achieved using a finite element based adaptive moving mesh approach, where an unstructured triangular mesh is adapted towards high activity regions, including those around defects. Spatial convergence studies show the adaptive method to be optimally convergent using quadratic triangular finite elements. The full effectiveness of the method can be seen when solving a challenging two-dimensional dynamic Pi-cell problem involving the creation, movement, and annihilation of defects.

ABSTRACT. Most of the approaches currently used to model liquid crystals (LCs) at continuum and mesoscopic length-scales are grid-based. As a result, their resolution - and their use of computational effort - is usually uniform across the simulated volume. As a consequence, a lot of CPU-time is spent calculating very standard and boring behaviours. If, to address this, variable resolution grids ARE used (e.g. to resolve a defect region) their ability to track a moving feature is limited.

Mesh-free methods such as Smooth Particle Hydrodynamics (SPH), were first introduced for the simulation of galaxies. They are now also used to model highly dynamic phenomena (e.g. tidal flows and breaking waves) utilising moving reference points which can dynamically annihilate or split in response to local system conditions. They are, though, prone to instabilities if care is not taken with boundary conditions. In this talk, I'll set out the basis of an LC implementation of SPH and the pros and cons of its further use.

ABSTRACT. In this talk we deal with equilibrium configurations of one-dimensional particle system with non-convex nearest-neighbour and next-to-nearest-neighbour interactions and its passage to the continuum. We first review classical and more recent results about the topic and then we discuss the case of external forces applied to the system. In particular, we derive compactness results for a Γ-development of the energy and we study the structure of the minimizers of the Γ-limit. Finally, we give some ideas on how to obtain the first order Γ-limit. This is based on a joint work with Julian Fischer and Anja Schlömerkemper.

ABSTRACT. We analyze a finite-difference approximation of a functional of Ambrosio-Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $\delta$ is smaller than the ellipticity parameter $\epsilon$, we show the $\Gamma$-convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no $L^p$ fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort.

ABSTRACT. We present some discrete models for crystals with surface scaling of the interaction energy. We assume that at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising transitions between different energy wells. In particular, changes of orientation can be penalised: this may replace the positive-determinant constraint usually required when only nearest neighbours are accounted for. From joint works in collaboration with R. Alicandro and M. Palombaro

ABSTRACT. A periodic crystal is modeled as a periodic set of zero-sized points in 3-space. Such a periodic set is usually given by a unit cell (a parallelepiped defined by three edge-lengths and three angles) and a motif of points with fractional coordinates in this cell. Representing a crystal as a unit cell plus a motif is highly ambiguous. Hence a reliable comparison of rigid crystals should be based on isometry invariants that are preserved under any rigid motions and are independent of a unit cell and a motif. The talk will discuss complete and continuous classifications of periodic crystals up to isometry. Most past invariants of crystals such as symmetry groups are discontinuous under atomic vibrations. The first results are based on joint papers with Andy Cooper, Angeles Pulido, Herbert Edelsbrunner, Mathijs Wintraecken, Teresa Heiss, Phil Smith, Marco Mosca, Dan Widdowson, e.g. arxiv:2009.02488.

ABSTRACT. Crystal Structure Prediction aims to reveal the properties that stable crystalline arrangements of a molecule have without stepping foot in a laboratory, consequently speeding up the discovery of new functional materials. Since it involves producing large datasets that themselves have little structure, an appropriate classification of crystals could add structure to these datasets and further streamline the process. We focus on geometric invariants, in particular introducing the density fingerprint of a crystal. After exploring its computations via Brillouin zones, we go on to note the useful properties of this classification before describing some applications.

ABSTRACT. The fundamental model of a periodic structure is a periodic point set up to rigid motion or isometry. The recent paper (arXiv:2104.11046) defined isometry invariants (density functions), which are complete in general position and continuous under perturbations. This work introduces much faster isometry invariants (average minimum distances), which are also continuous and distinguish some sets that have identical density functions. We explicitly describe the asymptotic behaviour of the new invariants for a wide class of sets including non-periodic. The proposed near linear time algorithm processed a dataset of hundreds of thousands of real structures in a few hours on a modest desktop. The talk is based on the joint work (arXiv:2009.02488) with D.Widdowson, A.Pulido, V.Kurlin, A.Cooper.

ABSTRACT. Periodic point sets model solid crystalline materials(crystals) by representing every atom as a zero-sized point. These periodic sets are considered up to rigid motion or isometry, because crystal structures are determined in a rigid form. We introduce new isometry invariants (pointwise distance distributions), which are stronger than average minimum distances from our previous work(arXiv:2009.02488). We prove that these invariants are continuous under perturbations of points. This continuous distance and a fast algorithm allow us to visualise huge crystal datasets as minimum spanning trees.

ABSTRACT. Paperboard is an engineering material consisting of a number of separate sheets of paper, that have been bonded together. Experimental evidence shows that paperboard undergoing bending develops phenomenologically plastic hinges. We consider a nonlinearly elastic mathematical model for paperboard, allowing debonding of the sheets at a given cost per unit area. Analysis of our model predicts a number of different regimes, including some where bending is concentrated in delaminated hinges, where the mid-plane of each individual layer may deform isometrically. This is joint work with Sergio Conti (Bonn) and Julia Orlik (Kaiserslautern).

ABSTRACT. We discuss fracture in a one dimensional toy model. A chain of particles is considered, interacting through nearest-neighbour Lennard-Jones potentials which are stochastically distributed. The convex-concave structure of the Lennard-Jones interactions allows for fracture. We derive the variational limit in the framework of $\Gamma$-convergence of this particle system which turns out to involve a homogenized energy density. Further, we study the limit of a rescaled version of the functional with a limiting energy of Griffith's type, consisting of an elastic part and a jump contribution. In a further approach, fracture is studied at the level of the discrete energies.

ABSTRACT. The representative volume element (RVE) method is a popular method for approximation of homogenized properties of random materials. In recent times, a significant progress has been made in understanding the approximation error for linear elliptic equations and convex integral functionals. We study nonlinearly elastic randomly laminated composite materials, and thus homogenization of a nonconvex integral functional. Under appropriate assumptions on the energy density (frame indifference; minimality non-degeneracy and smoothness at identity) and on the rapid decay of correlations of the random material, we establish an optimal error estimate for the RVE approximation in a neighborhood of the set of rotations.

ABSTRACT. The interaction between a peregrine falcon and a dove is visibly non-reciprocal. Unlike the dogma preached by Newton’s third law, the actions they exert on each other are by no means equal and opposite. What happens to the well-established framework of phase transitions in non-reciprocal systems far from equilibrium? In this talk, I will answer this question by looking at three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Simple demonstrations with robots will be presented along with naturally occurring phenomena from various domains of science that share a common feature: reciprocity has no reason to exist. In all these cases, the emergence of unique time-dependent many-body phases can be captured by combining insights from non-Hermitian quantum mechanics and bifurcation theory. This approach lays the foundation for a general theory of critical phenomena in non-reciprocal matter.

ABSTRACT. Asymmetric wave propagation in nonreciprocal structures enables exciting opportunities to control and enhance light-matter interactions in extreme ways. In this talk, we present some of our recent efforts on different exciting topics at the frontier of this field. We discuss the implications of causality for nonreciprocal plasma media, such as the existence of a low-loss frequency window with anomalous nonmonotonic dispersion, which enables a giant and broadband nonreciprocal response. We also present our work on extreme field enhancements and giant broadband nonlinear effects with one-way modes, as well as our recent findings on drift-biased nonreciprocal media, the presence of exceptional points in their dispersion relation, and their implications for nonreciprocal nonmagnetic thermal radiation effects.

ABSTRACT. Acoustic and elastic metamaterials with time- and space-dependent material properties have received great attention recently as a means to induce nonreciprocity, or directional control, of propagating mechanical waves. One potential application of nonreciprocal acoustics research is the circulator, which achieves one-way sound transmission through a network of ports. In this work, an elastic wave circulator is proposed and investigated, which is composed of a thin elastic ring with three semi-infinite elastic waveguides attached, creating a three-port network. Nonreciprocity is achieved for both flexural and extensional waves by modulating the elastic modulus of the ring in a rotating fashion. Two numerical models are derived and implemented to compute the elastic wave field in the circulator. The first is an approximate model based on coupled mode theory, which makes use of the known mode shapes of the non-modulated system. The second model is a finite element approach that includes a Fourier expansion in the harmonics of the modulation frequency and radiation boundary conditions at the ports. The coupled mode model is compared with the finite element approach, and the conditions on the modulation parameters that enable a high degree of nonreciprocity for extensional and flexural modes are then discussed.

ABSTRACT. Magnetic materials, such as ferromagnets and ferrites, are known to exhibit certain light-matter interactions with nonreciprocal effects, such as Faraday rotation, making them ideal for the design of non-reciprocal devices such as isolators, phase shifters, and many others. However, a major problem with such magnetic materials is they typically induce high absorption in frequency ranges of interest, an effect that is detrimental to the performance of these devices. Recently, it has been demonstrated that magnetic-dielectric photonic crystals are exceptional at producing enhanced gyroscopic effects while reducing overall absorption by several orders of magnitude and over broad frequency ranges in comparison to their constituent homogeneous magnetic materials. In this talk, we introduce and discuss this phenomenon while presenting some of the mathematical tools and techniques which are being used to analyze the macroscopic effects of these materials with a focus on one-directional periodically layered magnetic-dielectric composites. Numerical results will also be presented demonstrating the persistence of certain gyroscopic effects for large loss parameters arising from the conductivity of the magnetic layers.

ABSTRACT. Atomistic-to-continuum coupling methods are a class of computational multiscale methods which combine the accuracy of the atomistic model and the efficiency of the continuum model for the computation of defects in crystal solids. Such methods can be efficiently implemented by adaptivity and achieve (quasi-)optimal balance between accuracy and efficiency. In this talk, we will present a simplified a posteriori error estimator for a consistent a/c coupling method in 2D. Such error estimator is essentially a simplified version of the residual based error estimator for the a/c method which avoids the computation of model error in the continuum bulk. We will show both analytically and numerically that the model error in the continuum region, which is expensive in computational due to the discrepency of the finite element mesh and the reference lattice, is of higher order compared with other sauce of error and thus can be omitted. Numerical experiments are also given to demonstrate the efficiency of the simplified error estimator compared with the classical residual based error estimator for the adaptive computation of crystal defects.

ABSTRACT. Quasiperiodic structures, related to irrational numbers, are a class of important and widely existing systems, including quasicrystals, incommensurate systems, defects, interfacial problems. Due to the irrational numbers, the quasiperiodic system is a space-filling structure without decay, which results in difficulty in numerical computation. A traditional method is using a periodic system to approximate the quasiperiodic system. It produces a Diophantine approximation error. We will propose an efficient method to avoid the Diophantine approximation and obtain high accurate quasiperiodic solutions in this talk. We also apply the novel method to material computation, including soft quasicrystals, quasiperiodic quantum systems, and quasiperiodic interfaces.

ABSTRACT. We study the convergence of the elastic deformation from an atomistic model to a continuum model based on Cauchy-Born rule for crystalline solids, where point defects are allowed to exist. We prove, under certain sharp stability conditions at zero temperature of the perfect lattice, that the solids are stable when temperature and defect concentration are both low. Based on the stability conditions at zero/finite temperature and with/without defects, we show that the defected version of Cauchy-Born rule gives a correct nonlinear elasticity model in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with free energy functionals obtained from the Cauchy-Born rule. Both static and dynamic problems are considered. The results are focused on the simple crystals and can be easily extended to complex ones.

ABSTRACT. We prove the convergence from the atomistic model to the Peierls-Nabarro (PN) model of complex lattice in the two-dimensional bilayer system. We show that the displacement field of the dislocation solution of the PN model converges to the dislocation solution of the atomistic model with second-order accuracy. Consistency of the PN model and stability of the atomistic model are essential in the proof.

ABSTRACT. Recently, a novel type of a multiscale simulation, called Relative Resolution (RelRes), was introduced. In a single system, molecules switch their resolution in terms of their relative separation, with near neighbors interacting via fine-grained potentials yet far neighbors interacting via coarse-grained potentials; notably, these two potentials are analytically parameterized by a multipole approximation. This multiscale approach is consequently able to correctly retrieve across state space, the structural and thermal, as well as static and dynamic, behavior of various nonpolar mixtures. Our current work focuses on the practical implementation of RelRes in LAMMPS, specifically for the commonly used Lennard-Jones potential. By examining various correlations and properties of several alkane liquids, including complex solutions of alternate cooligomers and block copolymers, we confirm the validity of this automated LAMMPS algorithm. Most importantly, we demonstrate that this RelRes implementation gains almost an order of magnitude in computational efficiency, as compared with conventional simulations. We thus recommend this novel LAMMPS algorithm for anyone studying systems governed by Lennard-Jones interactions.

ABSTRACT. Bottom-up coarse-graining of polymers is commonly performed by matching structural order parameters such as distributions of bond lengths, bending and dihedral angles and pair distribution functions to corresponding properties from atomistic simulations. In this study, we introduce the distribution of nearest-neighbors as an additional order parameter in the concept of local density potentials. We describe how the inverse-Monte Carlo method provides a framework for forcefield development that is capable of overcoming challenges associated with the inevitable coupling present in such systems. The technique is developed using polyisoprene melts as a prototype and we demonstrate how thermodynamic and conformational properties can be captured at different state points provided that the forcefield is refined. We evaluate both the single-particle and the collective dynamics of these coarse-grain models, demonstrating that all forcefields result to similar acceleration relative to dynamics of atomistic systems. Finally, we discuss extensions to inhomogeneous systems where multibody non-bonded interactions present significant advantages in the context of coarse-grain modeling of polymers.

ABSTRACT. I will discuss the use of diffusion maps combined with the quasistationary distribution to create an effective framework for enhanced samplng of molecular systems with rare events.  The QSD gives localized control of the diffusion map which in turn reveals local collective variables that can be used to enhance sampling.   An iterative algorithm can be constructed to target these collective variables for enhancement (biasing), using say metadynamics, with transitions signalled by changes in the diffusion map spectrum.     This is joint work with Zofia Trstanova and Tony Lelievre.

ABSTRACT. The aim of molecular dynamics is to study the time-evolution of a molecular system in order to deduce various macroscopic properties. To do so, one needs to sample the Boltzmann-Gibbs measure. A classic process used in this scope is the overdamped Langevin dynamics. Such process has good theoretical properties, but one practical issue arises, that of metastability: the system can remain trapped in energetic wells for long periods of time, and the system's law's relaxation towards the equilibrium can be far too slow. In order to avoid metastability, one can rely on the Adaptive Biasing Force (ABF) method, whose convergence as already been proven in the conservative case (when the interaction force can be written as a gradient).

In this talk, we will present a study of the ABF method’s robustness under generic forces. We first ensure the flat histogram property is satisfied, and then introduce a fixed point problem yielding the existence of a stationary state. Using classical entropy techniques, we prove the exponential convergence of both biasing force and law of the process as time goes to infinity. We will eventually quickly present the work in progress regarding the implementation of the ABF method within the Tinker–HP software.

ABSTRACT. We propose a novel data-driven approach to solving a classical statistical mechanics problem: given data on collective motion of particles, characterise the set of free energies associated with the particle system. We demonstrate empirically that the particle data contains all the information necessary to infer the free energy of the underlying physical system. Unlike traditional physical modelling, which seeks to construct analytic approximations, our proposed approach leverages modern Bayesian computational capabilities to accomplish the same goal in a purely data-driven fashion. The Bayesian paradigm permits us to combine first principles with simulation data to obtain uncertainty-quantified predictions of the free energy, in the form of a probability distribution over the family of free energies, consistent with the observed particle data.

ABSTRACT. The computational cost of usual Monte Carlo methods for sampling a posteriori laws in Bayesian inference scales linearly with the number of data points. One option to reduce it to a fraction of this cost is to resort to mini-batching in conjunction with unadjusted discretizations of Langevin dynamics, in which case only a random fraction of the data is used to estimate the gradient. However, this leads to an additional noise in the dynamics and hence a bias on the invariant measure which is sampled by the Markov chain. We advocate using the so-called Adaptive Langevin dynamics, which is a modification of standard inertial Langevin dynamics with a dynamical friction which automatically corrects for the increased noise arising from mini-batching. We investigate in particular the practical relevance of the assumptions underpinning Adaptive Langevin (constant covariance for the estimation of the gradient), which are not satisfied in typical models of Bayesian inference; and show how to extend the approach to more general situations.

ABSTRACT. We propose a general framework for coarse-graining of multiscale diffusions with parameter uncertainties. Specifically, we consider slow-fast systems with uncertain drift or diffusion coefficients and derive an averaged or homogenised equation that represents a worst-case scenario for any given (possibly path-dependent) quantity of interest. We do so by reformulating the slow-fast system as an optimal control problem in which the unknown parameter plays the role of a control variable that can take any values in the parameter set. For systems with unknown diffusion coefficient, the underlying stochastic control problem admits an interpretation in terms of a stochastic differential equation driven by a G-Brownian motion. We discuss convergence of the (worst-case) slow process with respect to a nonlinear expectation on the probability space induced by the G-Brownian motion, and we illustrate the theoretical findings with simple numerical examples from stochastic turbulence modelling, one of which exhibits metastability for certain parameter regimes.

ABSTRACT. The study of the essential dynamics of molecular dynamics along a given reaction coordinate function has gained considerable research attentions in the past years. In this talk, I will discuss the effective dynamics of a high-dimensional diffusion process, obtained by conditional expectations. In contrast to the original process, the effective dynamics is a lower-dimensional object and provides information about the dynamics along the reaction coordinate function. I will discuss its properties, numerical algorithms, as well as its applications in understanding the dynamics of molecular systems (e.g. timescales calculations). This talk is based on joint work with Carsten Hartmann, Tony Lelievre, and Christof Schuette.

ABSTRACT. Chiral ferromagnets have spatially modulated magnetic order exemplified by helices, spirals, and skyrmion crystals. The theoretical understanding of these states is based on a competition of a strong Heisenberg exchange interaction favoring uniform magnetization and a weaker Dzyaloshinskii-Moriya (DM) interaction promoting twists in magnetization. We offer a geometric approach, in which chiral forces are a manifestation of curvature in spin parallel transport [1]. The resulting theory is a gauged version of the Heisenberg model, with the DM vectors serving as background SO(3) gauge fields. This geometrization of chiral magnetism is akin to the treatment of gravity in general relativity, where gravitational interactions are reduced to a curvature of spacetime. The geometric perspective provides a simple way to define a conserved spin current in the presence of spin-orbit interaction. The gauge-dependent nature of the DM term raises questions about its linear dependence on the (gauge-independent) spin current [2-3]. We also show that the gauged Heisenberg model in d=2 has a skyrmion-crystal ground state for a magic value of an applied magnetic field.

[1] D. Hill et al., arXiv:2008.08681. [2] T. Kikuchi, Phys. Rev. Lett. 116, 247201 (2016). [3] F. Freimuth et al., Phys. Rev. B 96, 054403 (2017).

ABSTRACT. Chiral skyrmions are particle-like solutions of the Landau–Lifshitz equation for magnets with the Dzyaloshinskii–Moriya (DM) interaction, characterized by a topological number. We study the profile and the energy of an axially symmetric skyrmion in the asymptotic limits of small and large DM parameter (alternatively, large and small anisotropy). We give exact formulas for the profile of the core, of the domain wall, and of the far-field. The derived solutions show the different length scales that are present in the skyrmion profiles. The matching of the fields leads to the skyrmion radius as a function of the DM parameter. The Belavin-Polyakov (BP) solution of the pure exchange model is shown to play the role of a universal limit of profiles for small skyrmions. The picture is created of a chiral skyrmion that is born out of a BP solution with an infinitesimally small radius, as the DM parameter is increased from zero. In the case of large DM interaction, the skyrmion radius becomes large and diverges to infinity as the DM parameter approaches a certain value. The skyrmion profile enters in formulas for the energy and for dynamical phenomena, for example, skyrmion translation, rotation, and breathing modes.

ABSTRACT. Magnetic skyrmions are a prime example of topologically non-trivial spin textures observed in a variety of magnetic materials. The orthodox theory of skyrmions in ultrathin ferromagnetic layers with interfacial Dzyaloshinskii-Moriya interaction (DMI) relies on a model that accounts for the dipolar interaction through an effective anisotropy term, neglecting long-range effects. However there have been a growing body of experimental and numerical evidence that points to a need to take into account the long-range dipolar energy in the models describing magnetic skyrmions. The above considerations put into question the validity of the commonly used assumption that the long-range contribution of the dipolar interaction is negligible. Here we use rigorous mathematical analysis to develop a skyrmion theory that takes into account the full dipolar energy in the thin film regime and provides analytical formulas for compact skyrmion radius, rotation angle and energy [2,3]. We demonstrate that the DMI threshold at which a compact skyrmion loses its Néel character is a factor of 3 higher than that for a single domain wall. The estimation of this reorientation thickness is important for applications as the skyrmion angle affects its current-induced dynamics.

ABSTRACT. A wide diversity of new skyrmion solutions in 2D chiral magnets has been reported recently. A naturally arising question is related to their dynamical properties, particularly the mobility of these skyrmions under impart of spin-transfer torque. Only a few particular magnetic solitons were studied earlier, but the systematic studies were missing. The analysis of many different solutions allowed us to reveal intriguing features of the skyrmion dynamics. In particular, when the driving force has the form of Zhang-Li torque, the distribution of the skyrmions in the velocity space exhibits evident geometrical properties. The analysis of the connection between the symmetry of the skyrmions and their dynamics allows to split up all skyrmions into three main classes: i) topologically trivial, ii) high-symmetrical, and iii) low-symmetrical skyrmions. Interestingly, the velocities distribution for each class represents a circle or a set of circles that degenerate into points at certain conditions. Based on numerical simulations of the Landau-Lifshitz equation and semianalytical Thiele's approach, we succeeded in explaining all the features of this distribution and its evolution under the variation of the internal and external parameters of the physical system.

ABSTRACT. Molecular dynamics (MD) is a cornerstone of computational materials science due to its high predictive power. However, the high computational cost of MD rather severely limits accessible length and time scales. While massively parallel computers are extremely adept at extending length scales, conventional domain decomposition techniques do little to extend times at modest sizes. About 20 years ago, Voter proposed a technique called Parallel Replica Dynamics that can overcome this limitation by parallelizing in the time domain, thereby allowing one to efficiently exploit large computing resources. I will recount evolution of the method and illustrate using practical examples how our ability to directly access long timescales has improved with each new generation of computers and algorithms. I will also discuss how the understanding of the mathematical underpinnings of the methods has yielded new insights into the behavior of systems evolving through rare events.

ABSTRACT. We will examine the nonlinear theory for smectics introduced by Brener and Marchenko. The main result is a sharp lower bound on the energy. The sharp lower bound corresponds to an equipartition of energy between compression and bending strains and was previously observed only when the Gaussian curvature of the layers vanishes. The 3D model contains as a special case the well studied 2D Aviles-Giga model, and the consequences of this connection will be discussed as well.

ABSTRACT. We present the derivation via Gamma-convergence of a two-dimensional energy functional, for modeling the effects of an electric field in a thin bent-core liquid crystal sample, in the ferromagnetic SmA-like phase. Starting from a three-dimensional sample, we prove that under proper rescaling, in the small thickness limit, the electric self-interactions give rise to boundary terms.

ABSTRACT. Topological defect plays an important role in the physics of liquid crystals. Although a large amount of previous studies is devoted to compute the stable defect structures in liquid crystals as a consequence of geometric frustration, how do we search for the entire family tree of all possible solutions without unwanted random guesses? Here we introduce a saddle dynamics method to construct the solution landscape. The method can not only identify the transition state between energy minima, but also reveal the relationships between all stationary states. As illustration, we solve the Landau-de Gennes energy to construct the defect landscapes of confined nematic liquid crystals.

ABSTRACT. We study the global well-posedness of the Allen-Cahn Ohta-Nakazawa model with two fixed nonlinear volume constraints. Utilizing the gradient flow structure of its free energy, we prove the existence and uniqueness of the solution by following De Giorgi's minimizing movement scheme.

ABSTRACT. Cells receive chemical signals at localized surface receptors, process the data and make decisions on where to move or what to do. Receptors occupy only a small fraction of the cell surface area, yet they exhibit exquisite sensory capacity. In this talk I will give an overview of the mathematics of this phenomenon and discuss recent results focusing on receptor organization. In many cell types, receptors have very particular spatial organization or ​clustering​ - the biophysical role of which is not fully understood. In this talk I will explore how the number and configuration of receptors allows cells to deduce directional information on the source of diffusing particles. This involves a wide array of mathematical techniques from asymptotic analysis, homogenization theory, computational PDEs and Bayesian statistical methodologies. Our results show that receptor organization plays a large role in how cells decode their environmental situation and infer the location of distant sources.

ABSTRACT. In this talk, I will present a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws. The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility. I will focus on a phase field formulation of the optimal control problem, hence exploiting the well-developed mathematical theory for the optimal control of semilinear PDEs. The solver for the discretised PDEs is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement. An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally. Furthermore, parallelisation as well as an adaptive step gradient update for the control are employed to further improve efficiency. I will present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency with applications, including 3D cell migration.

ABSTRACT. Many complex nonlinear systems have intrinsic structures such as energy dissipation or conservation, and satisfy certain global constraints, such as conservations of volume, mass, and/or surface area. I shall discuss how to design efficient numerical approaches using the SAV approach with Lagrange multipliers to preserve these structures. We shall present several applications, including in particular a phase-field model of vesicle membranes which conserves volume and surface area in addition to energy dissipation.

ABSTRACT. Many of the mathematical models in this mini-symposium involve a phase field variable. We show recent results on the accuracy of time stepping schemes for these problems. There are some subtle issues that are highlighted when the schemes are considered with a local error tolerance in the limit as the order parameter goes to zero. We also show some benchmark solutions for phase field models from materials science that can be used to assess the performance of new schemes.

ABSTRACT. We consider the propagation of acoustic waves in a particular periodic medium which consists of the plane $\mathbb{R}^2$ minus an infinite set of equispaced hexagonal perfect conductor obstacles. The distance between each obstacle is supposed to be small. We introduce two types of unbounded lineic defects, the zigzag one and the armchair one, by changing the distance between two obstacles from each side of a line having respectively a zigzag form or an armchair one. Our aim is to find guided modes, that is to say solutions of the homogeneous wave equation propagating along the defect. It is well known that the guided modes are related to a spectral problem. We exhibit conditions on the perturbations which ensure existence of guided modes. We show in particular that the conditions which ensure existence of guided modes are completely different if the perturbation follows a zigzag line and an armchair one. More precisely, the zigzag perturbations exhibit a certain stability (whose precise definition will be given) that the armchair perturbations have not. We will illustrate all the theoretical results by numerical simulations.

ABSTRACT. Twisted bilayer graphene (TBG) has been experimentally observed to exhibit almost flat bands when the twisting occurs at certain magic angle.s We show that in the approximation of vanishing AA-coupling, the magic angles (at which there exist entirely flat bands) are given as the eigenvalues of a non-hermitian operator, and that all bands start squeezing exponentially fast as the angle θ tends to 0. In particular, as the interaction potential changes, the dynamics of magic angles involves the non-physical complex eigenvalues. Using our new spectral characterization, we show that the equidistant scaling of inverse magic angles, is special for the choice of tunnelling potentials in the continuum model, and is not protected by symmetries. While we also show that the protection of zero-energy states holds in the continuum model as long as particle-hole symmetry is preserved, we observe that the existence of flat bands and the exponential squeezing are special properties of the chiral model. Finally, we analyze the density of states, also in the presence of an external magnetic field, and study applications to superconductivity. Joint work with M. Embree, M. Lemm, J. Wittsten,  X. Zhu, and M. Zworski

ABSTRACT. We will discuss some recent results on the spectral theory of some single and multi-layer graphene models in magnetic fields, including Cantor spectrum, spectral decomposition, Dirac cones.

ABSTRACT. Following the work by Xia et. al. for modelling smectic A liquid crystals, we provide further analysis to their proposed double Landau--de Gennes model, which couples a second order PDE for nematic tensor order parameter Q and a fourth order PDE for smectic density variation u. We first illustrate the existence result and then proceed to the a priori error analysis for both Q and u. More specifically, C^0 interior penalty methods are used to numerically solve the fourth order problem. Finally, we demonstrate the derived theoretical results by numerical experiments.

ABSTRACT. In recent years, energy-minimization finite-element methods have been proposed for the computational modeling of equilibrium states of several types of liquid crystals. Here, we present a four-field formulation for models of smectic A liquid crystals, based on the free-energy functionals proposed by Pevnyi, Selinger, and Sluckin, and by Xia et al. The Euler-Lagrange equations for these models include fourth-order terms acting on the smectic order parameter (or density variation of the LC). While H^2 conforming or C^0 interior penalty methods can be used to discretize such terms, we investigate introducing the gradient of the smectic order parameter as an explicit variable, and constraining its value using a Lagrange multiplier. In this talk, we will discuss analysis of the finite-element discretization, and the construction of optimal linear solvers for the saddle-point systems that result from discretization and linearization of the models.

ABSTRACT. The focus of this talk will be on the applications of the Full Approximation Storage (FAS) scheme and the Fast Subspace Descent (FASD) scheme in the modeling of equilibrium configurations for nematic liquid crystals under free elastic effects based on the Frank-Oseen free-energy model, subject to the unit-length constraint of the director. The FASD scheme is a generalization of the classical FAS scheme, and it can be thought of as an inexact version of nonlinear multigrid solvers based on space decomposition and subspace correction. Both methods allows greater control of solutions on smaller subspaces of the problem which means linearizing and solving smaller systems, and error analysis demonstrates that both are convergent methods. Thus, we consider a finite-element approach to discretize the constrained optimization problem, and compare the performance of the FAS nonlinear solver with FASD. To the best of our knowledge, application and comparison of these two methods on the Frank-Oseen free-energy model is a novel approach. We illustrate the algorithms' performance by solving experiments with a wide range of physical parameters as well as simple and patterned boundary conditions. This is joint work with James H. Adler, Xiaozhe Hu, and Tim Atherton.

ABSTRACT. This paper deals with the application of a finite-strain homogenization framework to develop constitutive models for anisotropic dielectric elastomer composites (DECs) consisting of initially aligned, rigid dielectric inclusions whose centers are distributed randomly in an elastomeric matrix. For this purpose, we make use of a partial decoupling strategy [1] to handle the strong coupling of the mechanical and electric effects in the composite. The homogenized electro-elastic energy of the composite is written in terms of a purely mechanical problem incorporating the effects of prescribed torques on the inclusions, together with a electrostatic problem evaluated in the deformed configuration of the composite. The model predicts the existence of certain non-symmetric “extra” Cauchy stresses—arising in the composite beyond the purely electrical (Maxwell) and purely mechanical stresses—which can be directly linked to changes in the effective electric susceptibility of the composite with the rotation of the inclusions. These stresses provide a mechanism for either enhancing or hindering the possible development of macroscopic instabilities in these materials [2].

References [1] P. Ponte Castañeda, M.H. Siboni, Int. J. Non-Linear Mech. 47, 293 (2012) [2] M.H. Siboni, P. Ponte Castañeda, Mech, Res. Commun. 96, 75 (2019)

ABSTRACT. The conductivity tensor for the stationary linear Hall effect and the permittivity tensor for the linear optical Faraday effect are mathematically analogous. We review our theoretical and experimental work on three-dimensional (3D) Hall effect metamaterials. This includes unbounded effective 3D metamaterial Hall coefficients, sign reversal of the effective Hall coefficient using chainmail-like 3D metamaterials, as well as tailoring of the effective anisotropic Hall matrix. The latter is exemplified by the parallel Hall effect.

References: Phys. Rev. X 5, 021030 (2015); Appl. Phys. Lett. 107, 132103 (2015); Phys. Rev. Lett. 118, 016601 (2017); Phys. Rev. Appl. 7, 044001 (2017); Phys. Rev. Lett. 120, 149702 (2018); New J. Phys. 20, 083034 (2018); Phys. Rev. Mater. 3, 015204 (2019).

ABSTRACT. We investigate wave propagation in dynamic laminates, i.e., composites where the elastic moduli and mass density are modulated in space and time in a plane, periodic and progressive fashion. The modulation breaks time-invariance thus introducing a bias in space-time and ultimately leading to the failure of reciprocity. Specifically, at low frequencies, the effective behavior of dynamic laminates in d+1 dimensions is proven to be of the Willis type with a non-negligible Willis coupling, that is a coupling between stress and velocity as well as between momentum and strain. Closed-form expressions of the Willis coupling as well as of the other effective constitutive tensors are found and analyzed.

ABSTRACT. Using homogenization, Willis discovered that the momentum in elastic composites is macroscopically coupled with the strain through a constitutive tensor. The now-termed Willis tensor not only enlarges the design space of metamaterials, but is also necessary for the effective description to be physical. In this talk, I will show how additional tensors of Willis type emerge by generalizing the homogenization theory of Willis to thermoelastic-, piezomagnetic- or piezoelectric media. I will provide examples for the latter case that demonstrate an electromomentum coupling. I will further show that this coupling is needed for obtaining a physically valid effective model. Finally, I will give a proof-of concept of how this coupling can be used in an active device for wave control.

ABSTRACT. We present a non-parametric Bayesian approach for developing structure-property models for grain boundaries (GBs) with built-in uncertainty quantification (UQ). Using this method we infer a structure-property model for H diffusivity in [100] tilt GBs in Ni at 700K based on molecular dynamics (MD) data. We then leverage these results to perform uncertainty propagation (UP) for mesoscale simulations of the effective diffusivity of polycrystals to investigate the interaction between structure-property model uncertainties and GB network structure. We observe a fundamental interaction between crystallographic correlations and spatial correlations in GB networks that causes certain types of microstructures (those with large populations of J2- and J3-type triple junctions) to exhibit intrinsically larger uncertainty in their effective properties. We also investigate the influence of different types of input data (bicrystal vs. polycrystal) and observe evidence of a transition between a data-rich regime in which bicrystals yield more accurate results to a data-limited regime in which polycrystals provide improved inferences.

ABSTRACT. The quantification and propagation of uncertainty in microstructure modeling of materials remain a relatively unexplored aspect of computational materials science approaches. We study both parameter and parametric uncertainties in the numerical solution of the Cahn-Hilliard model for the description of local and non-local interactions with a smooth physical potential coupled to the mechanical equilibrium equation. We first establish the probabilistic uncertainty in the description of the smooth physical potential (statistical integrity of chemical potential, a physically relevant logarithmic model) for the material under consideration by a Markov Chain Monte Carlo-based inference of the parameters of this potential model. We then use a sampling scheme preserving marginal distributions and pairwise correlations to propagate uncertainties across a high-dimensional model input space. We evaluate the first- and second-order moments in the quantities of interest that we interpret from the microstructure images. We correlate this with the expected macroscopic behavior of the material under study, in this case, mass scattering versus interface scattering in thermoelectric material.

ABSTRACT. Gaussian processes (GPs) are ubiquitously used in sciences and engineering for metamodeling, uncertainty quantification, or Bayesian analyses. Standard GPs, however, can only handle numerical or quantitative variables. In this talk, I will introduce latent map Gaussian processes (LMGPs) that inherit the attractive properties of GPs but are also applicable to mixed data that have both quantitative and qualitative inputs. I will elaborate on the core idea of LMGPs which consists of learning a low-dimensional manifold where all qualitative inputs are represented by some latent quantitative features. Through a wide range of analytical and real-world examples, I will demonstrate the advantages of LMGPs over state-of-the-art methods in terms of accuracy and versatility. In particular, I will show that LMGPs (1) can handle variable-length inputs, (2) have a nice neural network interpretation, and (3) can assimilate multi-fidelity data without imposing any hard constraints on how low and high fidelity data sources are related.

ABSTRACT. Multi-scale design is a cutting-edge research area to meet the increasing need for high-performance materials in electronics, energy, biomedical, and structural applications. The research in this field is rapidly expanding with the fabrication of adaptive thermal response materials, energetic composites, and materials for green energy applications. In this study, we present an overview of multi-scale design methods that link micro and macro (component) scales to calculate the mechanical properties of various materials, such as metals, metallic alloys, composites, and meta-materials. The material properties in macro-scale are sensitive to microstructural variations, which can be introduced by the experimental uncertainty arising from the fluctuations in thermal and stress gradients during processing/manufacturing. In particular, the coupled interactions between processing, microstructure, and structural-level response are captured using a concurrent approach that integrates high-fidelity material models into uncertainty quantification and machine learning (ML) schemes. With the integration into an automated optimization framework and incorporation of supervised ML techniques, we significantly reduced the computation time, effectively captured the outcomes of stochasticity in material design, and simplified achieving multiple material designs with optimized properties. Using this numerical framework, we can effectively address the tedious multi-scale design requirements of high-performance materials.

ABSTRACT. An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. This however leads to a slower convergence. A second alternative is to use a fixed step-size and run independent replicas of the algorithm and average these. A third option is to run replicas of the algorithm and allow them to interact. It is unclear which of these options works best. To address this question, we reduce the problem of the computation of an exact minimizer with noisy gradient information to the study of stochastic mirror descent with interacting particles. We study the convergence of stochastic mirror descent and make explicit the tradeoffs between communication and variance reduction. We provide theoretical and numerical evidence to suggest that interaction helps to improve convergence and reduce the variance of the estimate.

ABSTRACT. It is well known that stochasticity can play a fundamental role in various biochemical processes. Isothermal, well-mixed systems can be adequately modeled by Markov processes and, for such systems, methods such as Gillespie’s algorithm are typically employed. While such schemes are exact and easy to implement, the computational cost of simulating such systems becomes prohibitive as the frequency of the reaction events increases. This has motivated numerous coarse-grained schemes, where the “fast” reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation for systems where all reactants are present in large concentrations, the approximation breaks down when the fast chemical species exist in small concentrations. In this talk, we present a hybrid scheme for simulating well-mixed stochastic kinetics, using Gillespie–type dynamics to simulate the network in regions of low reactant concentration, and chemical Langevin dynamics when the concentrations of all species are large. These two regimes are coupled via an intermediate region in which a “blended” jump-diffusion model is introduced. We will also discuss the extension of these methods for simulating spatial reaction kinetics models, blending together partial differential equation with compartment based approaches, as well as compartment based approaches with individual particle models.

ABSTRACT. The generation of synthetic data that follow the distribution of real data has become a pervasive tool in several industrial and engineering fields including material science applications. One of the most successful families of generative models is the so called Generative Adversarial Nets (GANs). GANs are notoriously difficult to train and several issues concerning their stability and convergence exist. Inspired by statistical mechanics and information theory, we develop a rigorous and general framework for constructing divergences which interpolate between $f$-divergences and integral probability metrics. The proposed divergences inherit important properties from both families. We perform GAN training via divergence minimization using variational representations. We will present numerical convergence results for heavy-tailed distributions under the proposed divergence as well as performance improvements over the standard GAN models in image generation.

ABSTRACT. We present information-theoretic, non-parametric Uncertainty Quantification methods for testing the robustness of data-informed probabilistic graphical models (PGMs). They are typically assembled by combining expert-built mathematical models with multi-sourced data. PGMs allow to build complex probabilistic models by assigning different data inputs to individual nodes and expressing their dependencies through conditional distributions. They are classified into Markov Random Fields and Bayesian Networks (BNs). Learning PGMs from domain knowledge and data, more often sparse, multi-scaled and/or imperfect, incorporates diverse sources of uncertainty at different scales. This may impact the model predictive capability on targeted quantities of interest. We develop systematic UQ methods for PGMs to handle these uncertainties and provide guarantees for the predictive reliability of a data-informed baseline PGM, by perturbing model component(s) and/or graph structure. We rank model components of a BN baseline by the impact individual perturbations have on the prediction errors of the quantities of interest. Then, we can systematically improve the most error-prone components of the BN by mitigating its prediction error with better modeling or better data. We demonstrate our methods on systems from statistical mechanics, medical diagnostics and fuel cell material design.

ABSTRACT. Solution-phase synthesis can produce fcc metal nanocrystals with a variety of non-equilibrium shapes, such as nanocubes or octahedra, tailored to optimize properties for plasmonics or catalysis. However, these structures are metastable, so the nanocrystal ultimately undergoes surface diffusion mediated evolution to its equilibrium Wulff shape. We develop stochastic atomistic-level modeling of such reshaping incorporating a realistic description of the diffusion of undercoordinated surface atoms, and also perform complementary coarse-grained modeling. Evolution involves transfer of atoms from corners or edges to nucleate new layers of {100} side facets for nanocubes, and {111} side-facets for octahedra. Typical synthesized structures have some degree of truncation of edges and corners, and our analysis shows that the effective barrier for reshaping depends strongly on this degree of truncation. Results compare well with TEM experiments for Pd nanocrystals. References: ACS Nano 14 (2020) 8551; Chem. Rev. 119 (2019) 6670.

ABSTRACT. The thermodynamic stability of groups of atoms forming a nanoparticle can be estimated by comparing free energies. Free energy is defined in statistical mechanics as a high-dimensional integral in phase space. This integral cannot be evaluated directly but instead must be sampled numerically. Free energy calculation is well established for homogenous systems when structural order is high (i.e., for solids) by the Einstein crystal method, when order is low (gases or fluids) by thermodynamic integration, and for growing crystal nuclei via path sampling techniques. Here, I will discuss new theoretical methods that combine the advantages of existing methods and significantly extend them. No assumption about the degree of order in the system has to be made. The methods are particular useful for partially ordered systems as found in aggregates of atoms and colloids in solution during early stages of structure formation. I will present the theoretical motivation, briefly discuss numerical implementation, and an application to estimate the free-energy landscape of clusters. Extensions are the possibility to perform free-energy calculations without a reference state and to separate different contributions to free energy.

ABSTRACT. The derivation of variational models describing the epitaxial growth of thin films in the framework of the theory of Stress-Driven Rearrangement Instabilities (SDRI) will be presented, and the state of the art of the mathematical results described. By working in the context of both continuum and molecular mechanics, not only free boundary problems, but also atomistic models will be considered, and the discrete-to-continuum passage rigorously investigated in the intent to also provide a microscopical justification of the theory. An overview of the mathematical results achieved with various co-authors will be presented.

ABSTRACT. Diffusion of supported 3D nanoclusters (NCs) followed by coalescence leads to coarsening of ensembles of supported NCs, is a key pathway for degradation of supported metal catalysts. The dependence of the NC diffusion coefficient, DN, on size N (in atoms) is the key factor controlling Smoluchowski ripening kinetics, and traditional treatments assumed simple monotonic decrease with increasing size. We analyze a stochastic model for diffusion of (100)-epitaxially supported fcc NCs mediated by diffusion of atoms around the surface of the NC. Surface diffusion barriers under various environments are chosen to accurately describe Ag [Lai and Evans, Phy. Rev. Materials 3 (2019) 026001]. KMC simulations reveal a complex oscillatory variation of DN with N. Local minima DN not always correspond to N = Nc the equilibrium Winterbottom NC structure with closed-shell. Local maximum generally corresponds to N = Nc + 3. The oscillatory behavior is expected to disappear for larger N above O(10^2). Behavior has similarities but also basic differences from that for 2D supported NCs [Lai et al Phys. Rev. B 96 (2017) 235406]. Through detailed analysis of the energetics of the 3D NC diffusion pathway, we elucidate the above behavior as well as observed trends in effective diffusion barrier.

ABSTRACT. This work presents a variant of Fleming-Viot particle systems, which are a standard way to approximate the law of a Markov process with killing as well as related quantities. Classical Fleming-Viot particle systems proceed by simulating N trajectories, or particles, according to the dynamics of the underlying process, until one of them is killed. At this killing time, the particle is instantaneously branched on one of the (N-1) other ones, and so on until a fixed and finite final time T. In our variant, we propose to wait until K particles are killed and then rebranch them independently on the (N-K) alive ones. Specifically, we focus our attention on the large population limit and the regime where K/N has a given limit when N goes to infinity. In this context, we establish consistency and asymptotic normality results. The variant we propose is motivated by applications in rare event estimation problems through its connection with adaptive multilevel splitting techniques.

ABSTRACT. In the two decades since its inception transition path sampling (TPS) has become a common rare event technique in the molecular simulation toolbox. The great advantage of path sampling is that by focusing on the dynamical pathways undergoing the activated event, TPS avoids the problem of the choice of good collective variables to describe the reaction coordinate. Over the years many variants of TPS have been developed. Strangely enough, only recently large-scale TPS simulations on complex systems became feasible, such as clathrate nucleation and protein dissociation. Recently, we introduced several new approaches to enhance transition path sampling. The first is Nested TPS, which can fully cover trajectory space. The second is Virtual Interface Exchange TPS, which approximates the reweighted path ensemble by making use of the rejected pathways in a form of waste recycling, thus allowing direct evaluation of rate constants and free energy landscapes. Thirdly, we developed a new path sampling scheme for non-equilibrium dynamics. Finally, we introduced a method to incorporate experimental dynamical observables as constraints in the obtained path ensembles using the Maximum Caliber framework. I will give an overview of these recent developments, and discuss and illustrate the advantages and limitations of the respective algorithms.

ABSTRACT. The Adaptive Multilevel Splitting (AMS) is a powerful and versatile method to estimate rare events probabilities. The idea of the algorithm is to split the phase space into cells and calculate the probability to pass from one cell to another, a process that is done on the fly in order to reduce the variance of the estimation. Like any other splitting method, AMS gives an estimator for the probability of occurrence, when starting from a set of initial conditions. This probability can then be used to compute an estimation of the transition time at equilibrium, but only if the initial condition’s set represent the equilibrium. Here two problems are encountered: the choice of the distribution of initial conditions and its sampling. For the latter, it appears that the samples which contribute the most to the rare event probability estimator are typically not the most likely ones. This implies a large variance of the estimator. In this study we propose an adaptive importance sampling technique, that combined to AMS, allow us to sample the initial points both correctly and efficiently. We apply this method to a molecular toy case and show a significant gain in computational cost.

ABSTRACT. Computing averages with respect to probability measures, for instance the Gibbs-Boltzmann measure, on level-sets of reaction coordinates (submanifolds) is crucial in both free-energy calculations and computing closed projections of diffusion processes. In recent years, various numerical schemes based on reversible constrained stochastic dynamics have been proposed in the literature to address this problem. In this work we present a non-reversible generalisation of a projection-based scheme developed by Zhang [ESAIM: M2AN, 54 (2020), pp. 391–430]. This scheme consists of two steps – starting from a state on the submanifold, we first update the state using a non-reversible stochastic differential equation which takes the state away from the submanifold, and in the second step we project the state back onto the manifold using the long-time limit of a ordinary differential equation. In addition to proving the consistency of the scheme, we illustrate that the non-reversible scheme outperforms its reversible counterpart in terms of asymptotic variance.

ABSTRACT. This talk will introduce novel variational models in weighted Sobolev spaces with non-standard weights. The classical analysis tools do not apply in this setting. We introduce novel weighted Sobolev spaces and establish that the standard fractional order Sobolev spaces are a special case. We also derive novel trace results. The analysis is complemented with a finite element method with applications in imaging science.

ABSTRACT. Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator characterized by a doubly-variable fractional order and possibly truncated interactions. Under certain conditions on the model parameters and on the regularity of the fractional order we show that the corresponding Poisson problem is well-posed. We also introduce a finite element discretization and describe an efficient implementation of the finite-element matrix assembly in the case of piecewise constant fractional order. Through several numerical tests, we illustrate the improved descriptive power of this new operator across media interfaces. Furthermore, we present one-dimensional and two-dimensional h-convergence results that show that the variable-order model has the same convergence behavior as the constant-order model.

ABSTRACT. In this talk, I will introduce the recently developed meshfree methods based on the radial basis function to solve problems with the variable-order fractional Laplacian. The proposed methods take advantage of the analytical Laplacian of the radial basis functions so as to accommodate the discretization of the classical and variable-order fractional Laplacian in a single framework and avoid the large computational cost for numerical evaluation of the fractional derivatives. Moreover, our methods are simple and easy to handle complex geometry and local refinements, and their computer program implementation remains the same for any dimension d. The effects of variable-order fractional Laplacian will also be discussed.

ABSTRACT. We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s in [0.5,1). We also present an application of the advection-diffusion equation to anomalous transport of solutes.

ABSTRACT. Geometrically frustrated solids with a non-Euclidean reference metric are ubiquitous in biology and are becoming increasingly relevant in technological applications. Often the incompatibility results from surface accretion of mass as in tree growth or dam construction. We formulate a large-strain theory of continuous surface growth and discuss various means to control the acquired incompatibility by tailoring the deposition protocol.

ABSTRACT. We will report on recent advances in deriving effective theories for thin sheets consisting of multiple layers with (slightly) mismatching equilibria. While the regime of finite bending energy is well understood by now, the talk will focus on energy scaling regimes beyond Kirchhoff's theory leading to linearized Kirchhoff, von Kármán and linearized von Kármán functionals with a spontaneous curvature term. We will also investigate optimal energy configurations and find that the von Kármán scaling is critical for their generic shape.

ABSTRACT. Living organisms have mastered the dynamic control of residual stresses within sheets to induce shape transformation and locomotion. For instance, the spatiotemporal pattern of action-potential in a heart results in a dynamical stress field that leads to shape changes. Such out-of-equilibrium structures inspired the emerging field of soft robotics. However, state-of-the-art attempts to replicate this ability in synthetic materials are rudimentary.

In this talk, I will present the first autonomously shape-shifting sheets: thin sheets of responsive gel that shrink and swell in response to the phase of an oscillatory chemical (Belousov-Zhabotinsky) reaction[1]. Propagating reaction-diffusion fronts induce localized deformation of the gel. I will show that these localized deformations prescribe a spatiotemporal pattern of Gaussian-curvature, leading to time-periodic global shape changes. Next, I will present the computational tools and experimental protocols needed to control this system: principally the relationship between the Gaussian-curvature and the reaction phase, and optical imprinting of the wave pattern. Finally, I will present our current journey towards autonomous soft swimmers.

Together, these results demonstrate a route for modeling and developing fully autonomous soft machines mimicking some of the locomotive capabilities of living organisms [1] IL, Robert Deegan, and Eran Sharon, Phys. Rev. Lett. 2020 125(17) 178001.

ABSTRACT. Self-organized complex structures in nature, e.g., viral capsids, hierarchical biopolymers, and bacterial flagella, offer efficiency, adaptability, robustness, and multi-functionality. Can we program the self-assembly of three-dimensional (3D) complex structures using simple building blocks, and reach similar or higher level of sophistication in engineered materials? Here we present an analytic theory for the self-assembly of polyhedral nanoparticles (NPs) based on their crystal structures in non-Euclidean space. We show that the unavoidable geometrical frustration of these particle shapes, combined with competing attractive and repulsive interparticle interactions, lead to controllable self-assembly of structures of complex order. Applying this theory to tetrahedral NPs, we find high-yield and enantiopure self-assembly of helicoidal ribbons, exhibiting qualitative agreement with experimental observations. We expect that this theory will offer a general framework for the self-assembly of simple polyhedral building blocks into rich complex morphologies with new material capabilities such as tunable optical activity, essential for multiple emerging technologies.

ABSTRACT. Magnetic skyrmions are localized magnetic textures in magnetic films, behaving as particles and topologically different from the uniform ferromagnetic state. In metallic magnetic multilayers (MML) with perpendicular magnetic anisotropy (PMA), non-collinear chiral spin textures are stabilized by interfacial Dzyaloshinskii-Moriya interaction (DMI), which favours a unique sense of magnetization rotation. Magnetic skyrmions in MML were identified to be extremely promising for applications, as well as of fundamental interest [1]. In this presentation, I will describe some of our recent experimental results to prepare magnetic multilayered systems in which ferromagnetic [2] or antiferromagnetic skyrmions [3] are stabilized at room temperature and manipulated by current pulses through spin torque effects. I will also address more specifically the role of the dipolar fields in the MML that are not only responsible for an increase of the skyrmion diameter but also for the stabilization of hybrid 3D chiral textures [4]. [1] A. Fert, N. Reyren and V. Cros, Nat. Rev. Materials 2, 17031 (2017); [2] C. Moreau-Luchaire et al., Nat. Nanotech, 11, 444 (2016); W. Legrand et al, Nat. Materials 19, 34 (2020); W. Legrand et al, Sci. Adv. 4, eaat0415 (2018)

ABSTRACT. Skyrmions are topologically nontrivial patterns in the magnetization of extremely thin ferromagnets. Typically thought of as stabilized by the so-called Dzyaloshinskii-Moriya interaction (DMI), or antisymmetric exchange interaction, arising in such materials, they are of great interest in the physics community due to possible applications in memory devices.

In this talk, we will characterize skyrmions as local minimizers of a two-dimensional limit of the full micromagnetic energy, augmented by DMI and retaining the nonlocal character of the stray field energy. In the regime of dominating Dirichlet energy, we will provide rigorous predictions for their size and "wall angles". The main tool is a quantitative stability result for harmonic maps of degree 1 from the plane to the two-dimensional sphere, relating the energy excess of any competitor to the homogeneous H-distance to the closest harmonic map.

ABSTRACT. In classical Ferromagnetic Resonance (FMR) experiments, a microwave (AC) magnetic field is applied to a macroscopic ferromagnetic body subject to a strong constant magnetic field. At low AC power levels, only the spatially uniform mode of the magnetic precession is excited. When AC power increases above a threshold value, the uniform mode is nonlinearly coupled to spatially non-uniform spin wave modes and this leads to deviations from spatial uniformity. However, in submicron-scale ferromagnetic bodies, the geometric confinement substantially suppresses the nonlinear spin-wave interactions present in bulk ferromagnets. This allows the excitation of large-amplitude quasi-uniform precessions of magnetization, as it was theoretically predicted in Ref.[1] and recently experimentally confirmed [2]. The experiments show that the large angle quasi-uniform precession exhibit additional oscillations which manifest themselves in the form of nutations. These nutations can be probed and excited by applying additional appropriately tuned microwave signals. This results in new means for controlling highly nonlinear magnetization dynamics in nanostructures, opening interesting applicative opportunities in the context of magnetic nanotechnologies [2].

[1] G.Bertotti et al., "Spin-Wave Instabilities in Large-Scale Nonlinear Magnetization Dynamics", PRL 87, 217203 (2001) [2] Y.Li et al., "Nutation Spectroscopy of a Nanomagnet Driven into Deeply Nonlinear Ferromagnetic Resonance", PRX 9,041036 (2019)