Schedule for: 22w5102 - Theoretical and Applied Aspects for nonlocal Models

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We will present some nonlocal variants of nonlinear conservation laws and their mathematical properties such as the preservation of entropy conditions. We will focus on models involving a finite range of nonlocal interactions that can lead to traditional conservation laws in the local limit. We will also consider their applications to traffic flows with connected vehicles. We will illustrate how the proper use of nonlocal information can play important roles in both theory and application.

A fruitful procedure to construct a nonlocal model consists of replacing the classical gradient by a nonlocal one. Nonlocal gradients are diverse, and are specified through a kernel. The choice of the kernel defines the nonlocal gradient and the associated functional space. A particular choice gives rise to Riesz $s$-fractional gradient, which satisfies some natural properties required for a gradient-like operator and has a degree of differentiabiliy of order $0 < s < 1$. However, they are not adequate for nonlinear elasticity since they require the domain to be the whole space. In this talk we present a truncated version of Riesz fractional gradient which is suitable in nonlinear elasticity. We explain the theory of existence of minimizers based on polyconvexity of the stored energy function.

Due to the nonlocality of force interactions and the evolution of discontinuities, material instability in peridynamics exhibits features that are not present in the local theory of elastodynamics. Among these, stress waves can be stable at some wavelengths and propagation directions but unstable in others. At the unstable wavelengths, waveforms grow at a finite rate over time, enabling initial value problems to be solved meaningfully in many cases. This is unlike the local theory, in which elastically unstable materials can blow up instantaneously. In some situations, nonlocal material instability can be useful in modeling the evolution of failure realistically. Because waveforms in the unstable regime of wave propagation grow at a bounded rate over time, they can be useful in the modeling the details of crack nucleation. The process zone surrounding the tip of a growing crack can also contain material points where the underlying material model is unstable. This talk will present a summary of results on unstable wave growth and the role of material instability in the modeling of fracture and microstructure evolution within peridynamics. Recent results on the self- shaping of elastic peridynamic fibers, an effect that occurs due to a type of material instability, will also be discussed.

Peridynamics is a nonlocal reformulation of continuum mechanics that is suitable for representing fracture and failure. For practical engineering applications, precise application of boundary conditions is essential. However, nonlocal boundary conditions (sometimes called volume constraints), especially of Neumann type, remain poorly understood. We begin with a discussion of the nonlocal boundary (sometimes called a “collar”) within a nonlocal model and review some practical approaches to application of boundary conditions over this domain. We then focus our discussion on nonlocal diffusion models and present a new approach for nonlocal boundary conditions of Neumann-type supported by numerical convergence studies.

Local-to-nonlocal (LtN) coupling is a popular approach to combine the strengths of nonlocal and local modeling to attain efficient and accurate solutions of nonlocal problems. In LtN coupling, nonlocal models are employed in regions exhibiting phenomena not well represented by local PDE-based models, while coupled to those local models used elsewhere for computational expediency. An example of this in solid mechanics is the combination of the computational efficiency of classical continuum mechanics with the capability to simulate crack propagation of peridynamics. A main issue in LtN coupling is the appearance of coupling artifacts around coupling interfaces which pollute the solutions of coupled problems. Common artifacts include failure of passing patch tests in static problems and artificial wave reflections in dynamic problems. In this talk, we will present a coupling artifact that has been overlooked in the LtN coupling literature: the lack of overall equilibrium. We will analyze the problem of the overall equilibrium in LtN coupling and demonstrate that this artifact originates from the lack of balance between local and nonlocal tractions at coupling interfaces, which commonly results from the presence of high-order derivatives of displacements in the coupling zone. Our analysis will be presented through an approach to couple peridynamics and classical continuum mechanics called splice. Numerical results will be shown to confirm the analysis and demonstrate its applicability in the development of adaptive strategies for the modeling of problems with evolving fractures. This is a joint work with Greta Ongaro, Ugo Galvanetto, Tao Ni, and Mirco Zaccariotto.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

To be physically relevant, mathematical models must guarantee existence and uniqueness of solutions, as well as continuity with respect to the data. Thus, small changes in data or parameters will lead to appropriate changes in the solution. The kernel in nonlocal models adds flexibility in handling discontinuities by recording long range interactions. We consider the stability of the solution with regards to changes in the forcing term, the nonlocal boundary term, and the kernel(s) (a part of the operator itself) all incorporating heterogeneous kernels. The biharmonic operator appears in modeling deformations and damage in beams and plates. We extend these continuous dependence results to these nonlocal higher order operators in a version of the nonlocal biharmonic that iterates the nonlocal Laplacian.

We consider a nonlocal functional introduced by Burchard, Choksi, and Topaloglu and develop a numerical method based on a piecewise constant function approximation. The variation of the functional is a (block-)Töplitz operator which can be applied using fast Fourier transforms in $\mathcal{O}(N \log N)$ time, where $N$ is the number of degrees of freedom. Combined with a suitable iterative solver and an interior point optimization method, we numerically find minimizers of the nonlocal swarming functional and study their dependence on parameters.

Nonlinear diffusion models have a wide variety of applications, including but not limited to temperature dependent diffusion of materials and liquid movement through porous mediums. Integrating these operators to the nonlocal calculus framework allows us to decrease the regularity requirements for the solutions to nonlinear diffusion systems. Here, we show that the actions of a specific class of nonlinear nonlocal operators converge to the actions of their classical counterpart. Additionally, we consider the existence and convergence of solutions to nonlocal nonlinear systems with Dirichlet boundary conditions in the limit of the vanishing horizon.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider load controlled quasistatic evolutions of displacement fields inside a material body. Here we are concerned with extreme displacements inside damaging materials and propose a field theory for calculating them. Well posedness results for a nonlocal continuum model is established. The model can be viewed as a peridynamic one. We show local existence and uniqueness of quasistatic evolution for load paths originating at critical points associated with local energy minima. These belong to the strength domain of the material. The evolution of the displacements however is not constrained to lie inside the strength domain of the material. Unlike other quasistatic fracture/damage modeling approaches the elastic fields are not global minimizers of energy but local energy minimizers. Also, the notion of strength domain is used in the model which is not related to energy minimization but to material strength. The second variation around stationary points is used to show stability of the evolution and energy balance. A numerical method is implemented and simulations of load controlled evolutions are given.

The inverse Gaussian measure $d\gamma_{-d}=\pi^{d/2}e^{|x|^2}dx$ in $\mathbb{R}^d$ is used in the construction of self-expanders, which are special solutions describing singularities of geometric flows. It is an example of non-doubling and non-Ahlfors regular measure for which most of the classical harmonic analysis theory does not apply. We will show that this measure arises again naturally in the fundamental analysis of fractional powers of first order differential operators with zero order terms. Furthermore, we will show that the inverse Gaussian measure is one particular example of a much broader theory of special functions and orthogonal polynomials that we created. This is joint work with Martín Mazzitelli (Instituto Balseiro, Argentina) and José L. Torrea (Universidad Autónoma de Madrid, Spain).

Fractional calculus and its application to anomalous diffusion has recently received a tremendous amount of attention. In complex/heterogeneous material mediums, the long-range correlations or hereditary material properties are presumed to be the cause of such anomalous behavior. Owing to the revival of fractional calculus, these effects are now conveniently modeled by fractional-order differential operators and the governing equations are reformulated accordingly. In the first part of the talk, we plan to introduce both linear and nonlinear, fractional-order differential equations. As applications, we will develop new physical models for geophysical electromagnetism and a new notion of optimal control will be discussed. In the second part of the talk, we will focus on novel Deep Neural Networks (DNNs) based on fractional operators. We plan to discuss the approximation properties and apply them to image denoising and tomographic reconstruction problems. We will establish that these DNNs are also excellent surrogates to PDEs and inverse problems with multiple advantages over the traditional methods. If time permits, we will conclude the talk by showing some of our initial results on chemically reacting flows using DNNs which clearly shows the effectiveness of the proposed approach.

In this talk, we will briefly review the one-dimensional Peierls--Nabarro model for straight edge dislocations in crystals. A phase parameter is used to describe the ratio between the microscopic and mesoscopic scale, where the dislocations dynamics are characterized by a system of one-dimensional ODEs. We will then present our recent progress on the higher dimensional problem in which the dislocation curves are not straight edge dislocations. At the mesoscopic scale, we exhibit dislocation curves moving by mean curvature. This is joint work with Stefania Patrizi (UT Austin).

In both normal tissue and disease states, cells interact with one another, and other tissue components using adhesion proteins. These interactions are fundamental in determining tissue fates, and the outcomes of normal development, and cancer metastasis. Traditionally continuum models (PDEs) of tissues are based on purely local interactions. However, these models ignore important nonlocal effects in tissues, such as long-ranged adhesion forces between cells. In this talk, I focus on the nonlocal “Armstrong adhesion model” (2006) for adhering tissue (an example of an aggregation equation). Since its introduction, this approach has proven popular in applications to embyonic development and cancer modeling. Combining global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical properties of the non-local term, we prove a global bifurcation result for the non-trivial solution branches of the scalar Armstrong adhesion model. I will demonstrate how we used the equation's symmetries to classify the solution branches by the nodal properties of the solution's derivative. Joint work with Thomas Hillen (University of Alberta).

Obstacle-type problems associated with some fourth order elliptic operators arise naturally in the linearized Kirchhoff-Love theory for plate bending phenomena. Moreover, as first observed by Yang, they can be seen as extension problems in the spirit of Caffarelli-Silvestre extension of obstacle-type problems for the fractional Laplacian of order greater than one. In this talk, we present our recent work on problems of this type. In our approach, we use tools from calculus of variation and potential theory to investigate the well-posedness of the problem and the regularity of a solution. Our analysis of the structure of the free boundary is based on monotonicity formulas of Almgren- and Monneau-type. This talk is based on papers collaborated with D. Danielli and A. Petrosyan.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Discussions on external funding opportunities, writing better proposals and life in industry and national labs.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Nonlocal models are characterized by integral operators that embed lengthscales in their definition. As such, they are preferable to classical partial differential equation models in situations where the dynamics of a system is affected by the small scale behavior, yet the small scales would require prohibitive computational cost to be treated explicitly. In this sense, nonlocal models can be considered as coarse-grained, homogenized models that, without resolving the small scales, are still able to accurately capture the system’s global behavior. However, nonlocal models depend on “kernel functions” that are often hand tuned. We propose to learn optimal kernel functions from high fidelity data by combining machine learning algorithms, known physics, and nonlocal theory. This combination guarantees that the resulting model is mathematically well-posed and physically consistent. Furthermore, by learning the operator rather than a surrogate for the solution, these models generalize well to settings that are different from the ones used during training. We apply this learning technique to find homogenized nonlocal models for subsurface solute transport solely on the basis of breakthrough curves.

In this talk I will review some recent results concerning nonlocal interaction problems motivated by dislocation theory and I will discuss some related open problems.

In this talk I will consider a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. It has recently been proved by R. Frank and E. Lieb that the ball is the unique (up to translation) global minimizer for sufficiently large mass. After a review of the literature on this minimization problem, I will focus on the issue of the stability of the ball, in the sense of the positivity of the second variation of the energy with respect to smooth perturbations of the boundary of the ball. I will present a characterization of the range of masses for which the second variation is positive definite (large masses) or negative definite (small masses). Moreover, I will discuss the connection between the stability of the ball and its local/global minimality. This talk is based on a work in collaboration with R. Cristoferi and I. Topaloglu.

The ball enjoys several optimality properties. In particular, for small masses, the ball is the unique volume constrained minimizer of the functional given by the sum of the isotropic perimeter and a Riesz type of nonlocal energy. Moreover, balls are the unique volume constrained maximizers and the unique volume constrained critical point of the energy given by Riesz type of nonlocal energies. The goal of this seminar is to investigate the validity of the above two results in the case where the isotropic perimeter is substituted with a crystalline anisotropic perimeter, and when the class of competitors is restricted to polygons with a fixed number of sides, respectively. In particular, for the former we will investigate the case of general kernels in any dimension, while for the latter we will focus on the case of Riesz potentials on triangles and quadrilaterals in dimension two. This talk is based on joint works with Marco Bonacini (Università di Trento) and Ihsan Topaloglu (Virginia Commonwealth University).

We propose a variational learning strategy for the discovery of non-equilibrium equations, through the variational action density from which these equations may be derived. The strategy is based on the so-called Onsager’s variational principle, which may be written as a function of the free energy and dissipation potential, and utilizes neural network architectures that strongly guarantee thermodynamic consistency. The method is applied to three distinct illustrative examples, aimed at showcasing distinct important features of the strategy proposed. These encompass (i) the phase transformation occurring in coiled-coil protein, where the free energy density is non-convex, (ii) the discovery of a reduced order model for the dynamic response of a viscoelastic material, which utilizes the variational structure as a tool for approximation, and (iii) a linear and nonlinear diffusion model, where both evolution equations may be obtained from distinct free energies and dissipation potentials (i.e., the action is not unique).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

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 e!ects of variable-order fractional Laplacian will also be discussed.

The naive discretization of nonlocal operators leads to matrices with significant density, as compared to classical PDE equations. This makes the efficient solution of nonlocal models a challenging task. In this presentation, we will discuss on-going research into assembly and multilevel solution techniques that are suitable for nonlocal models.

We show several quantitative results that generalize known nonlocal rigidity relations for vector fields representing deformations of elastic media. We show that the distance in Lebesgue space of a deformation from a rigid motion is bounded by a multiple of a strain energy associated to the deformation. This non-convex energy is a nonlocal constitutive relation that represents the extent to which the deformation stretches and shrinks distances. This inequality can be thought of as a nonlinear fractional Poincaré-Korn inequality. We linearize this inequality and use it as a crucial ingredient in obtaining a true fractional Korn inequality for Lipschitz domains. This fractional Korn inequality states that a class of vector fields whose semi-norms involve the magnitude of directional difference quotients is in fact equivalent to the class of fractional Sobolev spaces. We also show the validity of these inequalities for more general interaction kernels of non-fractional type.

We propose an extension of the well-known Klausmeier model of vegetation to two plant species that consume water at different rates. Rather than competing directly, the plants compete through their intake of water, which is a shared resource between them. In semi-arid regions, the Klausmeier model produces vegetation spot patterns. We are interested in how the competition for water affects the co-existence and stability of patches of different plant species. We consider two plant types: a “thirsty” species and a “frugal” species, that only differ by the amount of water they consume per unit growth, while being identical in other aspects. We find that there is a finite range of precipitation rate for which two species can co-exist. Outside of that range (when the rate is either sufficiently low or high), the frugal species outcompetes the thirsty species. As the precipitation rate is decreased, there is a sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. The pattern consisting of only frugal spots is the most resilient. The next-most-resilient pattern consists of all-thirsty patches, with the mixed pattern being less resilient than either of the homogeneous patterns. We also examine numerically what happens for very large precipitation rates. We find that for a sufficiently high rate, the frugal plant takes over the entire range, outcompeting the thirsty plant.

In this talk we consider a measure-theoretical formulation of the training of NeurODEs in the form of a mean-field optimal control with the L2-regularization. We derive the first order optimality conditions for the NeurODEs training problem in the form of a mean-field maximum principle, and show that it admits a unique control solution, which is Lipschitz continuous in time. Some instructive numerical experiments are also provided.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

We introduce a spectral method to approximate the fractional Laplacian with zero exterior condition. Our approach is based on interpolation by tensor products of sinc-functions, which combine a simple representation in Fourier-space with fast enough decay to suitably approximate the bounded support of solutions to the Dirichlet problem. This yields a numerical complexity of $\mathcal{O}(N \log N)$ for the application of the operator to a discretization with $N$ degrees of freedom. Iterative methods can then be employed to solve the fractional partial differential equations with exterior Dirichlet condition. We show a number of example applications and that are in line with rates for finite element based approaches.

The Ohta-Kawasaki energy is widely employed in the analysis of copolymers. They take the form of a perimeter term, which models local, close range interactions, and a nonlocal interaction term, which models long range effects. The geometry of minimizers has been studied in 2D and 3D for binary systems, and in 2D for ternary systems. Two key hurdles make the study of 3D ternary systems significantly more challenging: first, the nonlocal term is a Coulomb type energy, which, unlike the 2D case, cannot be further simplified; second, there is no equivalent of the quantitative isoperimetric inequality in 3D with two mass constraints. In this talk we will present some recent results on the geometry of minimizers of 3D ternary systems.

Constitutive modeling based on the continuum mechanics theory has been a classical approach for modeling the mechanical responses of materials. However, when constitutive laws are unknown or when defects and/or high degrees of heterogeneity present, these classical PDE models may become inaccurate. In this talk we propose to use data-driven modeling which directly utilizes high-fidelity simulation and experimental measurements on displacement fields, to predict a material's response without the necessity of using conventional constitutive models. Nonlocal operator regression approaches will be presented, to obtain homogenized surrogates for material modeling. We show that by combining machine learning identifiability theory, known physics, and nonlocal theory, this combination guarantees that the resulting data-driven nonlocal model is mathematically well-posed, physically consistent, and converging as the data resolution increases. Lastly, we further equip the nonlocal operator regression approach with neural networks and leverage it to integral neural operators, which provide heterogenous surrogates for material responses. As an application, we employ the proposed approach to learn material models directly from digital image correlation (DIC) displacement tracking measurements, and show that the learnt model substantially outperforms conventional constitutive models in predicting complex material response.

Nonlocal models feature a finite length scale, referred to as the horizon, such that points separated by a distance smaller than the horizon interact with each other. Such models have proven to be useful in a variety of settings. However, due to the reduced sparsity of discretizations, they are also generally computationally more expensive compared to their local differential equation counterparts. We introduce a multifidelity Monte Carlo method that combines the nonlocal model of interest with surrogate models that use coarser grids and/or smaller horizons and/or interpolants and thus have lower costs. Using the multifidelity method, the overall computational cost of uncertainty quantification is reduced without compromising accuracy. It is shown for a nonlocal diffusion model and for a nonlocal shallow-water model that speedups of up to two orders of magnitude can be achieved using the multifidelity method to estimate the expectation of an output of interest.