Schedule for: 18w5090 - Spectral Geometry: Theory, Numerical Analysis and Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the 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.

I will give a very pedestrian overview of some problems in spectral geometry — a vast topic covering relations between eigenvalues of boundary value problems for the Laplacian (or for other differential operators) in a Euclidean domain or on a Riemannian manifold, and the underlying geometry. They will include some open problems of various degrees of difficulty.

Abstract: In this talk I will discuss the question of the maximization of the $k$-th eigenvalue of the Neumann-Laplacian under a volume constraint. After an introduction to the topic I will discuss the question of existence of optimal geometries. For now, there is no a general existence result, but one can prove existence of an optimal {\it (over) relaxed domain}, view as a density function. These results are an on-going work with E. Oudet. In the second part of the talk, I will focus on the low eigenvalues. The first non-trivial one is maximized by the ball, the result being due to Szego and Weinberger in the fifties. Concerning the second non-trivial eigenvalue, Girouard, Nadirashvili and Polterovich proved that the supremum in the family of planar simply connected domains of $R^2$ is attained by the union of two disjoint, equal discs. I will show that a similar statement holds in any dimension and without topological restrictions. This last result is jointly obtained with A. Henrot.

It is well-known that one cannot generally hear the shape of a drum: the metric of a compact surface is not uniquely determined by its Laplace-Beltrami spectrum. But one can still seek computational solutions to the inverse problem: given a sequence of eigenvalues, can we compute a surface whose Laplace-Beltrami spectrum approximates the sequence? I will discuss some numerical experiments related to this problem for the case of surfaces of sphere topology, whose discrete conformal parameterization leads to an especially simple formulation of the inverse problem.

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 the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

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!

A classical approach for surface classification is to find a compact algebraic representation for each surface that would be similar for objects within the same class and preserve dissimilarities between classes. We introduce self functional maps as a novel surface representation that satisfies these properties, translating the geometric problem of surface classification into an algebraic form of classifying matrices. The proposed map transforms a given surface into a universal isometry invariant form defined by a unique matrix. The suggested representation is realized by applying the functional maps framework to map the surface into itself. The key idea is to use two different metric spaces of the same surface for which the functional map serves as a signature. Specifically, in this lecture, we suggest the regular and the scale invariant surface laplacian operators to construct two families of eigenfunctions. The result is a matrix that encodes the interaction between the eigenfunctions resulted from two different Riemannian manifolds of the same surface. Using this representation, geometric shape similarity is converted into algebraic distances between matrices. In contrast to geometry understanding there is the emerging field of deep learning. Learning systems are rapidly dominating the areas of audio, textual, and visual analysis. Recent efforts to convert these successes over to geometry processing indicate that encoding geometric intuition into modeling, training, and testing is a non-trivial task. It appears as if approaches based on geometric understanding are orthogonal to those of data-heavy computational learning. We propose to unify these two methodologies by computationally learning geometric representations and invariants and thereby take a small step towards a new perspective on geometry processing. If time permits I will present examples of shape matching, facial surface reconstruction from a single image, reading facial expressions, shape representation, and finally definition and computation of invariant operators and signatures.

The transmission eigenvalue problem plays a central role in inverse scattering theory. This is a non-selfadjoint problem for a coupled pair of partial differential equations in a bounded domain corresponding to the support of the scattering object. Unfortunately, relatively little is known about the spectrum of this problem. In this talk I will consider the simplest case of the transmission eigenvalue problem for which the domain and eigenfunctions are spherically symmetric. In this case the transmission eigenvalue problem reduces to an eigenvalue problem for ordinary differential equations. Through the use of the theory of entire functions of a complex variable, I will show that there is a remarkable diversity in the behavior of the spectrum of this problem depending on the behavior of the refractive index near the boundary. Included in my talk will be results on the existence of complex eigenvalues, the inverse spectral problem and a remarkable connection (due to Fioralba Cakoni and Sagun Chanillo) between the location of transmission eigenvalues for automorphic solutions of the wave equation in the hyperbolic plane and the Riemann hypothesis.

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

Subspace iteration for computing eigenvalues and eigenvectors, a natural generalization of the power method, is among the most straight-forward methods to analyze and implement. A variant of subpace iteration, called FEAST, that uses rational filters to accelerate convergence toward a targeted invariant subspace (typically selected by enclosing the eigenvalues of interest by a simple contour), has gained significant interest in recent years, having been adopted as part of Intel's Math Kernel Library for matrix eigenvalue problems. Such rational filters are often derived from quadrature approximations of contour integrals---think of approximating Cauchy's integral formula for the indicator function of the region enclosed by the contour. The rational filter governs the iterative convergence of the method, so it is perhaps surprising that quadrature error plays little role in the analysis of the algorithm, and we will explain why this is so. Inspired by this approach, we consider filtered subspace iteration for (possibly) unbounded selfadjoint operators, having in mind differential operators as motivational examples. In this broader context, it is natural to consider errors in approximating the invariant subspace with respect to different norms, and we provide a fairly general framework for analyzing both iteration and discretization errors for eigenvalues and invariant subspaces. The bulk of the computational effort in the algorithm involves approximating the action of the resolvent at a few points along a contour enclosing the eigenvalues of interest. In our numerical examples, such finite-rank approximations of the resolvent are obtained by finite element discretizations.

In this talk,we analyze the connections between the nodal domains of the eigenfunctions of the Dirichlet-Laplacian and the partitions of the domain by $ k$ open sets $D_i$ which are minimal in the sense that the maximum over the $D_i$'s of the groundstate energy of the Dirichlet realization of the Laplacian is minimal. Instead of considering the maximum among the first eigenvalues, we can also consider the $p$-norm of the vector composed by the first eigenvalues of each subdomain.

Recently a novel family of eigensolvers, called spectral indicator methods (SIMs), was proposed. Given regions of the complex plane, SIMs compute indicators and use them to detect eigenvalues. Regions that contain eigenvalues are subdivided and the procedure is repeated until eigenvalues are isolated with a specified precision. In this talk, by a special way of using Cayley transformation and Krylov subspaces, a memory efficient eigensolver for sparse eigenvalue problems is proposed. The method uses little memory and is particularly suitable for the computation of many eigenvalues of large problems. The eigensolver is realized in Matlab and tested using various matrices.

Tangent vector fields on surfaces are linear operators acting on scalar functions. Taking this classical view as the starting point for the discretization of tangent vector fields on discrete surfaces, leads to interesting operator-based insights and applications. For example, geometric properties of the vector field can be expressed as algebraic properties of its matrix representation. We will present some theoretical properties and applications to geometry processing.

In recent years the subject of sharp inequalities for Laplace eigenvalues has received a lot of attention, in particular, due to its connection with minimal surfaces in spheres. In this talk we will explain this connection and give an overview of some recent estimates for Laplace eigenvalues on Riemannian surfaces.

Subdivision surfaces are a mainstream methodology in computer graphics and geometry processing to create smooth surfaces with a multiresolution hierarchy. The recent popularity of Isogeometric Analysis brought a renewed interest in such surfaces for the purpose of solving differential equations. We define subdivision methods for piecewise-constant directional fields and show how such methods can be used for robust and efficient vector field processing on subdivision surfaces.

This talk will be part-tutorial and part-research presentation. I will begin by summarizing some applications of spectral geometry, in particular the geometry of the Laplacian operator, appearing in the discrete geometry processing, computer graphics, and machine learning literatures. Using these applications as motivation, I will construct discretizations of the Laplacian suitable for calculations on triangulated surfaces, volumes bounded by a discretized surface, and point clouds. The talk will conclude with some of my own research in spectral geometry for surface correspondence, shape analysis, and applications.

We present fast integral solvers for the evaluation of waves and eigenstates. Based on novel fast high-order methods for evaluation of integral operators, these algorithms can accurately calculate eigenvalues and eigenfunctions in a variety of important settings, including setups leading to singular eigenfunctions and exponentially-decaying eigenfunctions as well as eigenfunctions of very high frequencies, and they can be used to confidently determine nodal lines and nodal domains for challenging configurations. Connections with high-order solvers for problems of electromagnetic scattering by large and complex three-dimensional structures will be mentioned. Applications to a variety of spectral problems, including Zaremba eigenvalue problems, scattering poles and near-singular systems, Steklov eigenvalue problems, and boundary-perturbative eigensolvers will be described.

We study eigenvalue asymptotics for a class of Steklov problems, possibly mixed with Dirichlet and/or Neumann boundary conditions, on planar domains with piecewise smooth boundary and with finitely many corners. This includes the famous "sloshing problem" as well as the Steklov problem on polygons. Two interesting features of this problem, which I will explain, are the surprisingly precise asymptotics we can obtain (with error decreasing as the spectral parameter increases) and a connection to a scattering problem on the Steklov portion of the boundary. This is joint work with M. Levitin (Reading), L. Parnovski (UCL), and I. Polterovich (Montreal).

In this talk we discuss the behavior of Laplace eigenfunctions when restricted to a fixed submanifold by studying the averages given by the integral of the eigenfunctions over the submanifold. In particular, we show that the averages decay to zero when working on a surface with Anosov geodesic flow regardless of the submanifold (curve) that one picks. This is based on joint works with John Toth and Jeffrey Galkowski.

Verified computing is a newly developed methodology to estimate all errors in numerical computing and provide mathematically rigorous results. Recently, there have been several newly developed verified computing methods to give guaranteed eigenvalue estimation for differential operators. In this talk, I will explain basic concepts about verified computing and give a survey on guaranteed eigenvalue estimation methods. Particularly, the newly developed verified eigenvalue estimation method based on finite element method (FEM) will be introduced in detail. Such a method has been successfully applied to various differential operators, for example, the Laplace, the Biharmonic, the Stokes, the Steklov operators. Also, applications of the guaranteed eigenvalue estimation in mathematical proof will be introduced. As an example, I will show the latest result on solution existence proof about the Navier-Stokes equation in 3D space.

How to analyze complex shapes, such as of the highly folded surface of the brain? This talk will show how spectral representations of shapes can benefit neuroimaging. Here, we exploit spectral coordinates derived from the eigenfunctions of the graph Laplacian. Methodologically, we address the inherent instability of spectral shape decompositions. This change of paradigm, exploiting spectral representations, enables an intrinsic processing of brain surfaces. Brain surface matching will be shown as an example.

The adaptive finite element method (AFEM) is a mature technique for the computation of approximate solutions to partial differential equations. AFEM's for eigenvalue problems have been successfully implemented and analysed in the case of various applications; as usually when dealing with eigenvalue problems, most available results deal with eigenmodes of multiplicity one. Only recently an active research field was focused on the approximation of multiple eigenvalues and, more generally, of cluster of eigenvalues. This new viewpoint opens new scenarios and raises several questions, some of which will be discussed during this talk. Another critical aspect about the theoretical analysis of AFEM's for eigenvalue problems concerns the approximation of eigenproblems in mixed form. We will discuss the optimal convergence of the AFEM applied to the Laplace eigenvalue problem in mixed form; our analysis applies to standard simplicial mixed schemes, in two and threes dimensions and is cluster-robust. The quasi-orthogonality property has been proved by using a suitable superconvergence. Some of the results presented in this talk are based on references [1,2]. [1] D. Boffi, R. G. Dur\'an, F. Gardini, and L. Gastaldi. A posteriori error analysis for nonconforming approximation of multiple eigenvalues. Mathematical Methods in the Applied Sciences, 40 (2017), no. 2, 350-369. [2] D. Boffi, D. Gallistil, F. Gardini, and L. Gastaldi. Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form. Mathematics of Computation, 86 (2017), no. 307, 2213-2237.

The goal of this talk is to study bounds on the Riesz means of mixed Steklov problems. The Riesz mean is a convex function of eigenvalues and has an important role and connection with other spectral quantities. We recall the results known in this direction for the Laplace eigenvalues. Then we introduce the mixed Steklov problem and state the main results. We also discuss some key ideas of the proof. This is joint work with Ari Laptev.

This talk introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the corresponding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing. The efficiency of the guaranteed error bounds involves the global mesh-size and is proven for the large class of graded meshes. Numerical examples demonstrate the reliability of the guaranteed error control even with inexact solve of the algebraic eigenvalue problem. This motivates an adaptive algorithm which monitors the discretisation error, the maximal mesh-size, and the algebraic eigenvalue error. The accuracy of the guaranteed eigenvalue bounds is surprisingly high with efficiency indices as small as 1.4. This is joint work with Carsten Carstensen.

Computer graphics and geometry processing study the representation, processing, and analysis of 3D shapes, with wide applications to brain imaging, computer vision, computer aided design and engineering, and so on. Intrinsic approaches, usually based on the Laplace-Beltrami operator, have been popular in computer graphics. However, intrinsic approaches cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we advocate using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for geometry processing and shape analysis. We consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.

The Jones eigenvalue problem is an overdetermined problem, where the Neumann eigenvalue problem for linear elasticity is coupled with a constraint on the normal trace of the displacement along the boundary. This eigenvalue problem presents interesting features, not least of which is the sensitive dependance on boundary geometry. We prove the existence of eigenpairs of this eigenvalue problem on Lipschitz domains in 2D and 3D, and use numerical methods to approximate the eigenpairs on some simple geometries.

A variety of tasks in inverse problems and data analysis can be formulated as the variational problem of minimizing the Dirichlet energy of a function that takes values in a certain target set and possibly satisfies additional constraints. These additional constraints may be used to enforce fidelity to data or other structural constraints arising in the particular problem considered. I'll present diffusion generated methods for solving this problem for a wide class of target sets and prove some stability and convergence results. I’ll give examples of how these methods can be used for the geometry processing task of generating quadrilateral meshes, finding Dirichlet partitions, constructing smooth orthogonal matrix valued functions, and solving inverse problems for target-valued maps. This is joint work with Dong Wang and Ryan Viertel.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.