Abstract : Eigenvalue problems of partial differential equations have many important applications in science and engineering, e.g., design of solar cells for clean energy, calculation of electronic structure in condensed matter, extraordinary optical transmission, non-destructive testing, photonic crystals, and biological sensing. This mini-symposium focuses on the computation modeling and numerical analysis for PDE eigenvalue problems. It intends to bring the leading researchers to discuss the recent developments and build collaborations among participants of various backgrounds.
Organizer(s) : Hengguang Li, Xuefeng Liu, Jeffrey Ovall, Jiguang Sun
Abstract : Transmission eigenvalues can be determined from multistatic scattering data over a range of frequencies and have been suggested as target signatures in inverse scattering. To avoid the need for data over a range of frequencies other artificial eigenvalue problems can be derived by modifying the far field operator. We shall consider “modified transmission eigenvalues” for thin structures. Numerical results will show that a few eigenvalues of each type can be determined from multi-static scattering data.
[01744] Inverse eigenvalue problems for inferring crystal structure from neutron scattering data
Format : Talk at Waseda University
Author(s) :
Guannan Zhang (Oak Ridge National Laboratory)
Abstract : We are interested in inferring the atomic structure of crystal materials from neutron scattering data. The atomic structure is modeled by a parameterized Hamiltonian matrix and the measurements in neutron scattering experiments are the eigenvalues and functionals of the eigenvectors of the Hamiltonian matrix. The goal is to find the optimal parameter in the Hamiltonian matrix to match the scattering data. The main challenge is that the spectrum of the Hamiltonian matrix is very sensitive to its parameters, which leads to a very rough loss landscape. To address this issue, we propose a new loss function that utilize the characteristic polynomial of the Hamiltonian matrix as the loss function. When a matrix has the same eigenvalues as the measurement data, the characteristic loss is zero. Our experiments show that the new loss has a much smoother landscape for an optimization algorithm to find a satisfactory solution. We have demonstrated the effectiveness of our approach in solving the crystal field parameters from neutron scattering data.
[01689] Application of Prolate Eigensystem to Born Inverse Scattering
Format : Talk at Waseda University
Author(s) :
Shixu Meng (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Abstract : This talk discusses the application of generalized prolate spheroidal wave functions and eigenvalues (in short prolate eigensystem) to Born inverse scattering problems. We first establish a Picard criterion to reconstruct the contrast. Further motivated by a Sturm-Liouville theory associated with the prolate eigensystem, we give a spectral cutoff regularization for noisy data and an explicit stability estimate for contrast in $H^s$, $0
[01864] Computational tools for exploring eigenvector localization
Format : Talk at Waseda University
Author(s) :
Jeffrey Ovall (Portland State University)
Robyn Reid (Portland State University)
Abstract : It is well-known that waves can often be decomposed as an infinite sum
where each term in the sum is a product of a function varying
only in time and a function varying only in space. The spatial functions are often called
standing waves in this context, and are eigenvectors
of a spatial differential operator associated with
the medium through which the waves are propagating.
It is not as well-known that properties of the
medium can cause some eigenvectors to be strongly spatially localized.
A practical consequence of eigenvector localization is that waves at certain
frequencies can be ``trapped'' at some location or ``channelled''
along some favorable path. Such features are of interest in the
design of structures having desired acoustic or electromagnetic
properties: sound-mitigating outdoor barriers and next generation
organic LEDs and solar cells are examples of this design principle in
action. There remain many open problems related to understanding and
exploiting this kind of localization, and we will discuss a
computational approach that we expect will provide useful insight. More
specifically, we focus on the issue of eigenvector localization, outlining our
computational approach and providing theoretical, heuristic, and
empirical support for it through several examples (with many
pictures).
[01877] LOWER EIGENVALUE BOUNDS FOR THE HARMONIC AND BI-HARMONIC OPERATOR
Format : Talk at Waseda University
Author(s) :
Carsten Carstensen (Humboldt-Universitaet zu Berlin umboldt-Universitaet zu Berlin)
Sophie Puttkammer (Humboldt-Universitaet zu Berlin umboldt-Universitaet zu Berlin)
Abstract : Like guaranteed upper eigenvalue bounds with conforming finite element methods, guaranteed lower eigenvalue
bounds (GLB) follow from min-max principles. Part 1 recalls GLB for the simplest second-order and fourth-order
eigenvalue problems from a simple post-processing. The maximal mesh-size therein destroys nive adaptive
mesh-refining and motivates a new methodology. Prt 2 present a new method with fine-tuned stabilization
for the direct computation of GLB. Part 3 studies an optimal adaptive mesh-refining algorithm.
[01904] Reduced order models for parametric PDE eigenvalue problems
Format : Talk at Waseda University
Author(s) :
Daniele Boffi (KAUST)
Abstract : It is well known that the approximation of parametric eigenvalue problems offer much more challenges than the the corresponding source problems. This is due in particular to the lack of smoothness of the solutions with respect to the parameters. Multiplicities, clusters, and crossing of eigenvalues must be dealt with in an appropriate way in order achieve meaningful and accurate solutions.
We discuss how to track the eigensolutions in presence of multidimensional parameters and we propose new ideas for the model order reduction of eigenvalues problems.
[01686] Verification of guaranteed lower eigenvalue bounds form a hybrid-high order method
Format : Talk at Waseda University
Author(s) :
Carsten Carstensen (Humboldt-Universität zu Berlin)
Benedikt Gräßle (Humboldt-Universität zu Berlin)
Ngoc Tien Tran (Friedrich-Schiller-Universität Jena)
Abstract : A new class of skeletal methods provides direct guaranteed lower eigenvalue bounds $($GLB$)$ under verifiable assumptions on the maximal mesh-size and discretisation parameters.
The verification of the GLB condition requires the knowledge of some stability constants and its validity implies that the computed discrete eigenvalue is already a GLB. This talk discusses the explicit estimation of the stability constants for the hybrid-high order $($HHO$)$ eigenvalue solver of Carstensen-Ern-Puttkammer $[$Numer.Math. 149, 2021$]$ and its recent modification with an even simpler p-robust parameter selection.
We prove an a priori quasi-best approximation property and establish stabilization-free reliable and efficient a posteriori error control. Computer benchmarks provide striking numerical evidence for optimal high-order convergence rates of the associated adaptive mesh-refining algorithm.
[01859] Poisson solvers for the biharmonic eigenvalue problem with the Navier boundary condition
Format : Talk at Waseda University
Author(s) :
Baiju Zhang (Beijing Computational Science Research Center )
Hengguang Li (Wayne State University)
Zhimin Zhang (Wayne State University)
Abstract : Consider the biharmonic eigenvalue problem with the Navier boundary condition. The Ciarlet-Raviart mixed method solves this problem by decomposing the 4th-order operator into two Laplacians but can produce spurious eigenvalues in non-convex domains. To overcome this difficulty, we adopt a recently developed mixed method, which decomposes the biharmonic equation into three Poisson equations and still recovers the original solution. Using this idea, we design an efficient biharmonic eigenvalue algorithm, which contains only Poisson solvers. With this approach, eigenfunctions can be confined in the correct space and thereby spurious modes in non-convex domains are avoided. Numerical results will be reported to validate the algorithm.
[01910] Why Spectral Methods are preferred in PDE Eigenvalue Problems?
Format : Talk at Waseda University
Author(s) :
Zhimin Zhang (CSRC & WSU)
Abstract : When approximating PDE eigenvalue problems by numerical methods such as finite difference and finite element, it is common knowledge that only a small portion of numerical eigenvalues are reliable. As a comparison, spectral methods may perform extremely well in some situation, especially for 1-D problems. In addition, we demonstrate that spectral methods can outperform traditional methods and the state-of-the-art method in 2-D problems even with singularities.
[01726] High-precision guaranteed eigenvalue bounds using higher order finite elements and graded meshes
Format : Talk at Waseda University
Author(s) :
Xuefeng LIU (Niigata University)
Abstract : The projection residue error-based eigenvalue bound proposed by the author has a drawback that it is influenced by the worst-case projection error and is not able to take the advantage of non-uniform meshes such as graded meshes. To address these issues, we propose a new method based on the Kato-Lehmann-Goerisch theorem to obtain high-precision eigenvalue bounds that take full advantage of higher-order FEMs, graded meshes, and possible strong regularities of eigenfunctions.
[01925] Comparison of guaranteed lower eigenvalue bounds with three skeletal methods
Carsten Carstensen (Humboldt-Universitaet zu Berlin )
Benedikt Gräßle (Humboldt-Universität zu Berlin)
Abstract : The focus of this talk is the comparison of three specially tailored skeletal hybrid schemes which provide direct guaranteed lower eigenvalue bounds (GLB) for the Dirichlet eigenvalue problem of the Laplacian. While the scheme presented in (Carstensen-Zhai-Zhang2020) has established the groundwork with a first formulation of the conditions under which GLB can be computed with a hybridized discontinuous Galerkin (HDG) method, a further development in (Carstensen-Ern-Puttkammer2021) results in a modified hybrid-high order method with a simplified stabilization term. However, it involves two parameters whose choice can be unclear due to stability estimates with constants which depend on the polynomial degree $p$ of the approximation spaces and numerical computations show a lack of robustness. (Carstensen-Grässle-Tran2022,subm.) presents a different stabilization which uses a $p$-robust parameter. Numerical examples for all three methods and various polynomial degrees with optimal orders of convergence will be shown in this talk. The details of the qualitative differences in the computation of GLB and possible further practical improvements will be discussed.
[02081] Novel spectral methods using multivariate Muntz polynomials/functions for Schrodinger eigenvalue problems with singular potentials
Format : Talk at Waseda University
Author(s) :
Huiyuan Li (Institute of Software Chinese Academy of Sciences)
Abstract : In this talk, we first introduce multivariate Muntz ball polynomials and Muntz Hermite functions, and then propose novel spectral methods for solving the eigenvalue problems of the Schrodinger operators $[-\Delta + c/|x|^2] + z |x|^{q/p}$ and $-\nabla\cdot (|x|^{2\mu} \nabla) + c |x|^{2\mu-2}$. The Muntz polynomials/functions are tailored to fit the singularities of the eigenfunctions and are orthogonal with respect to the inner product associated with the underlying Schrodinger operator. Numerical experiments demonstrate the efficiency and the exponential order of convergence of our methods, and validate the superiority over other methods.
[02157] Continuity and differentiability of eigenvalues of Laplacian with respect to general domain perturbations
Format : Talk at Waseda University
Author(s) :
Takuya Tsuchiya (Ehime University)
Abstract : We consider the eigenvalue problems of Laplacian on bounded domains with Lipschitz boundaries. Suppose that a domain is smoothly perturbed, and the perturbation is parametrized in $t$.
In this talk, we discuss about continuity and differentiability of perturbed eigenvalues with respect to the parameter $t$.
[01803] GPU-accelerated high order mimetic finite difference methods for Maxwell equations and eigenproblems
Format : Talk at Waseda University
Author(s) :
Yan Xu (University of Science and Technology of China)
Abstract : In this paper, we consider the eigenvalue problem of the three-dimensional time-harmonic Maxwell equations. The problem is discretized by the general mimetic finite difference method (MFDM), which is based on $L^2$ de Rham complex and has a deep relation with finite element exterior calculus theory. The discretization for the shifted differential operator is also covered. The main challenge arises from the large null space of the approximate curl operator. We introduce an auxiliary scheme to reach nontrivial eigenpairs without going through null space. We design a multigrid-type preconditioner for the algorithm to reduce the iteration count of the iterative eigensolver. Most of the algorithm are basic matrix and vector operators, which are fine-grained parallelism and can be easily accelerated by GPU. Numerical examples of the band structures of three-dimensional photonic crystals are presented to demonstrate the capability and efficiency of the algorithm.
[01881] Compatible Approximation of Holomorphic Eigenvalue Problems
Format : Online Talk on Zoom
Author(s) :
Martin Halla (eorg-August Universität Göttingen, Institut für Numerische und Angewandte Mathematik)
Abstract : I consider Galerkin approximations of EVP for holomorphic operator functions, which arise e.g. from finite element discretizations of PDE eigenvalue problems. The convergence is ensured for "regular" approximations (Karma (1996)). This property is unconditionally satisfied for weakly coercive problems. However, for non weakly coercive problems there exist hardly any results. I present a technique to prove the regularity for such cases, which builds upon the weak T-coercivity of the continuous problem.
[01914] The new computational method on elastic transmission eigenvalue problem
Format : Online Talk on Zoom
Author(s) :
Yingxia Xi (Nanjing University of Science and Technology)
Xia Ji (Beijing Institute of Technology)
Shuo Zhang (Academy of Mathematics and Systems Science)
Abstract : We will present a finite element scheme for the elastic transmission eigenvalue problem written as a fourth order eigenvalue problem. The scheme uses piecewise cubic polynomials and obtains optimal convergence rate. Compared with other low-degree and nonconforming finite element schemes, the scheme inherits the continuous bilinear form which does not need extra stabilizations and is thus simple to implement.
