# Registered Data

## [00761] Recent Advances on quadrature methods for integral equations and their applications

**Session Date & Time**:- 00761 (1/3) : 3C (Aug.23, 13:20-15:00)
- 00761 (2/3) : 3D (Aug.23, 15:30-17:10)
- 00761 (3/3) : 3E (Aug.23, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: Numerical methods based on integral equations are powerful tools for simulating physical systems that arise in fluid mechanics, acoustics, electromagnetics, and many other fields. A crucial component of any efficient integral equation solver is a specialized quadrature method for the discretization of the underlying integral operators. This mini-symposium will focus on the main challenges in quadrature research including accurate evaluation of singular and near-singular integrals associated with surface and volume potentials, adaptive discretization in complex geometries, and the efficient implementation of quadrature schemes in practical applications.**Organizer(s)**: Anna-Karin Tornberg, Hai Zhu, Bowei Wu**Classification**:__65R20__,__65N38__**Speakers Info**:- Ludvig af Klinteberg (Mälardalen University)
- Joar Bagge (KTH)
- Daniel Fortunato (Flatiron Institute)
- Adrianna Gillman (University of Colorado Boulder)
- Abinand Gopal (Yale University)
- Federico Izzo (KTH)
- Carlos Pérez-Arancibia (University of Twente)
- Zewen Shen (University of Toronto)
- Chiara Sorgentone (Sapienza University of Rome)
- Shravan Veerapaneni (University of Michigan)
- Bowei Wu (University of Massachusetts Lowell)
- Hai Zhu (Flatiron Institute)

**Talks in Minisymposium**:**[02383] Is polynomial interpolation in the monomial basis unstable?****Author(s)**:**Zewen Shen**(University of Toronto)- Kirill Serkh (University of Toronto)

**Abstract**: Polynomial interpolation in the monomial basis is a key step in a number of popular quadrature methods for integral equations. We will show that, despite its ill-conditioning, the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the Vandermonde matrix has a condition number smaller than the reciprocal of machine epsilon. We will also explore some applications of our analysis.

**[03009] Near singularity errors in boundary integrals: identification, estimation and swapping****Author(s)**:**Ludvig af Klinteberg**(Mälardalen University)

**Abstract**: Evaluating a layer potential close to its source geometry poses numerical difficulties due to the presence of a near singularity. In this presentation I will discuss how the singularity can be understood in terms of its complexified preimage in the parametrization of the source geometry. Finding the preimage numerically allows us to both estimate quadrature errors to high precision, and derive new and more capable quadrature methods.

**[03610] Quadrature errors for layer potentials near surfaces with spherical topology****Author(s)**:**Chiara Sorgentone**(Sapienza, Università di Roma)- Anna-Karin Tornberg (KTH Royal Institute of Technology)

**Abstract**: Numerical simulations often involve 3D objects with spherical topology, e.g. rigid particles, drops, vesicles. When the underlying numerical method is based on boundary integral equations, standard quadrature rules can yield large errors in computing the layer potentials if the distance between the surfaces is too small and the associated integrals become nearly singular. We will present numerical and analytical approaches to efficiently evaluate the quadrature error estimates for these situations.

**[03612] Euler-Maclaurin formulas for near-singular integrals****Author(s)**:**Bowei Wu**(University of Massachusetts Lowell)

**Abstract**: Near-singular integrals frequently arise in fluid dynamics, material science, and many other scientific applications, where close fluid-structure interactions are common. Numerical approximation of near-singular integrals thus has practical importance. Approximating near-singular integrals using regular quadrature methods is in general inefficient and expensive. But more efficient quadrature rules can be developed by modifying regular quadrature rules using an error correction approach. We introduce new generalized Euler-Maclaurin formulas that are tailored to a family of near-singular functions. High-order accurate modified Trapezoidal quadrature rules are then derived based on these formulas.

**[04278] A new boundary integral equation solver for problems in exteriors of open arcs****Author(s)**:**Abinand Gopal**(Yale University)- Shidong Jiang (Flatiron Institute )
- Vladimir Rokhlin (Yale University)

**Abstract**: When solving a constant-coefficient elliptic PDE, it is often convenient to first reformulate the problem as a boundary integral equation. This is usually done by representing the solution as an unknown density function times a kernel function integrated over the boundary of the domain. The choice of kernel is usually dictated by the boundary conditions and made such that the resulting equation for the density is a second kind Fredholm integral equation. However, when the problem is posed on the exterior of an arc in 2D or a surface in 3D, this becomes more complicated and the usual single layer and double layer representations run into difficulties. In this talk, we present a new solver for this regime. We use a representation based on the composition of standard layer potential with a hypersingular operator and compute the kernel of the composite operator directly. We then solve the resulting linear system with a direct solver.

**[04432] A High-Order Close Evaluation Scheme of Helmholtz Layer Potentials in 3D****Author(s)**:**Hai Zhu**(Flatiron Institute)- Shidong Jiang (Flatiron Institute)

**Abstract**: We present an efficient high-order discretization scheme for the evaluation of the Helmholtz layer potentials on smooth surfaces in three dimensions. The scheme is panel based and applies an analytical surface to line integral conversion on each panel to evaluate single layer, double layer, adjoint double layer, and hypersingular potentials accurately. A new basis approximation scheme tailed for Helmholtz kernels is proposed. Both nearly singular and singular cases are supported via the same recursive framework.

**[04640] An adaptive discretization technique for boundary integral equations in the plane****Author(s)**:**Adrianna Gillman**(University of Colorado, Boulder)- Yabin Zhang (Westlake University)

**Abstract**: Typically the discretization of integral equations on two dimensional complex geometries involves the use of a panel based quadrature (such as variants of Gaussian quadrature). The placement of the panels is often ad hoc and based on being able to integrate quantities such as arc-length and/or curvature to a desired accuracy. These quantities do not necessarily correspond to what is needed to achieve accuracy in the solution to a partial differential equation. Alternatively, a refinement strategy based on looking at relative error and wisely choosing which part of the geometry to refine can be done but this involves global solves which can be prohibitively expensive. In this talk, we will present an adaptive discretization technique which is guaranteed to achieve the desired accuracy and does not require the inversion of a full discretized integral equation at each step in the refinement process. Numerical results will illustrate the performance of the method.

**[04717] Density interpolation methods for volume integral operators****Author(s)**:**Carlos Perez-Arancibia**(University of Twente)- Thomas G. Anderson (Rice University)
- Luiz M. Faria (INRIA/ENSTA Paris)
- Marc Bonnet (CNRS/ENSTA Paris)

**Abstract**: This talk outlines a novel class of high-order methods for the efficient numerical evaluation of volume potentials (VPs) defined by volume integrals over complex geometries. Inspired by the Density Interpolation Method (DIM) for boundary integral operators, the proposed methodology leverages Green’s third identity and a local polynomial interpolation of the density function to recast a given VP as a linear combination of surface-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated inside and outside the integration domain using existing methods (e.g. DIM), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules to integrate over structured or unstructured domain decompositions without local numerical treatment at and around the kernel singularity. The proposed methodology is flexible, easy to implement, and fully compatible with well-established fast algorithms such as the Fast Multipole Method and H-matrices, enabling VP evaluations to achieve linearithmic computational complexity. To demonstrate the merits of the proposed methodology, we applied it to the Nyström discretization of the Lippmann-Schwinger volume integral equation for frequency-domain Helmholtz scattering problems in piecewise-smooth variable media.

**[04811] Special quadrature via line extrapolation, with application to Stokes flow****Author(s)**:**Joar Bagge**(KTH Royal Institute of Technology)- Anna-Karin Tornberg (KTH Royal Institute of Technology)

**Abstract**: In integral equations, special quadrature methods are needed to perform singular or nearly singular integration. We consider one such method, sometimes called the "Hedgehog method", based on extrapolation (or interpolation) along a line. Different strategies for selecting the placement of sampling points along the line are investigated. We consider extrapolation using polynomials or rational functions. The resulting methods are compared with quadrature by expansion (QBX) in the context of Stokes flow containing rigid rodlike particles.

**[05076] Corrected trapezoidal rules for boundary integral methods on non-parametrized surfaces****Author(s)**:- Olof Runborg (KTH Royal Institute of Technology)
- Richard Tsai (The University of Texas at Austin)
- Yimin Zhong (Auburn University )
**Federico Izzo**(KTH Royal Institute of Technology)

**Abstract**: We present higher-order quadratures for a family of boundary integral operators with application to the linearized Poisson-Boltzmann equation. Using the Implicit Boundary Integral formulation, surface point singularities in a layer potential extend along the surface normal lines. In this volumetric setting, we use the trapezoidal rule, and develop higher-order quadratures by correcting it in nodes close to the singularity line with weights dependent on the singularity type and geometrical information extracted from the non-parametrized surface.

**[05159] A fully adaptive, high-order, fast Poisson solver for complex two-dimensional geometries****Author(s)**:**Daniel Fortunato**(Flatiron Institute)- David B Stein (Flatiron Institute)
- Alex H Barnett (Flatiron Institute)

**Abstract**: We present a new framework for the fast solution of inhomogeneous elliptic boundary value problems in domains with smooth boundaries. High-order solvers based on adaptive box codes or the fast Fourier transform can efficiently treat the volumetric inhomogeneity, but require care to be taken near the boundary to ensure that the volume data is globally smooth. We avoid function extension or cut-cell quadratures near the boundary by dividing the domain into two regions: a bulk region away from the boundary that is efficiently treated with a truncated free-space box code, and a variable-width boundary-conforming strip region that is treated with a spectral collocation method and accompanying fast direct solver. Particular solutions in each region are then combined with layer potentials to yield the global solution. The resulting solver has an optimal computational complexity of $O(N)$ for an adaptive discretization with $N$ degrees of freedom. We demonstrate adaptive resolution of volumetric data, boundary data, and geometric features across a wide range of length scales, to typically 10-digit accuracy.

**[05303] Recursive product integration schemes for volume potentials on irregular domains****Author(s)**:**shravan veerapaneni**- Hai Zhu (Flatiron Institute)

**Abstract**: We will discuss a new volume potential evaluation scheme for Gaussian and Laplace kernel in complex domains. The volume integral is computed by applying Green's theorem to convert these smooth or singular domain integrals on a volume mesh to a set of line integrals on the boundary skeleton of the volume mesh. This new approach allows easier integral-equation based solver implementation in complex domains, with much fewer restrictions on leaf level box refinement.