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1. Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

   https://doi.org/10.1007/s10208-018-09407-7  

   Kanagawa, Motonobu; Sriperumbudur, Bharath K.; Fukumizu, Kenji (January 2019, Foundations of Computational Mathematics)

   This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights, and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand. 

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2. Fast algorithms using orthogonal polynomials

   https://doi.org/10.1017/S0962492920000045  

   Olver, Sheehan; Slevinsky, Richard Mikaël; Townsend, Alex (May 2020, Acta Numerica)

   We review recent advances in algorithms for quadrature, transforms, differential equations and singular integral equations using orthogonal polynomials. Quadrature based on asymptotics has facilitated optimal complexity quadrature rules, allowing for efficient computation of quadrature rules with millions of nodes. Transforms based on rank structures in change-of-basis operators allow for quasi-optimal complexity, including in multivariate settings such as on triangles and for spherical harmonics. Ordinary and partial differential equations can be solved via sparse linear algebra when set up using orthogonal polynomials as a basis, provided that care is taken with the weights of orthogonality. A similar idea, together with low-rank approximation, gives an efficient method for solving singular integral equations. These techniques can be combined to produce high-performance codes for a wide range of problems that appear in applications. 

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3. Error analysis for the implicit boundary integral method

   https://doi.org/10.1007/s10543-024-01051-8  

   Zhong, Yimin; Ren, Kui; Runborg, Olof; Tsai, Richard (January 2025, BIT Numerical Mathematics)

   Abstract The implicit boundary integral method (IBIM) provides a framework to construct quadrature rules on regular lattices for integrals over irregular domain boundaries. This work provides a systematic error analysis for IBIMs on uniform Cartesian grids for boundaries with different degrees of regularity. First, it is shown that the quadrature error gains an additional order of$$\frac{d-1}{2}$$ $\frac{d-1}{2}$from the curvature for a strongly convex smooth boundary due to the “randomness” in the signed distances. This gain is discounted for degenerated convex surfaces. Then the extension of error estimate to general boundaries under some special circumstances is considered, including how quadrature error depends on the boundary’s local geometry relative to the underlying grid. Bounds on the variance of the quadrature error under random shifts and rotations of the lattices are also derived. 

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4. Lagrangian–Eulerian multidensity topology optimization with the material point method

   https://doi.org/10.1002/nme.6668  

   Li, Yue; Li, Xuan; Li, Minchen; Zhu, Yixin; Zhu, Bo; Jiang, Chenfanfu (March 2021, International Journal for Numerical Methods in Engineering)

   Abstract In this paper, a hybrid Lagrangian–Eulerian topology optimization (LETO) method is proposed to solve the elastic force equilibrium with the Material Point Method (MPM). LETO transfers density information from freely movable Lagrangian carrier particles to a fixed set of Eulerian quadrature points. This transfer is based on a smooth radial kernel involved in the compliance objective to avoid the artificial checkerboard pattern. The quadrature points act as MPM particles embedded in a lower‐resolution grid and enable a subcell multidensity resolution of intricate structures with a reduced computational cost. A quadrature‐level connectivity graph‐based method is adopted to avoid the artificial checkerboard issues commonly existing in multiresolution topology optimization methods. Numerical experiments are provided to demonstrate the efficacy of the proposed approach. 

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5. Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals

   https://doi.org/10.1007/s10543-023-00984-w  

   Mitchell, William; Natkin, Abbie; Robertson, Paige; Sullivan, Marika; Yu, Xuechen; Zhu, Chenxin (July 2023, BIT Numerical Mathematics)

   Abstract Gauss–Legendre quadrature, Clenshaw–Curtis quadrature and the trapezoid rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals that arise in three-dimensional viscous fluid flow. 

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