Optimization of fast algorithms for global Quadrature by Expansion using target-specific expansions

Abstract

We develop an algorithm for the asymptotically fast evaluation of layer potentials close to and on the source geometry, combining Geometric Global Accelerated QBX ('GIGAQBX') and target-specific expansions. GIGAQBX is a fast high-order scheme for evaluation of layer potentials based on Quadrature by Expansion ('QBX') using local expansions formed via the Fast Multipole Method (FMM). Targetspecific expansions serve to lower the cost of local expansion evaluation, reducing the computational effort from O((p + 1) 2 ) to O(p + 1) in three dimensions, without any accuracy loss compared with conventional expansions, but with the loss of source/target separation in the expansion coefficients. GIGAQBX is a 'global' QBX scheme, meaning that the potential is mediated entirely through expansions for points close to or on the boundary. In our scheme, this single global expansion is decomposed into two parts that are evaluated separately: one part incorporating near-field contributions using target-specific expansions, and one part using conventional spherical harmonic expansions of far-field contributions, noting that convergence guarantees only exist for the sum of the two sub-expansions. By contrast, target-specific expansions were originally introduced as an acceleration mechanism for 'local' QBX schemes, in which the far-field does not contribute to the QBX expansion. Compared with the unmodified GIGAQBX algorithm, we show through a reproducible, time-calibrated cost model that the combined scheme yields a considerable cost reduction for the near-field evaluation part of the computation. We support the effectiveness of our scheme through numerical results demonstrating performance improvements for Laplace and Helmholtz kernels. * wala1@illinois.edu † andreask@illinois.edu arXiv:1811.01110v2 [math.NA] 5 Aug 2019 stage-1 node QBX center Figure 1: Depiction of QBX center placement for a discretization over a triangular element. The QBX centers for on-surface evaluation are spawned in the normal direction at the discretization nodes termed 'stage-1' nodes in [32], which serve as targets for on-surface evaluation. See also Section 2.5.While Π is a generally accurate approximation of Sµ for target points away from the source geometry Γ, accuracy substantially decreases in the region close to Γ or on Γ itself. A recent approach to resolving this problem is based on Quadrature by Expansion (QBX, [20]). By using the broadly applicable assumption that the underlying kernel is a (locally) analytic function of the target point x, and leveraging the strengths of smooth high-order quadrature rules for one-and two-dimensional functions, QBX achieves high-order accuracy without sacrificing generality. The key idea behind QBX is that, to extend the applicability of smooth high-order quadrature to all points x ∈ R 3 , the potential (2) can be expanded in a local expansion about a center c ∈ R 3 \ Γ, and this expansion may recover the value of the potential by analytic continuation in regions where smooth quadrature is not adequate. A pote...

Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries

“…A different approach that has been taken to accelerate QBX is to couple it to a customized FMM, which has been done in two dimensions [38][39][40] and more recently in three dimensions. 41,42 This coupling is a natural step to take since the FMM uses expansions of the same kind as QBX, but it requires nontrivial modifications to the FMM. The resulting method has complexity O(N) and works for any smooth geometry.…”

“…This separation is necessary in the QBX-FMM methods, although target-specific expansions can also be used in QBX-FMM methods to lower the computational cost. 42 Some of the recent work have focused on automating the parameter selection based on a given error tolerance, resulting in the adaptive QBX method. 44 The results have so far not been generalized to three dimensions.…”

High-order close evaluation of Laplace layer potentials: A differential geometric approach

“…where i = 0, 1, 2, 3 corresponds is the index of the quaternion and each of the vectors q (k,l) i are k th degree polynomials in r, which can be derived using quaternion product as (30) q…”

“…quadrature-by-expansion (QBX) scheme, originally proposed in [3,14], the DLP is approximated at centers away from M using high-order local expansions, which are valid at points closer to or on M. Extension of QBX to three-dimensional problems was recently explored in [22,29,30]. A related algorithm is the hedgehog scheme of [18], which in turn is an extension of the earlier work by Ying et al [34].…”

Scalable simulation of realistic volume fraction red blood cell flows through vascular networks

“…This approach is versatile and efficient due to only requiring discretization of RBCs and blood vessel surfaces, while achieving high-order convergence and optimal complexity implementation due to fast summation methods [21,38,43,44,47,48,59]. To solve elliptic partial differential equations, BIE approaches have been successful in several application domains [6,49,50,57]. However, to our knowledge, there has been no work combining a Stokes boundary solver on arbitrary complex geometries in 3d with a collision detection and resolution scheme to simulate RBC flows at large scale.…”