Gary R. Marple

Alex H. Barnett, Gary Marple, Shravan Veerapaneni et al.

We present a spectrally-accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. Applications include computing the effective permeability of composite media (homogenization), and microfluidic chip design. Our basic idea is to exploit a small least squares solve to apply periodicity without ever handling periodic Green's functions. We exhibit fast solvers for the two-dimensional (2D) doubly-periodic Neumann Laplace problem (flow around insulators), and Stokes non-slip fluid flow problem, that for inclusions with smooth boundaries achieve 12-digit accuracy, and can handle thousands of inclusions per unit cell. We split the infinite sum over the lattice of images into a directly-summed "near" part plus a small number of auxiliary sources which represent the (smooth) remaining "far" contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rank-1 or rank-3 correction and a Schur complement, leaves a well-conditioned square system which can be solved iteratively using fast multipole acceleration plus a low-rank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, nor particle meshes are required), allows adaptivity, and is essentially dimension- and PDE-independent, so would generalize without fuss to 3D and to other non-oscillatory elliptic problems such as elastostatics. We incorporate recently developed spectral quadratures that accurately handle close-to-touching geometries. We include many numerical examples, and provide a software implementation.

Gary R. Marple, David Gorsich, Paramsothy Jayakumar et al.

A mobility map, which provides maximum achievable speed on a given terrain, is essential for path planning of autonomous ground vehicles in off-road settings. While physics-based simulations play a central role in creating next-generation, high-fidelity mobility maps, they are cumbersome and expensive. For instance, a typical simulation can take weeks to run on a supercomputer and each map requires thousands of such simulations. Recent work at the U.S. Army CCDC Ground Vehicle Systems Center has shown that trained machine learning classifiers can greatly improve the efficiency of this process. However, deciding which simulations to run in order to train the classifier efficiently is still an open problem. According to PAC learning theory, data that can be separated by a classifier is expected to require $\mathcal{O}(1/ε)$ randomly selected points (simulations) to train the classifier with error less than $ε$. In this paper, building on existing algorithms, we introduce an active learning paradigm that substantially reduces the number of simulations needed to train a machine learning classifier without sacrificing accuracy. Experimental results suggest that our sampling algorithm can train a neural network, with higher accuracy, using less than half the number of simulations when compared to random sampling.

Gary Marple, Alex Barnett, Adrianna Gillman et al.

This paper presents a new boundary integral equation (BIE) method for simulating particulate and multiphase flows through periodic channels of arbitrary smooth shape in two dimensions. The authors consider a particular system---multiple vesicles suspended in a periodic channel of arbitrary shape---to describe the numerical method and test its performance. Rather than relying on the periodic Green's function as classical BIE methods do, the method combines the free-space Green's function with a small auxiliary basis, and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms, and handle a large number of vesicles in a geometrically complex channel. Spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures, while a first-order semi-implicit scheme evolves particles by treating the vesicle-channel interactions explicitly. New constraint-correction formulas are introduced that preserve reduced areas of vesicles, independent of the number of time steps taken. By using two types of fast algorithms, (i) the fast multipole method (FMM) for the computation of the vesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) a fast direct solver for the BIE on the fixed channel geometry, the computational cost is reduced to $O(N)$ per time step where $N$ is the spatial discretization size. Moreover, the direct solver inverts the wall BIE operator at $t = 0$, stores its compressed representation and applies it at every time step to evolve the vesicle positions, leading to dramatic cost savings compared to classical approaches. Numerical experiments illustrate that a simulation with $N=128, 000$ can be evolved in less than a minute per time step on a laptop.