Publications

A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such matrices can be transformed into a Cauchy-like form that has hierarchical low rank structure. The rank structure of this matrix is explained, and it is shown that the ranks of the relevant submatrices grow only logarithmically with the number of columns of the matrix. A fast rank-structured hierarchical approximation method based on this analysis is developed, along with a hierarchical least-squares solver for these and related systems. This result is a direct method for inverting nonuniform discrete transforms with a complexity that is usually nearly linear with respect to the degrees of freedom in the problem. This solver is benchmarked against various iterative and direct solvers in the setting of inverting the one-dimensional type-II (or forward) transform, for a range of condition numbers and problem sizes (up to (4 10

Hamiltonian Monte Carlo (HMC) is the mainstay of applied Bayesian inference for differentiable models. However, HMC still struggles to sample from hierarchical models that induce densities with multiscale geometry: a large step size is needed to efficiently explore low curvature regions while a small step size is needed to accurately explore high curvature regions. We introduce the delayed rejection generalized HMC (DR-G-HMC) sampler that overcomes this challenge by employing dynamic step size selection, inspired by differential equation solvers. In generalized HMC, each iteration does a single leapfrog step. DR-G-HMC sequentially makes proposals with geometrically decreasing step sizes upon rejection of earlier proposals. This simulates Hamiltonian dynamics that can adjust its step size along a (stochastic) Hamiltonian trajectory to deal with regions of high curvature. DR-G-HMC makes generalized HMC competitive by decreasing the number of rejections which otherwise cause inefficient backtracking and prevents directed movement. We present experiments to demonstrate that DR-G-HMC (1) correctly samples from multiscale densities, (2) makes generalized HMC methods competitive with the state of the art No-U-Turn sampler, and (3) is robust to tuning parameters.

We study level set teleportation, an optimization routine which tries to accelerate gradient descent (GD) by maximizing the gradient norm over a level set of the objective. While teleportation intuitively speeds-up GD via bigger steps, current work lacks convergence theory for convex functions, guarantees for solving the teleportation operator, and even clear empirical evidence showing this acceleration. We resolve these open questions. For convex functions satisfying Hessian stability, we prove that GD with teleportation obtains a combined sub-linear/linear convergence rate which is strictly faster than GD when the optimality gap is small. This is in sharp contrast to the standard (strongly) convex setting, where teleportation neither improves nor worsens convergence. To evaluate teleportation in practice, we develop a projected-gradient method requiring only Hessian-vector products. We use this to show that gradient methods with access to a teleportation oracle out-perform their standard versions on a variety of problems. We also find that GD with teleportation is faster than truncated Newton methods, particularly for non-convex optimization.

Using a state-of-the-art numerical scheme, we show that the Higgs mode under excitation exhibits chirped oscillations and exponential decay when fluctuations are included. This is in stark contrast to conventional BCS collisionless dynamics which predict power-law decay and the absence of chirping. The chirped amplitude mode enables us to determine the local modification of the effective potential even when the system is in a long-lived prethermal state. We then show that this chirped amplitude mode is an experimentally observable quantity since the photoinduced (super)current in pump-probe experiments serves as an efficient proxy for the order parameter dynamics, including the chirped dynamics. Our result is based on the attractive Hubbard model using dynamical mean-field theory within the symmetry-broken state after a excitation across the superconducting gap. Since the collective response involves long timescales, we extend the hierarchical low-rank compression method for nonequilibrium Green's functions to symmetry-broken states and show that it serves as an efficient representation despite long-lived memory kernels.

Stokes flows with near-touching rigid particles induce near-singular lubrication forces under relative motion, making their accurate numerical treatment challenging. With the aim of controlling the accuracy with a computationally cheap method, we present a new technique that combines the method of fundamental solutions (MFS) with the method of images. For rigid spheres, we propose to represent the flow using Stokeslet proxy sources on interior spheres, augmented by lines of image sources adapted to each near-contact to resolve lubrication. Source strengths are found by a least-squares solve at contact-adapted boundary collocation nodes. We include extensive numerical tests, and validate against reference solutions from a well-resolved boundary integral formulation. With less than 60 additional image sources per particle per contact, we show controlled uniform accuracy to three relative digits in surface velocities, and up to five digits in particle forces and torques, for all separations down to a thousandth of the radius. In the special case of flows around fixed particles, the proxy sphere alone gives controlled accuracy. A one-body preconditioning strategy allows acceleration with the fast multipole method, hence close to linear scaling in the number of particles. This is demonstrated by solving problems of up to 2000 spheres on a workstation using only 700 proxy sources per particle.

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 Laplace 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. With an efficient two-dimensional (2D) implementation we demonstrate adaptive resolution of volumetric data, boundary data, and geometric features across a wide range of length scales, to typically 10-digit accuracy. The cost of all boundary corrections remains small relative to that of the bulk box code. The extension to 3D is expected to be straightforward in many cases because the strip

We present a generalization of the discrete Lehmann representation (DLR) to three-point correlation and vertex functions in imaginary time and Matsubara frequency. The representation takes the form of a linear combination of judiciously chosen exponentials in imaginary time, and products of simple poles in Matsubara frequency, which are universal for a given temperature and energy cutoff. We present a systematic algorithm to generate compact sampling grids, from which the coefficients of such an expansion can be obtained by solving a linear system. We show that the explicit form of the representation can be used to evaluate diagrammatic expressions involving infinite Matsubara sums, such as polarization functions or self-energies, with controllable, high-order accuracy. This collection of techniques establishes a framework through which methods involving three-point objects can be implemented robustly, with a substantially reduced computational cost and memory footprint.

We introduce a Fourier-based fast algorithm for Gaussian process regression in low dimensions. It approximates a translationally invariant covariance kernel by complex exponentials on an equispaced Cartesian frequency grid of \(M\) nodes. This results in a weight-space \(M\times M\) system matrix with Toeplitz structure, which can thus be applied to a vector in \({\mathcal O}(M \log{M})\) operations via the fast Fourier transform (FFT), independent of the number of data points \(N\). The linear system can be set up in \({\mathcal O}(N+M \log{M})\) operations using nonuniform FFTs. This enables efficient massive-scale regression via an iterative solver, even for kernels with fat-tailed spectral densities (large \(M\)). We provide bounds on both kernel approximation and posterior mean errors. Numerical experiments for squared-exponential and Matérn kernels in one, two, and three dimensions often show 1–2 orders of magnitude acceleration over state-of-the-art rank-structured solvers at comparable accuracy. Our method allows two-dimensional Matérn-\(\frac{3}{2}\) regression from \(N=10^9\) data points to be performed in two minutes on a standard desktop, with posterior mean accuracy \(10^{-3}\). This opens up spatial statistics applications 100 times larger than previously possible.

This work addresses the question of uniqueness and regularity of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand -- whose precise form derives directly from the theory of perfect plasticity -- behaves quadratically close to the origin and grows linearly once a specific threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field -- the Cauchy stress in the terminology of perfect plasticity -- which allows us to define characteristic lines, and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study BF, we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data devoid of any regularity properties, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data. We also show a partial regularity result for the minimizer.

The Gibbs sampler, also known as the coordinate hit-and-run algorithm, is a Markov chain that is widely used to draw samples from probability distributions in arbitrary dimensions. At each iteration of the algorithm, a randomly selected coordinate is resampled from the distribution that results from conditioning on all the other coordinates. We study the behavior of the Gibbs sampler on the class of log-smooth and strongly log-concave target distributions supported on ℝ