Publications

We consider the quantum electrodynamics of single photons in arrays of one-way waveguides, each containing many atoms. We investigate both chiral and antichiral arrays, in which the group velocities of the waveguides are the same or alternate in sign, respectively. We find that in the continuum limit, the one-photon amplitude obeys a Dirac equation. In the chiral case, the Dirac equation is hyperbolic, while in the antichiral case it is elliptic. This distinction has implications for the nature of photon transport in waveguide arrays. Our results are illustrated by numerical simulations.

We consider infinite-horizon discounted Markov decision processes and study the convergence rates of the natural policy gradient (NPG) and the Q-NPG methods with the log-linear policy class. Using the compatible function approximation framework, both methods with log-linear policies can be written as inexact versions of the policy mirror descent (PMD) method. We show that both methods attain linear convergence rates and $\tilde{\mathcal{O}}(1/\epsilon^2)$ sample complexities using a simple, non-adaptive geometrically increasing step size, without resorting to entropy or other strongly convex regularization. Lastly, as a byproduct, we obtain sublinear convergence rates for both methods with arbitrary constant step size.

The aggregation of epitopes that are also able to bind major histocompatibility complex (MHC) alleles raises questions around the potential connection between the formation of epitope aggregates and their affinities to MHC receptors. We first performed a general bioinformatic assessment over a public dataset of MHC class II epitopes, finding that higher experimental binding correlates with higher aggregation-propensity predictors. We then focused on the case of P10, an epitope used as a vaccine candidate against Paracoccidioides brasiliensis that aggregates into amyloid fibrils. We used a computational protocol to design variants of the P10 epitope to study the connection between the binding stabilities towards human MHC class II alleles and their aggregation propensities. The binding of the designed variants was tested experimentally, as well as their aggregation capacity. High-affinity MHC class II binders in vitro were more disposed to aggregate forming amyloid fibrils capable of binding Thioflavin T and congo red, while low affinity MHC class II binders remained soluble or formed rare amorphous aggregates. This study shows a possible connection between the aggregation propensity of an epitope and its affinity for the MHC class II cleft.

When factorized approximations are used for variational inference (VI), they tend to underestimate the uncertainty -- as measured in various ways -- of the distributions they are meant to approximate. We consider two popular ways to measure the uncertainty deficit of VI: (i) the degree to which it underestimates the componentwise variance, and (ii) the degree to which it underestimates the entropy. To better understand these effects, and the relationship between them, we examine an informative setting where they can be explicitly (and elegantly) analyzed: the approximation of a Gaussian,~p, with a dense covariance matrix, by a Gaussian,~q, with a diagonal covariance matrix. We prove that q always underestimates both the componentwise variance and the entropy of p, \textit{though not necessarily to the same degree}. Moreover we demonstrate that the entropy of q is determined by the trade-off of two competing forces: it is decreased by the shrinkage of its componentwise variances (our first measure of uncertainty) but it is increased by the factorized approximation which delinks the nodes in the graphical model of p. We study various manifestations of this trade-off, notably one where, as the dimension of the problem grows, the per-component entropy gap between p and q becomes vanishingly small even though q underestimates every componentwise variance by a constant multiplicative factor. We also use the shrinkage-delinkage trade-off to bound the entropy gap in terms of the problem dimension and the condition number of the correlation matrix of p. Finally we present empirical results on both Gaussian and non-Gaussian targets, the former to validate our analysis and the latter to explore its limitations.

Omicron BA.1 is a highly infectious variant of SARS-CoV-2 that carries more than thirty mutations on the spike protein in comparison to the Wuhan wild type (WT). Some of the Omicron mutations, located on the receptor-binding domain (RBD), are exposed to the surrounding solvent and are known to help evade immunity. However, the impact of buried mutations on the RBD conformations and on the mechanics of the spike opening is less evident. Here, we use all-atom molecular dynamics (MD) simulations with metadynamics to characterize the thermodynamic RBD-opening ensemble, identifying significant differences between WT and Omicron. Specifically, the Omicron mutations S371L, S373P, and S375F make more RBD interdomain contacts during the spike's opening. Moreover, Omicron takes longer to reach the transition state than WT. It stabilizes up-state conformations with fewer RBD epitopes exposed to the solvent, potentially favoring immune or antibody evasion.

This manual describes the use of the Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Toolkit for Advanced Optimization (TAO) for the numerical solution of partial differential equations and related problems on high-performance computers. PETSc/TAO is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all distributed memory communication.

The acoustic inverse obstacle scattering problem consists of determining the shape of a domain from measurements of the scattered far field due to some set of incident fields (probes). For a penetrable object with known sound speed, this can be accomplished by treating the boundary alone as an unknown curve. Alternatively, one can treat the entire object as unknown and use a more general volumetric representation, without making use of the known sound speed. Both lead to strongly nonlinear and nonconvex optimization problems for which recursive linearization provides a useful framework for numerical analysis. After extending our shape optimization approach developed earlier for impenetrable bodies, we carry out a systematic study of both methods and compare their performance on a variety of examples. Our findings indicate that the volumetric approach is more robust, even though the number of degrees of freedom is significantly larger. We conclude with a discussion of this phenomenon and potential directions for further research.

A new scheme is presented for imposing periodic boundary conditions on unit cells with arbitrary source distributions. We restrict our attention here to the Poisson, modified Helmholtz, Stokes and modified Stokes equations. The approach extends to the oscillatory equations of mathematical physics, including the Helmholtz and Maxwell equations, but we will address these in a companion paper, since the nature of the problem is somewhat different and includes the consideration of quasiperiodic boundary conditions and resonances. Unlike lattice sum-based methods, the scheme is insensitive to the unit cell's aspect ratio and is easily coupled to adaptive fast multipole methods (FMMs). Our analysis relies on classical “plane-wave” representations of the fundamental solution, and yields an explicit low-rank representation of the field due to all image sources beyond the first layer of neighboring unit cells. When the aspect ratio of the unit cell is large, our scheme can be coupled with the nonuniform fast Fourier transform (NUFFT) to accelerate the evaluation of the induced field. Its performance is illustrated with several numerical examples.

Recently the SP (Stochastic Polyak step size) method has emerged as a competitive adaptive method for setting the step sizes of SGD. SP can be interpreted as a method specialized to interpolated models, since it solves the interpolation equations. SP solves these equation by using local linearizations of the model. We take a step further and develop a method for solving the interpolation equations that uses the local second-order approximation of the model. Our resulting method SP2 uses Hessian-vector products to speed-up the convergence of SP. Furthermore, and rather uniquely among second-order methods, the design of SP2 in no way relies on positive definite Hessian matrices or convexity of the objective function. We show SP2 is competitive both in experiments and in theory.
We show SP2 is very competitive on matrix completion, non-convex test problems and logistic regression. We also provide a convergence theory on sums-of-quadratics.

This is a handbook of simple proofs of the convergence of gradient and stochastic gradient descent type methods. We consider functions that are Lipschitz, smooth, convex, strongly convex, and/or Polyak-Łojasiewicz functions. Our focus is on