Publications

Speckle or multiplicative noise is a critical issue in coherence-based imaging systems, such as synthetic aperture radar and optical coherence tomography. Existence of speckle noise considerably limits the applicability of such systems by degrading their performance. On the other hand, the sophistications that arise in the study of multiplicative noise have so far impeded theoretical analysis of such imaging systems. As a result, the current acquisition technology relies on heuristic solutions, such as oversampling the signal and converting the problem into a denoising problem with multiplicative noise. This paper attempts to bridge the gap between theory and practice by providing the first theoretical analysis of such systems. To achieve this goal the log-likelihood function corresponding to measurement systems with speckle noise is characterized. Then employing compression codes to model the source structure, for the case of under-sampled measurements, a compression-based maximum likelihood recovery method is proposed. The mean squared error (MSE) performance of the proposed method is characterized and is shown to scale as O(klognm−−−−−√) , where k , m and n denote the intrinsic dimension of the signal class according to the compression code, the number of observations, and the ambient dimension of the signal, respectively. This result, while in contrast to imaging systems with additive noise in which MSE scales as O(klognm) , suggests that if the signal class is structured (i.e., k≪n ), accurate recovery of a signal from under-determined measurements is still feasible, even in the presence of speckle noise. Simulation results are presented that suggest image recovery under multiplicative noise is inherently more challenging than additive noise, and that the derived theoretical results are sharp

We introduce a fast direct solver for variable-coefficient elliptic partial differential equations on surfaces based on the hierarchical Poincaré-Steklov method. The method takes as input an unstructured, high-order quadrilateral mesh of a surface and discretizes surface differential operators on each element using a high-order spectral collocation scheme. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in O(N N) operations for a mesh with N elements. The resulting fast direct solver may be used to accelerate high-order implicit time-stepping schemes, as the precomputed operators can be reused for fast elliptic solves on surfaces. On a standard laptop, precomputation for a 12th-order surface mesh with over 1 million degrees of freedom takes 17 seconds, while subsequent solves take only 0.25 seconds. We apply the method to a range of problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace-Beltrami problem, the Hodge decomposition of a tangential vector field, and some time-dependent nonlinear reaction-diffusion systems.

We perform a data-driven dimensionality reduction of the scale-dependent four-point vertex function characterizing the functional renormalization group (FRG) flow for the widely studied two-dimensional \(t−t′\) Hubbard model on the square lattice. We demonstrate that a deep learning architecture based on a neural ordinary differential equation solver in a low-dimensional latent space efficiently learns the FRG dynamics that delineates the various magnetic and d-wave superconducting regimes of the Hubbard model. We further present a dynamic mode decomposition analysis that confirms that a small number of modes are indeed sufficient to capture the FRG dynamics. Our Letter demonstrates the possibility of using artificial intelligence to extract compact representations of the four-point vertex functions for correlated electrons, a goal of utmost importance for the success of cutting-edge quantum field theoretical methods for tackling the many-electron problem.

We investigate the effects of non-zero spatial curvature on cosmic inflation in the light of cosmic microwave background (CMB) anisotropy measurements from the Planck 2018 legacy release and from the 2015 observing season of BICEP2 and the Keck Array. Even a small percentage of non-zero curvature today would significantly limit the total number of e-folds of the scale factor during inflation, rendering just-enough inflation scenarios with a kinetically dominated or fast-roll stage prior to slow-roll inflation more likely. Finite inflation leads to oscillations and a cutoff towards large scales in the primordial power spectrum and curvature pushes them into the CMB observable window. Using nested sampling, we carry out Bayesian parameter estimations and model comparisons taking into account constraints from reheating and horizon considerations. We confirm the preference of CMB data for closed universes with Bayesian odds of over 100:1 and with a posterior on the curvature density parameter of ΩK,0=−0.051±0.017 for a curvature extension of LCDM and ΩK,0=−0.031±0.014 for Starobinsky inflation. Model comparisons of various inflation models give similar results as for flat universes with the Starobinsky model outperforming most other models.

Single particle cryo-electron microscopy has become a critical tool in structural biology over the last decade, able to achieve atomic scale resolution in three dimensional models from hundreds of thousands of (noisy) two-dimensional projection views of particles frozen at unknown orientations. This is accomplished by using a suite of software tools to (i) identify particles in large micrographs, (ii) obtain low-resolution reconstructions, (iii) refine those low-resolution structures, and (iv) finally match the obtained electron scattering density to the constituent atoms that make up the macromolecule or macromolecular complex of interest. Here, we focus on the second stage of the reconstruction pipeline: obtaining a low resolution model from picked particle images. Our goal is to create an algorithm that is capable of ab initio reconstruction from small data sets (on the order of a few thousand selected particles). More precisely, we seek an algorithm that is robust, automatic, able to assess particle quality, and fast enough that it can potentially be used to assist in the assessment of the data being generated while the microscopy experiment is still underway.

Simulations with adaptive time-dependent bias enable an efficient exploration of the conformational space of a system. However, the dynamic information is altered by the bias. Infrequent metadynamics recovers the transition rate of crossing a barrier, if the collective variables are ideal and there is no bias deposition near the transition state. Unfortunately, these conditions are not always fulfilled. To overcome these limitations, and inspired by single-molecule force spectroscopy, we use Kramers’ theory for calculating the barrier-crossing rate when a time-dependent bias is added to the system. We assess the efficiency of collective variables parameter by measuring how efficiently the bias accelerates the transitions. We present approximate analytical expressions of the survival probability, reproducing the barrier-crossing time statistics and enabling the extraction of the unbiased transition rate even for challenging cases. We explore the limits of our method and provide convergence criteria to assess its validity.

We introduce a numerical method for the solution of the time-dependent Schrödinger equation with a smooth potential, based on its reformulation as a Volterra integral equation. We present versions of the method both for periodic boundary conditions, and for free space problems with compactly supported initial data and potential. A spatially uniform electric field may be included, making the solver applicable to simulations of light-matter interaction. The primary computational challenge in using the Volterra formulation is the application of a spacetime history dependent integral operator. This may be accomplished by projecting the solution onto a set of Fourier modes, and updating their coefficients from one time step to the next by a simple recurrence. In the periodic case, the modes are those of the usual Fourier series, and the fast Fourier transform (FFT) is used to alternate between physical and frequency domain grids. In the free space case, the oscillatory behavior of the spectral Green's function leads us to use a set of complex-frequency Fourier modes obtained by discretizing a contour deformation of the inverse Fourier transform, and we develop a corresponding fast transform based on the FFT. Our approach is related to pseudospectral methods, but applied to an integral rather than the usual differential formulation. This has several advantages: it avoids the need for artificial boundary conditions, admits simple, inexpensive, high-order implicit time marching schemes, and naturally includes time-dependent potentials. We present examples in one and two dimensions showing spectral accuracy in space and eighth-order accuracy in time for both periodic and free space problems.

Partial differential equations (PDEs) play a dominant role in the mathematical modeling of many complex dynamical processes. Solving these PDEs often requires prohibitively high computational costs, especially when multiple evaluations must be made for different parameters or conditions. After training, neural operators can provide PDEs solutions significantly faster than traditional PDE solvers. In this work, invariance properties and computational complexity of two neural operators are examined for transport PDE of a scalar quantity. Neural operator based on graph kernel network (GKN) operates on graph-structured data to incorporate nonlocal dependencies. Here we propose a modified formulation of GKN to achieve frame invariance. Vector cloud neural network (VCNN) is an alternate neural operator with embedded frame invariance which operates on point cloud data. GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN. However, GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN.

With the funding provided by this award, we developed numerical codes for the study of magnetically confined plasmas for fusion applications. Accordingly, our work can be divided into two separate categories: 1) the design and analysis of novel numerical methods providing high accuracy and high efficiency; 2) the study of the equilibrium and stability of magnetically confined plasmas with some of these numerical codes, as well as the study of the nature of the turbulent behavior which may arise in the presence of instabilities. We first developed new numerical schemes based on integral equation methods for the computation of steady-state magnetic configurations in fusion experiments, providing high accuracy for the magnetic field and its derivatives, which are required for stability and turbulence calculations. We employed different integral formulations depending on the application of interest: axisymmetric or non-axisymmetric equilibria, force-free or magnetohydrodynamic equilibria, fixed-boundary equilibria or free-boundary equilibria. While efficient, these methods do not yet apply to plasma boundaries which are not smooth, a situation which is fairly common in magnetic confinement experiments. To address this temporary weakness, we also constructed a new steady-state solver based on the Hybridizable Discontinuous Galerkin (HDG) method, which provides full geometric flexibility. In addition to these numerical tools focused on steady-states, we also contributed to the improvement of the speed and accuracy of codes simulating the plasma dynamics of fusion plasmas, by developing a novel velocity space representation for the efficient solution of kinetic equations, which most accurately describe the time evolution of hot plasmas in fusion experiments. Using the tools discussed above, we studied several questions pertaining to the equilibrium and stability of magnetically confined plasmas. In particular, we derived a new simple model for axisymmetric devices called tokamaks, to predict how elongated a fusion plasma can be before it becomes unstable and collapses. We also looked at the effect of the shape of the outer plasma surface on key properties of the steady-state magnetic configurations, and how these properties impact turbulence in fusion plasmas, and the corresponding transport of momentum. Likewise, we studied the role of large localized flows on the steady-state magnetic configurations, and how they may influence plasma stability and turbulence. Non-axisymmetric steady-state magnetic configurations are inherently more complex than axisymmetric steady-state configurations, and the subject of ongoing controversies regarding the regularity of the equations determining such steady-states, and their solutions. Implementing an existing NYU code in a new geometry, we studied the nature of the singularity of the solutions observed in the code, and methods to eliminate them. Our main conclusion is that by appropriately tailoring the plasma boundary, it is possible to eliminate the singularities otherwise appearing in our simulations, and to obtain steady-states which appear to be smooth. To gain further insights on incompletely understood turbulence phenomena, we proposed a new reduced model capturing most of these phenomena, which is simple enough to not require expensive numerical simulations on massive supercomputers to investigate them. We demonstrated the strong similarity between our simulations and published results obtained from computationally expensive simulations, and plan to rely on our reduced model to identify the key mechanisms determining the evolution and strength turbulent driven transport in fusion plasmas. Finally, we proposed a new framework for tokamak reactor design studies, enabling us to consider the relative merits of steady-state versus pulsed fusion reactors. We found that pulsed fusion reactors may benefit most from recent advances in magnet technology, and the availability of very high field magnets. As such, they may become more desirable than steady-state tokamak reactors for cost efficient electricity generation.

A new scheme is proposed to construct a $\mathcal{C}^n$ function extension for smooth functions defined on a smooth domain $D\in \mathbb{R}^d$. Unlike the PUX scheme, which requires the extrapolation of the volume grid via an expensive ill-conditioned least squares fitting, the scheme relies on an explicit formula consisting of a linear combination of function values in $D,$ which only extends the function along the normal direction. To be more precise, the $\mathcal{C}^n$ extension requires only $n+1$ function values along the normal directions in the original domain and ensures $\mathcal{C}^n$ smoothness by construction. When combined with a shrinking function and a smooth window function, the scheme can be made stable and robust for a broad class of domains with complex smooth boundary.