arXiv subjects

Abbas Ourmazd

Publications and source records attributed to Abbas Ourmazd.

9 recordsLinked to original sources

arXiv2022

Data Assimilation in Operator Algebras

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a non-abelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to new computational data assimilation schemes that are (i) automatically positivity-preserving; and (ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to data assimilation of the Lorenz 96 multiscale system and the El Nino Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

math.ST↗

arXiv2021

Embedding classical dynamics in a quantum computer

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides a new operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space, $\mathcal H $, and an embedding of classical observables into self-adjoint operators on $\mathcal H$. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of $\mathcal H$, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve an exponential quantum computational advantage, we project the state of the quantum system to a density matrix on a $2^n$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through a quantum circuit of size $O(n)$. Furthermore, the circuit features a state preparation stage, also of size $O(n)$, and a quantum Fourier transform stage of size $O(n^2)$, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions as $n\to\infty$. We present simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.

quant-ph↗

arXiv2019

Megahertz single-particle imaging at the European XFEL

The emergence of high repetition-rate X-ray free-electron lasers (XFELs) powered by superconducting accelerator technology enables the measurement of significantly more experimental data per day than was previously possible. The European XFEL will soon provide 27,000 pulses per second, more than two orders of magnitude more than any other XFEL. The increased pulse rate is a key enabling factor for single-particle X-ray diffractive imaging, which relies on averaging the weak diffraction signal from single biological particles. Taking full advantage of this new capability requires that all experimental steps, from sample preparation and delivery to the acquisition of diffraction patterns, are compatible with the increased pulse repetition rate. Here, we show that single-particle imaging can be performed using X-ray pulses at megahertz repetition rates. The obtained results pave the way towards exploiting high repetition-rate X-ray free-electron lasers for single-particle imaging at their full repetition rate.

physics.ins-det↗

arXiv2018

Spatiotemporal pattern extraction by spectral analysis of vector-valued observables

We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called Vector-valued Spectral Analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data, and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto-Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.

math.DS↗

arXiv2018

Vector-Valued Spectral Analysis of Space-Time Data

Identifying coherent spatiotemporal patterns generated by complex dynamical systems is a central problem in many science and engineering disciplines. Here, we combine ideas from the theory of operator-valued kernels with delay-embedding techniques in dynamical systems to develop of a method for objective identification of spatiotemporal coherent patterns, without prior knowledge of the state space and/or the dynamical laws of the system generating the data. A key aspect of this method is that it operates on a space of vector-valued observables using a kernel measure of similarity that takes into account both temporal and spatial degrees of freedom (in contrast, classical kernel techniques such as PCA utilize aggregate measures of similarity between 'snapshots'). As a result, spectral decomposition of data via our approach yields a significantly more efficient and physically meaningful representation of complex spatiotemporal signals than conventional methods based on scalar-valued kernels. We demonstrate this behavior with applications to an oscillator model and sea surface temperature data from a climate model.

physics.data-an↗

arXiv2011

The Symmetries of Image Formation by Scattering. I. Theoretical Framework

We perceive the world through images formed by scattering. The ability to interpret scattering data mathematically has opened to our scrutiny the constituents of matter, the building blocks of life, and the remotest corners of the universe. Here, we deduce for the first time the fundamental symmetries underlying image formation. Intriguingly, these are similar to those of the anisotropic "Taub universe"' of general relativity, with eigenfunctions closely related to spinning tops in quantum mechanics. This opens the possibility to apply the powerful arsenal of tools developed in two major branches of physics to new problems. We augment these tools with graph-theoretic means to recover the three-dimensional structure of objects from random snapshots of unknown orientation at four orders of magnitude higher complexity than previously demonstrated. Our theoretical framework offers a potential link to recent observations on face perception in higher primates. In a later paper, we demonstrate the recovery of structure and dynamics from ultralow-signal random sightings of systems with no orientational or timing information.

physics.comp-ph↗

arXiv2011

The symmetries of image formation by scattering. II. Applications

We show that the symmetries of image formation by scattering enable graph-theoretic manifold-embedding techniques to extract structural and timing information from simulated and experimental snapshots at extremely low signal. The approach constitutes a physically-based, computationally efficient, and noise-robust route to analyzing the large and varied datasets generated by existing and emerging methods for studying structure and dynamics by scattering. We demonstrate three-dimensional structure recovery from X-ray diffraction and cryo-electron microscope image snapshots of unknown orientation, the latter at 12 times lower dose than currently in use. We also show that ultra-low-signal, random sightings of dynamically evolving systems can be sequenced into high quality movies to reveal their evolution. Our approach offers a route to recovering timing information in time-resolved experiments, and extracting 3D movies from two-dimensional random sightings of dynamic systems.

physics.comp-ph↗

arXiv2010

Bayesian algorithms for recovering structure from single-particle diffraction snapshots of unknown orientation: a comparison

The advent of X-ray Free Electron Lasers promises the possibility to determine the structure of individual particles such as microcrystallites, viruses and biomolecules from single-shot diffraction snapshots obtained before the particle is destroyed by the intense femtosecond pulse. This program requires the ability to determine the orientation of the particle giving rise to each snapshot at signal levels as low as ~10-2 photons/pixel. Two apparently different approaches have recently demonstrated this capability. Here we show they represent different implementations of the same fundamental approach, and identify the primary factors limiting their performance.

cond-mat.soft↗

arXiv2008

Structure from Fleeting Illumination of Faint Spinning Objects in Flight with Application to Single Molecules

There are many instances when the structure of a weakly-scattering spinning object in flight must be determined to high resolution. Examples range from comets to nanoparticles and single molecules. The latter two instances are the subject of intense current interest. Substantial progress has recently been made in illuminating spinning single particles in flight with powerful X-ray bursts to determine their structure with the ultimate goal of determining the structure of single molecules. However, proposals to reconstruct the molecular structure from diffraction "snapshots" of unknown orientation require ~1000x more signal than available from next-generation sources. Using a new approach, we demonstrate the recovery of the structure of a weakly scattering macromolecule at the anticipated next-generation X-ray source intensities. Our work closes a critical gap in determining the structure of single molecules and nanoparticles by X-ray methods, and opens the way to reconstructing the structure of spinning, or randomly-oriented objects at extremely low signal levels. Other potential applications include low-dose electron microscopy, ultra-low-signal tomography of non-stationary objects without orientational information, and the study of heavenly bodies.

physics.bio-ph↗