Plug and play: a general approach for the fusion of sensor and machine learning models
Regularized or Bayesian inversion has revolutionized our ability to reconstruct images from incomplete data.
97 MATHEMATICS AND COMPUTING↗
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TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems
Deep learning (DL), in particular deep neural networks, by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet—a Tikhonov neural network—which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers’s equation, and 50 for inverse initial conditions for 2D Navier–Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. Furthermore, this is possible owing to the model-constrained term, replications, and randomization.
97 MATHEMATICS AND COMPUTING↗
Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems
The action principle by Low for the classic Vlasov–Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler–Poincaré formulation of Vlasov–Maxwell-type systems, effectively starting with Low's action and using constrained variations for the Eulerian description of particle motion, has been known for a while Cendra et al, it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler–Poincaré machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov–Darwin system Sugama et al. The present exposition discusses a generic class of local Vlasov–Maxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler–Poincaré framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear- and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. Here, we also discuss what happens if no symmetries exist. Finally, two explicit examples will be given—the classic Vlasov–Maxwell and the drift-kinetic Vlasov–Maxwell—and the results expressed in the language of regular vector calculus for familiarity.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Alternatives to Contour Visualizations for Power Systems Data
Electrical grids are geographical and topological structures whose voltage states are challenging to represent accurately and efficiently for visual analysis. The current common practice is to use colored contour maps, yet these can misrepresent the data. We examine the suitability of four alternative visualization methods for depicting voltage data in a geographically dense distribution system - Voronoi polygons, H3 tessellations, S2 tessellations, and a network-weighted contour map. We find that Voronoi tessellations and network-weighted contour maps more accurately represent the statistical distribution of the data than regular contour maps.
MATHEMATICS AND COMPUTING,POWER TRANSMISSION AND D↗
An HP Adaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws
This dissertation addresses various issues for model classes of hyperbolic conservation laws. The basic approach developed in this work employs a new family of adaptive, hp-version, finite element methods based on a special discontinuous Galerkin formulation for hyperbolic problems. The discontinuous Galerkin formulation admits high-order local approximations on domains of quite general geometry, while providing a natural framework for finite element approximations and for theoretical developments. The use of hp-versions of the finite element method makes possible exponentially convergent schemes with very high accuracies in certain cases; the use of adaptive hp-schemes allows h-refinement in regions of low regularity and p-enrichment to deliver high accuracy, while keeping problem sizes manageable and dramatically smaller than many conventional approaches. The use of discontinuous Galerkin methods is uncommon in applications, but the methods rest on a reasonable mathematical basis for low-order cases and has local approximation features that can be exploited to produce very efficient schemes, especially in a parallel, multiprocessor environment. The place of this work is to first and primarily focus on a model class of linear hyperbolic conservation laws for which concrete mathematical results, methodologies, error estimates, convergence criteria, and parallel adaptive strategies can be developed, and to then briefly explore some extensions to more general cases. Next, we provide preliminaries to the study and a review of some aspects of the theory of hyperbolic conservation laws. We also provide a review of relevant literature on this subject and on the numerical analysis of these types of problems.
Bey, Kim S.↗
Uncertainty-driven dynamics for active learning of interatomic potentials
Machine learning (ML) models, if trained to data sets of high-fidelity quantum simulations, produce accurate and efficient interatomic potentials. Active learning (AL) is a powerful tool to iteratively generate diverse data sets. In this approach, the ML model provides an uncertainty estimate along with its prediction for each new atomic configuration. If the uncertainty estimate passes a certain threshold, then the configuration is included in the data set. Here we develop a strategy to more rapidly discover configurations that meaningfully augment the training data set. The approach, uncertainty-driven dynamics for active learning (UDD-AL), modifies the potential energy surface used in molecular dynamics simulations to favor regions of configuration space for which there is large model uncertainty. The performance of UDD-AL is demonstrated for two AL tasks: sampling the conformational space of glycine and sampling the promotion of proton transfer in acetylacetone. The method is shown to efficiently explore the chemically relevant configuration space, which may be inaccessible using regular dynamical sampling at target temperature conditions.
97 MATHEMATICS AND COMPUTING↗
Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation
Many material and biological samples in scientific imaging are characterized by nonlocal repeating structures. These are studied using scanning electron microscopy and electron tomography. Sparse sampling of individual pixels in a two-dimensional image acquisition geometry, or sparse sampling of projection images with large tilt increments in a tomography experiment, can enable high speed data acquisition and minimize sample damage caused by the electron beam. In this paper, we present an algorithm for electron tomographic reconstruction and sparse image interpolation that exploits the nonlocal redundancy in images. We adapt a framework, termed plug-and-play priors, to solve these imaging problems in a regularized inversion setting. The power of the plug-and-play approach is that it allows a wide array of modern denoising algorithms to be used as a “prior model” for tomography and image interpolation. We also present sufficient mathematical conditions that ensure convergence of the plug-and-play approach, and we use these insights to design a new nonlocal means denoising algorithm. Finally, we demonstrate that the algorithm produces higher quality reconstructions on both simulated and real electron microscope data, along with improved convergence properties compared to other methods.
47 OTHER INSTRUMENTATION↗
Online Primal-Dual Methods With Measurement Feedback for Time-Varying Convex Optimization
This paper addresses the design and analysis of feedback-based online algorithms to control systems or networked systems based on performance objectives and engineering constraints that may evolve over time. The emerging timevarying convex optimization formalism is leveraged to model optimal operational trajectories of the systems, as well as explicit local and network-level operational constraints. Departing from existing batch and feed-forward optimization approaches, the design of the algorithms capitalizes on an online implementation of primal-dual projected-gradient methods; the gradient steps are, however, suitably modified to accommodate feedback from the system in the form of measurements - hence, the term 'online optimization with feedback.' By virtue of this approach, the resultant algorithms can cope with model mismatches in the algebraic representation of the system states and outputs, they avoid pervasive measurements of exogenous inputs, and they naturally lend themselves to a distributed implementation. Under suitable assumptions, analytical convergence claims are established in terms of dynamic regret. Furthermore, when the synthesis of the feedback-based online algorithms is based on a regularized Lagrangian function, Q-linear convergence to solutions of the time-varying optimization problem is shown.
97 MATHEMATICS AND COMPUTING↗
Nonlinear problems in flight dynamics
A comprehensive framework is proposed for the description and analysis of nonlinear problems in flight dynamics. Emphasis is placed on the aerodynamic component as the major source of nonlinearities in the flight dynamic system. Four aerodynamic flows are examined to illustrate the richness and regularity of the flow structures and the nature of the flow structures and the nature of the resulting nonlinear aerodynamic forces and moments. A framework to facilitate the study of the aerodynamic system is proposed having parallel observational and mathematical components. The observational component, structure is described in the language of topology. Changes in flow structure are described via bifurcation theory. Chaos or turbulence is related to the analogous chaotic behavior of nonlinear dynamical systems characterized by the existence of strange attractors having fractal dimensionality. Scales of the flow are considered in the light of ideas from group theory. Several one and two degree of freedom dynamical systems with various mathematical models of the nonlinear aerodynamic forces and moments are examined to illustrate the resulting types of dynamical behavior. The mathematical ideas that proved useful in the description of fluid flows are shown to be similarly useful in the description of flight dynamic behavior.
Chapman, G. T.↗
Non-commutative disintegrations: Existence and uniqueness in finite dimensions
Motivated by advances in categorical probability, we introduce non-commutative almost everywhere (a.e.) equivalence and disintegrations in the setting of C*-algebras. We show that C*-algebras (resp. W*-algebras) and a.e. equivalence classes of 2-positive (resp. positive) unital maps form a category. We prove non-commutative disintegrations are a.e. unique whenever they exist. We provide an explicit characterization for when disintegrations exist in the setting of finite-dimensional C*-algebras, and we give formulas for the associated disintegrations.
97 MATHEMATICS AND COMPUTING↗
C{sup ∗}-Quadratic Quantization
In the first part of the paper we introduce a new parametrization for the manifold underlying quadratic analogue of the usual Heisenberg group introduced in Accardi et al. (Infin Dimens Anal Quantum Probab Relat Top 13:551–587, 2010) which makes the composition law much more transparent. In the second part of the paper the new coordinates are used to construct an inductive system of ∗-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode quadratic Weyl algebra. We prove that the inductive limit ∗-algebra is factorizable and has a natural localization given by a family of ∗-sub-algebras each of which is localized on a bounded Borel subset of ℝ. Moreover, we prove that the family of quadratic analogues of the Fock states, defined on the inductive family of ∗-algebras, is projective hence it defines a unique state on the limit ∗-algebra. Finally we complete this ∗-algebra under the (minimal regular) C{sup ∗}-norm thus obtaining a C{sup ∗}-algebra.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Singularly Perturbed System of Parabolic Equations in the Critical Case
We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudary-layer functions. The construction of the asymptotics is based on the regularization method for singularly perturbed problems developed by S. A. Lomov and adapted to singularly perturbed parabolic equations with two viscous boundaries by A. S. Omuraliev.
97 MATHEMATICS AND COMPUTING↗
Rapidly varying solutions of a second-order differential equation with regularly and rapidly varying nonlinearities
For a second-order differential equation whose the right-hand side contains the sum of terms with regularly and rapidly varying nonlinearities, we establish the conditions of existence and asymptotic representations of rapidly varying solutions, as the argument tends to a singular point.
97 MATHEMATICS AND COMPUTING↗
Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format
Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial directions is deemed to be practically infeasible beyond a relatively small number of dimensions, e.g., three or four. In fact, the computational burden in terms of storage and floating point operations scales exponentially with the number of dimensions. This phenomenon is known as the curse of dimensionality and motivated the development of alternative methods such as the Monte Carlo method. The tensor product approach can be very effective for high-dimensional numerical integration if we can resort to an accurate low-rank tensor-train representation of the integrand function. In this work, we discuss this approach and present numerical evidence showing that it is very competitive with the Monte Carlo method in terms of accuracy and computational costs up to several hundredths of dimensions if the integrand function is regular enough and a sufficiently accurate low-rank approximation is available.
97 MATHEMATICS AND COMPUTING↗
Iterative Joint Ptychography-Tomography with Total Variation Regularization
In order to determine the 3D structure of a thick sample, researchers have recently combined ptychography (for high resolution) and tomography (for 3D imaging) in a single experiment. 2-step methods are usually adopted for reconstruction, where the ptychography and tomography problems are often solved independently. In this paper, we provide a novel model and ADMM-based algorithm to jointly solve the ptychography-tomography problem iteratively, also employing total variation regularization. The proposed method permits large scan stepsizes for the ptychography experiment, requiring less measurements and being more robust to noise with respect to other strategies, while achieving higher reconstruction quality results.
2-step methods↗
Exploring the Use of Alfven Waves in Magnetometer Calibration at Geosynchronous Orbit
An Alfven wave is a type magnetohydrodynamicwave that travels through a conducting fluid under the influence of a magnetic field. Researchers have successfully calculated offset vectors of magnetometers in interplanetary space by optimizing the offset to maximize certain Alfvenic properties of observed waves (Leinweber, Belcher). If suitable Alfven waves can be found in the magnetosphere at geosynchronous altitude then these techniques could be used to augment the overall calibration plan for magnetometers in this region such as on the GOES spacecraft, possibly increasing the time between regular maneuvers. Calibration maneuvers may be undesirable because they disrupt the activities of other instruments. Various algorithms to calculate an offset using Alfven waves were considered. A new variation of the Davis-Smith method was derived because it can be mathematically shown that the Davis-Smith method tolerates filtered data, which expands potential applications. The variant developed was designed to find only the offset in the plane normal to the main field because the overall direction of Earth's magnetic field rarely changes, and theory suggests the Alfvenic disturbances occur transverse to the main field. Other variations of the Davis-Smith method encounter problems with data containing waves that propagate in mostly the same direction. A searching algorithm was then designed to look for periods of time with potential Alfven waves in GOES 15 data based on parameters requiring that disturbances be normal to the main field and not change field magnitude. Final waves for calculation were hand-selected. These waves produced credible two-dimensional offset vectors when input to the Davis-Smith method. Multiple two-dimensional solutions in different planes can be combined to get a measurement of the complete offset. The resulting three dimensional offset did not show sufficient precision over several years to be used as a primary calibration method, but reflected changes in the offset fairly well, suggesting that the method could be helpful in monitoring trends of the offset vector when maneuvers cannot be used.
calibration↗
ECP Software Technology Capability Assessment Report
The Exascale Computing Project (ECP) Software Technology (ST) Focus Area is responsible for developing critical software capabilities that will enable successful execution of ECP applications, and for providing key components of a productive and sustainable Exascale computing ecosystem that will position the US Department of Energy (DOE) and the broader high performance (HPC) community with a firm foundation for future extreme-scale computing capabilities. This ECP ST Capability Assessment Report (CAR) provides an overview and assessment of current ECP ST capabilities and activities, giving stakeholders and the broader HPC community information that can be used to assess ECP ST progress and plan their own efforts accordingly. ECP ST leaders commit to updating this document on regular basis (every six to 12 months).
97 MATHEMATICS AND COMPUTING↗
ECP Software Technology Capability Assessment Report
The Exascale Computing Project (ECP) Software Technology (ST) Focus Area is responsible for developing critical software capabilities that will enable successful execution of ECP applications, and for providing key components of a productive and sustainable Exascale computing ecosystem that will position the US Department of Energy (DOE) and the broader high performance (HPC) community with a firm foundation for future extreme-scale computing capabilities. This ECP ST Capability Assessment Report (CAR) provides an overview and assessment of current ECP ST capabilities and activities, giving stakeholders and the broader HPC community information that can be used to assess ECP ST progress and plan their own efforts accordingly. ECP ST leaders commit to updating this document on regular basis (every six to 12 months).
97 MATHEMATICS AND COMPUTING↗