Original Operator Research Articles

The adjoint Rayleigh and Orr-Sommerfeld equations: Green function and eigenmodes

The Rayleigh and Orr-Sommerfeld equations are ODEs which arise from the linearized Euler and Navier-Stokes equation around a shear flow. In this paper, we consider the adjoints of the Rayleigh and Orr-Sommerfeld equations on [0,∞) with respect to the complex L2 product. In the viscous case, we consider a family of viscosity-dependent Navier boundary conditions, which in the limit corresponds to the no-slip condition. We rigorously establish existence and asymptotic properties of their eigenvalues, eigenmodes and Green functions away from critical layers. The adjoint operators are useful because they also allow us to deduce properties about the kernels and images of the original operators.

A lightweight joint metric detection approach on YOLO for hot spots in photovoltaic modules

The hot spot effect is one of the primary causes of damage to photovoltaic (PV) modules and a significant factor contributing to the decline in their power generation capacity. Thermal imaging inspection of PV modules is an indispensable aspect of PV plant operation and maintenance. This article introduces a lightweight detection algorithm for hot spots in PV modules based on an enhanced version of YOLOv8. The algorithm incorporates a lightweight convolution operator, Volumetric Grid Spatial Cross Stage Partial, as a replacement for the original C2f operator, resulting in effective reduction of the model's parameter size and improved detection speed. Additionally, the Content-Aware ReAssembly of FEatures upsampling operator is utilized instead of the nearest-neighbor algorithm during the upsampling process, thereby minimizing the loss of hot spot feature information. Furthermore, a joint metric approach combining the Complete Intersection over Union metric and the normalized Wasserstein distance metric is proposed to calculate the localization loss of the target bounding boxes, thereby enhancing the regression accuracy of small hot spot prediction boxes. Experimental results demonstrate that the improved YOLOv8 network enables fast and accurate detection of hot spots. This method achieves an impressive average precision of 98.8% and a high detection speed of 218.3 fps.

Hamiltonian, geometric momentum and force operators for a spin zero particle on a curve: physical approach

The Hamiltonian for a spin zero particle that is confined to a 1D curve embedded in the 3D space is constructed. Confinement is achieved by starting with the particle living in a small tube surrounding the curve, and assuming an infinitely strong normal force that squeezes the thickness of the tube to zero, eventually pinning the particle to the curve. We follow the new approach that we applied to confine a particle to a surface, in that we start with an expression for the 3D momentum operators whose components along and normal to the curve directions are separately Hermitian. The kinetic energy operator expressed in terms of the momentum operator in the normal direction is then a Hermitian operator in this case. When this operator is dropped and the thickness of the tube surrounding the curve is set to zero, one automatically gets the Hermitian curve Hamiltonian that contains the geometric potential term as expected. It is demonstrated that the origin of this potential lies in the ordering or symmetrization of the original 3D momentum operators in order to render them Hermitian. The Hermitian momentum operator for the particle as it is confined to the curve is also constructed and is seen to be similar to what is known as the geometric momentum of a particle confined to a surface in that it has a term proportional to the curvature that is along the normal to the curve. The force operator of the particle on the curve is also derived, and is shown to reduce, for a curve with a constant curvature and torsion, to a -apparently- single component normal to the curve that is a symmetrization of the classical expression plus a quantum term. All the above quantities are then derived for the specific case of a particle confined to a cylindrical helix embedded in 3D space.

On rotated CMV operators and orthogonal polynomials on the unit circle

Split-step quantum walk operators can be expressed as a generalized version of CMV operators with complex transmission coefficients, which we call rotated CMV operators. Following the idea of Cantero, Moral and Velazquez's original construction of the original CMV operators from the theory of orthogonal polynomials on the unit circle (OPUC), we show that rotated CMV operators can be constructed similarly via a rotated version of OPUCs with respect to the same measure, and admit an analogous LM -factorization as the original CMV operators. We also develop the rotated second kind polynomials corresponding to the rotated OPUCs. We then use the LM -factorization of rotated alternate CMV operators to compute the Gesztesy–Zinchenko transfer matrices for rotated CMV operators.

The Hessian by blocks for neural network by backward propagation

The back-propagation algorithm used with a stochastic gradient and the increase in computer performance are at the origin of the recent Deep learning trend. For some problems, however, the convergence of gradient methods is still very slow. Newton's method offers potential advantages in terms of faster convergence. This method uses the Hessian matrix to guide the optimization process but increases the computational cost at each iteration. Indeed, although the expression of the Hessian matrix is explicitly known, previous work did not propose an efficient algorithm for its fast computation. In this work, we first propose a backward algorithm to compute the exact Hessian matrix. In addition, the introduction of original operators, for the calculation of second derivatives, facilitates the reading and allows the parallelization of the backward-looking algorithm. To study the practical performance of Newton's method, we apply the proposed algorithm to train two classical neural networks for regression and classification problems and display the associated numerical results.

Entropic transfer operators

We propose a new concept for the regularization and discretization of transfer and Koopman operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite-dimensional approximation of some transfer or Koopman operator which can be analyzed computationally. We prove that the spectrum of the discretized operator converges to the one of the regularized original operator, give a detailed analysis of the relation between the discretized and the original peripheral spectrum for a rotation map on the n-torus and provide code for three numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered.

Modeling, landscape analysis, and solving the capacitated single-allocation hub maximal covering problem using the GARVND hybrid algorithm

In this paper, a new variation to the Hub Maximal Covering problem called the Capacitated Single-Allocation Hub Maximal Covering Problem (CSAHMCP) is addressed for the first time and a novel binary-integer mathematical model has been proposed. To analyze the features of the problem’s solution space and to determine the most effective heuristic approach to solve it, a comprehensive Fitness Landscape Analysis is performed which reveals that the problem is of ‘rugged plain’ type and can best be solved using an algorithm with good exploration and exploitation capabilities at the same time. Accordingly, we proposed a new algorithm named GARVND, which is a hybrid of the Genetic Algorithm acting as the main framework for diversification, and a variant of the Variable Neighborhood Descent algorithm called ‘Reduced VND’ acting as a means to improve individual solutions (intensification) by searching in the spaces generated by four types of neighborhood operators. We also introduced two original crossover and mutation operators that can be used for any hub location solution method. The parameters of GARVND are tuned toward less computational times and higher solution quality by using the Taguchi experiment design technique. We solved 114 CSAHMCP instances (several times each) from the Australian Post (AP) dataset using the proposed GARVND, as well as by the Genetic Algorithm, Variable Neighborhood Search, Simulated Annealing, Tabu Search algorithms, and a hybrid algorithm consisting of Grey Wolf Optimizer and Simulated Annealing, in addition to the Gurobi exact solver. Extensive computational results and statistical analyses showed that GARVND yields favorable solutions, outperforming all the compared algorithms by producing smaller gaps with global optimal objective values. In addition, to illustrate the practical value of the CSAHMCP and the GARVND method, we performed a cost-flow sensitivity analysis for the AP Central Sydney network and came up with the best expansion scenarios to be adopted by the management.

STile: Searching Hybrid Sparse Formats for Sparse Deep Learning Operators Automatically

Sparse operators, i.e., operators that take sparse tensors as input, are of great importance in deep learning models. Due to the diverse sparsity patterns in different sparse tensors, it is challenging to optimize sparse operators by seeking an optimal sparse format, i.e., leading to the lowest operator latency. Existing works propose to decompose a sparse tensor into several parts and search for a hybrid of sparse formats to handle diverse sparse patterns. However, they often make a trade-off between search space and search time: their search spaces are limited in some cases, resulting in limited operator running efficiency they can achieve. In this paper, we try to extend the search space in its breadth (by doing flexible sparse tensor transformations) and depth (by enabling multi-level decomposition). We formally define the multi-level sparse format decomposition problem, which is NP-hard, and we propose a framework STile for it. To search efficiently, a greedy algorithm is used, which is guided by a cost model about the latency of computing a sub-task of the original operator after decomposing the sparse tensor. Experiments of two common kinds of sparse operators, SpMM and SDDMM, are conducted on various sparsity patterns, and we achieve 2.1-18.0× speedup against cuSPARSE on SpMMs and 1.5 - 6.9× speedup against DGL on SDDMM. The search time is less than one hour for any tested sparse operator, which can be amortized.

Estimation of the Born data in inverse scattering of layered media

The first term in the Born series, as we call the Born data, is linear with the scatterers. Here we present a scheme to map the total field data to the Born data in layered media using only the single-input single-output (SISO) setup. This nonlinear mapping is based on the reduced order model (ROM) approach, which constructs ROMs of the original wave operator. Normally, the construction of ROMs requires multi-input multi-output data. By introducing fictitious sensors, we estimate the Born data with SISO data in layered media. We give a simple way of using the time-domain Green’s function to estimate the received data for other fictitious sensors without calculating the complicated Sommerfeld integral. The resulting Born data contains only the single-scattering component, which can be helpful for many imaging applications. A numerical example is given incorporating the direct imaging back-propagation method. It validates the linearity of the Born data by providing an artifact-free image without the optimization process.

Quartic Gradient Flow

Saddle-point configurations, such as the Euclidean bounce and sphalerons, are known to be difficult to find numerically. In this letter we study a new method, Quartic Gradient Flow, to search for such configurations. The central idea is to introduce a gradient-flow-like equation in such a way that all the fluctuations around the saddle-point have eigenvalues that are square of the eigenvalues of the original quadratic operator. We illustrate how the method works for the Euclidean bounce and sphalerons.

Algorithmic monotone multiscale finite volume methods for porous media flow

Multiscale finite volume methods are known to produce reduced systems with multipoint stencils which, in turn, could give non-monotone and out-of-bound solutions. We propose a novel solution to the monotonicity issue of multiscale methods. The proposed algorithmic monotone (AM- MsFV/MsRSB) framework is based on an algebraic modification to the original MsFV/MsRSB coarse-scale stencil. The AM-MsFV/MsRSB guarantees monotonic and within bound solutions without compromising accuracy for various coarsening ratios; hence, it effectively addresses the challenge of multiscale methods' sensitivity to coarse grid partitioning choices. Moreover, by preserving the near null space of the original operator, the AM-MsRSB showed promising performance when integrated in iterative formulations using both the control volume and the Galerkin-type restriction operators. We also propose a new approach to enhance the performance of MsRSB for MPFA discretized systems, particularly targeting the construction of the prolongation operator. Results show the potential of our approach in terms of accuracy of the computed basis functions and the overall convergence behavior of the multiscale solver while ensuring a monotone solution at all times.

A new family of aggregation functions for intervals

Aggregation operators are unvaluable tools when different pieces of information have to be taken into account with respect to the same object. They allow to obtain a unique outcome when different evaluations are available for the same element/object. In this contribution we assume that the opinions are not given in form of isolated values, but intervals. We depart from two “classical” aggregation functions and define a new operator for aggregating intervals based on the two original operators. We study under what circumstances this new function is well defined and we provide a general characterization for monotonicity. We also study the behaviour of this operator when the departing functions are the most common aggregation operators. We also provide an illustrative example demonstrating the practical application of the theoretical contribution to ensemble deep learning models.

Instanton effects in twist-3 generalized parton distributions

The instanton vacuum picture is used to study hadronic matrix elements of the twist-3 (dimension-4, spin-1) QCD operators measuring the quark spin density and spin-orbit correlations. The QCD operators are converted to effective operators in the low-energy effective theory emerging after chiral symmetry breaking, in a systematic approach based on the diluteness of the instanton medium and the 1/Nc expansion. The instanton fields induce spin-flavor-dependent “potential” terms in the effective operators, complementing the “kinetic” terms from the quark field momenta. As a result, the effective operators obey the same equation-of-motion relations as the original QCD operators. The spin-orbit correlations are qualitatively different from naive quark model expectations.

UAV-based aeromagnetic surveys for orphaned well location: Emerging best practices

Historical oil and gas exploration and development in the northeastern United States left behind hundreds of thousands of orphaned oil and gas wells in areas where hydrocarbon extraction predated state and federal regulations on well abandonment. Abandoned by their original operators with incomplete records of location and well status, orphaned wells pose significant economic and environmental challenges associated with possible emissions of formation liquids and gases. Effective location and assessment of orphaned wells is a key first step in well site remediation; however, the task of locating orphaned wells has proven to be a formidable challenge in some of the most impacted areas. Specifically, in key areas of the northeastern United States, where the oil and gas industry first emerged in the late 19th and early 20th centuries, historical lease maps have proven largely inaccurate, and wide-area terrestrial geophysical surveys in difficult terrain characteristics and dense vegetation were found to be cost-prohibitive. In recent years, integration of miniaturized magnetic sensors with low-altitude uncrewed aerial vehicles (UAVs) facilitated successful implementation of wide-area aeromagnetic surveys targeting orphaned well identification in areas where piloted aeromagnetic surveys were previously infeasible due to challenging terrain and canopy conditions. Aeromagnetic data sets collected by ultra-low-altitude UAV surveys and processed to highlight magnetic anomalies associated with orphaned well casings have the potential to drastically reduce search areas for orphaned wells and complement ground surveys, thus helping turn the tide in orphaned well remediation efforts. As UAV-based aeromagnetic surveys gain recognition as a dependable wide-area surveying tool for remote identification of orphaned well sites, we highlight key advantages and important limitations of this methodology, underscore key practical challenges to successful implementation, and outline best practices in effective survey design and execution.

Projection-Grid Schemes on Irregular Grids for a Parabolic Equation

A family of projection-grid schemes has been constructed for approximating parabolic equations with a variable diffusion coefficient in tensor form. The schemes are conservative and retain the self-adjointness of the original differential operator and are destined for calculations on 3D irregular difference grids, including tetrahedral, mixed (grids of arbitrary polyhedra), and locally adaptive (octal-tree type).

Fully reliable iteration method based on simplification of the original operator

AbstractWe consider an iteration method for solving the problem with a complicated positive definite operator A by means of a simpler operator A0 (which inversion is much cheaper than inversion of A). The difference between A and A0 is controlled by some parameter. We prove contraction of the iteration operator and deduce fully computable two– sided a posteriori estimates.

Circuit-Depth Reduction of Unitary-Coupled-Cluster Ansatz by Energy Sorting.

Quantum computation represents a revolutionary approach to solving problems in quantum chemistry. However, due to the limited quantum resources in the current noisy intermediate-scale quantum (NISQ) devices, quantum algorithms for large chemical systems remain a major task. In this work, we demonstrate that the circuit depth of the unitary coupled cluster (UCC) and UCC-based ansatzes in the algorithm of the variational quantum eigensolver can be significantly reduced by an energy-sorting strategy. Specifically, subsets of excitation operators are first prescreened from the operator pool according to its contribution to the total energy. The quantum circuit ansatz is then iteratively constructed until convergence of the final energy to a typical accuracy. For demonstration, this method has been successfully applied to molecular and periodic systems. Particularly, a reduction of 50%-98% in the number of operators is observed while retaining the accuracy of the original UCCSD operator pools. This method can be straightforwardly extended to general parametric variational ansatzes.

Numerical simulations of pure quasi-P-waves in orthorhombic anisotropic media

The accurate simulation of anisotropic media is critical in seismic imaging and inversion. In recent years, some scholars have dedicated efforts to the study of precise elastic waves in anisotropic media; however, it is easy to separate P-wave and S-wave from elastic wave fields in isotropic media but difficult to separate them in anisotropic media. To address this issue, others have proposed pseudo-pure-wave equations based on the theory of wave-mode separation, but shear wave interference still exists. Therefore, we derived the first-order pure quasi-P-wave equation with no shear wave component in orthorhombic anisotropic media (ORT) which is common in the Earth’s crust and has very important research value. The presence of a pseudodifferential operator in the equation poses a challenge for solving. In order to solve the pure wave equation, we decomposed the original pseudodifferential operator into an elliptic differential operator and a scalar operator, both of which are easily solvable. In addition, we extended the equation from ORT media to tilted ORT (TORT) media. The example results indicate that our pure quasi-P-wave equation can yield a more stable and accurate P-wave field. The pure wave equation we propose can be applied in reverse time migration (RTM), the least squares RTM (LSRTM), and even the full waveform inversion (FWI).

High-fidelity realization of the AKLT state on a NISQ-era quantum processor

The AKLT state is the ground state of an isotropic quantum Heisenberg spin-1 model. It exhibits an excitation gap and an exponentially decaying correlation function, with fractionalized excitations at its boundaries. So far, the one-dimensional AKLT model has only been experimentally realized with trapped-ions as well as photonic systems. In this work, we successfully prepared the AKLT state on a noisy intermediate-scale quantum (NISQ) era quantum device. In particular, we developed a non-deterministic algorithm on the IBM quantum processor, where the non-unitary operator necessary for the AKLT state preparation is embedded in a unitary operator with an additional ancilla qubit for each pair of auxiliary spin-1/2’s. Such a unitary operator is effectively represented by a parametrized circuit composed of single-qubit and nearest-neighbor CX gates. Compared with the conventional operator decomposition method from Qiskit, our approach results in a much shallower circuit depth with only nearest-neighbor gates, while maintaining a fidelity in excess of 99.99% with the original operator. By simultaneously post-selecting each ancilla qubit such that it belongs to the subspace of spin-up |↑>, an AKLT state can be systematically obtained by evolving from an initial trivial product state of singlets plus ancilla qubits in spin-up on a quantum computer, and it is subsequently recorded by performing measurements on all the other physical qubits. We show how the accuracy of our implementation can be further improved on the IBM quantum processor with readout error mitigation.

Krasnoselskij-type algorithms for fixed point problems and variational inequality problems in Banach spaces

Existence and uniqueness as well as the iterative approximation of fixed points of enriched almost contractions in Banach spaces are studied. The obtained results are generalizations of the great majority of metric fixed point theorems, in the setting of a Banach space. The main tool used in the investigations is to work with the averaged operator Tλ instead of the original operator T. The effectiveness of the new results thus derived is illustrated by appropriate examples. An application of the strong convergence theorems to solving a variational inequality is also presented.