Good Approximation Research Articles

Iterative quantum algorithm for combinatorial optimization based on quantum gradient descent

Combinatorial optimization has wide and high-value applications in many fields of science and industry, but solving general combinatorial optimization problems is non-deterministic polynomial time (NP) hard. Many such problems can be mapped onto the ground-state-search problems of the Ising model. Here, an iterative quantum algorithm based on quantum gradient descent to solve combinatorial optimization problems is introduced, where the initial state of a quantum register evolves over several iterations to a good approximation of the Ising-Hamiltonian ground state. We verified the effectiveness of the proposed algorithm in solving the MaxCut problem for different types of undirected graphs by numerical simulations, and analyzed the robustness of the algorithm to errors by simulating random error and Gaussian error. We compared the performance of the algorithm with the quantum approximate optimization algorithm, and the results indicate that the proposed algorithm has comparable convergence performance. We also verified the feasibility of the algorithm by conducting experiments on a real quantum computer through the quantum cloud platform. Our work provides a potential method for solving combinatorial optimization problems on future quantum devices without the use of complex classical optimization loops.

The multiobjective traveling salesman–repairman problem with profits: design and implementation of a variable neighborhood descent algorithm for a real scenario

AbstractThis paper introduces a problem that can be seen as a combination of the traveling salesman problem with profits and the traveling repairman problem with profits, coined as the multi‐objective traveling salesman–repairman problem with profits (Mo‐TSRPP). The objective of the Mo‐TSRPP is to simultaneously optimize three objectives: the total cost, total latency, and total profit. Indirectly, the number of nodes visited is also considered although not as an objective itself since it is determined by the size of every efficient solution in the Pareto front. The Mo‐TSRPP emerges as a real‐world problem in which a freelancer, which repairs appliances, wants to plan the daily route. Moreover, the daily plan does not require to visit all customers. To solve the problem, first, a greedy randomized adaptive procedure is designed to generate a set of high‐quality nondominated solutions and then, a variable neighborhood descent algorithm is applied for further improving the initial set. This procedure allows us to attain a good approximation of the Pareto front. To prove the performance of the proposal a comparison is done against three well‐known evolutionary algorithms: NSGA‐II, SPEA‐2, and MOEA/D. Finally, a realistic problem is shown and solved to illustrate the potential of the algorithm.

Quantum mechanical Gaussian wavepackets of single relativistic particles

We study the evolutions of selected quasi-(1+1) dimensional wavepacket solutions to the Klein–Gordon equation for a relativistic charged particle in uniform motion or accelerated by a uniform electric field in Minkowski space. We explore how good the charge density of a Klein–Gordon wavepacket can be approximately described by a Gaussian state with the single-particle interpretation. We find that the minimal initial width of a wavepacket for a good Gaussian approximation in position space is about the Compton wavelength of the particle divided by its Lorentz factor at the initial moment. This can be considered as a manifestation of relativistic length contraction, which is also observed numerically in the spreading of the wavepacket’s charge density.

Spacecraft Materials’ Reflectivity and Surface Morphology: Aging Caused by Proton Irradiation

The radiation environment in Low Earth Orbit (LEO) is dominated by protons captured by Earth’s magnetic field in the Inner Van-Allen belt. Defunct satellites and other space debris objects can be resident in this environment for several decades and even centuries. So far, there is little knowledge about the impact of long-duration proton exposure to the surface morphology and reflectivity in LEO environment. We report on a laboratory test campaign exposing typical spacecraft materials with protons of 100 keV and 2.5 keV kinetic energy and a fluence corresponding to an in-orbit duration of 100 years and 120 years, respectively, in an 800 km sun-synchronous orbit. Although we find microscopic changes in surface morphology, reflectivity changes of all tested materials were smaller than 15%. This result brings positive news for on-going efforts to use optical methods, e.g. lightcurve measurements or active polarimetry, for characterizing space objects, since it suggests that data can - to a good approximation - be analyzed without accounting for proton induced aging effects that might affect the materials’ optical properties over time.

The Hodge decomposition of shell current on the Keda Torus eXperiment device

The Hodge decomposition is a valuable tool for uniquely decomposing total currents on the composite shell into three types: inductive current, halo current, and harmonic current, each with its specific physical meaning. During plasma disruptions, halo currents appear, essential for studying the wall’s thermal load and electromagnetic force. Furthermore, understanding halo currents is crucial for improving the existing methodologies by removing their effects on equilibrium reconstructions and instability analyses based on boundary magnetic probe data. On the Keda Torus eXperiment (KTX) device, radial and tangent halo currents can be simultaneously provided to locate the contact region during a minor disruption experimentally. Additionally, experimental results demonstrate that, in addition to the occurrence of halo current during minor disruption events, halo current is already present simultaneously with the generation of inductive current when a resistive wall mode exists. For devices that lack the capability to measure the two-dimensional shell current distribution on the entire shell, we propose a method to estimate inductive and halo currents only using a set of shell currents along the toroidal direction. This technique is demonstrated on the KTX device and provides an overall good approximation of the inductive and halo current distribution.

Combining contour and region for closed boundary extraction of a shape.

This study explored human ability to extract closed boundary of a target shape in the presence of noise using spatially global operations. Specifically, we investigated the contributions of contour-based processing using line edges and region-based processing using color, as well as their interaction. Performance of the subjects was reliable when the fixation was inside the shape, and it was much less reliable when the fixation was outside. With fixation inside the shape, performance was higher when both contour and color information were present compared to when only one of them was present. We propose a biologically-inspired model to emulate human boundary extraction. The model solves the shortest (least-cost) path in the log-polar representation, a representation which is a good approximation to the mapping from the retina to the visual cortex. Boundary extraction was framed as a global optimization problem with the costs of connections calculated using four features: distance of interpolation, turning angle, color similarity and color contrast. This model was tested on some of the conditions that were used in the psychophysical experiment and its performance was similar to the performance of subjects.

Entropy is a good approximation to the electronic (static) correlation energy

For an electronic system, given a mean field method and a distribution of orbital occupation numbers that are close to the natural occupations of the correlated system, we provide formal evidence and computational support to the hypothesis that the entropy (or more precisely −σS, where σ is a parameter and S is the entropy) of such a distribution is a good approximation to the correlation energy. Underpinning the formal evidence are mild assumptions: the correlation energy is strictly a functional of the occupation numbers, and the occupation numbers derive from an invertible distribution. Computational support centers around employing different mean field methods and occupation number distributions (Fermi–Dirac, Gaussian, and linear), for which our claims are verified for a series of pilot calculations involving bond breaking and chemical reactions. This work establishes a formal footing for those methods employing entropy as a measure of electronic correlation energy (e.g., i-DMFT [Wang and Baerends, Phys. Rev. Lett. 128, 013001 (2022)] and TAO-DFT [J.-D. Chai, J. Chem. Phys. 136, 154104 (2012)]) and sets the stage for the widespread use of entropy functionals for approximating the (static) electronic correlation.

Parametric finite element approximations for anisotropic surface diffusion with axisymmetric geometry

We consider parametric finite element approximations for the anisotropic surface diffusion flow in an axisymmetric setting. Based on the anisotropy function, we introduce a symmetric positive definite matrix with a suitable stabilizing function. This then gives rise to two novel weak formulations for the axisymmetric flow in terms of the generating curve of the interface. By using piecewise linear elements in space and backward Euler in time, we discretize the weak formulations to obtain different approximating methods. These include a linear approximation which has good mesh properties and nonlinear approximations which can be shown rigorously to satisfy the volume preservation or the unconditional energy stability on the discrete level. Extensive numerical results are reported to demonstrate the accuracy and efficiency of the introduced methods for computing the axisymmetric flow.

Approximating Kolmogorov complexity

It is well known that the Kolmogorov complexity function (the minimal length of a program producing a given string, when an optimal programming language is used) is not computable and, moreover, does not have computable lower bounds. In this paper we investigate a more general question: can this function be approximated? By approximation we mean two things: firstly, some (small) difference between the values of the complexity function and its approximation is allowed; secondly, at some (rare) points the values of the approximating function may be arbitrary. For some values of the parameters such approximation is trivial (e.g., the length function is an approximation with error d except for a O ( 2 − d ) fraction of inputs). However, if we require a significantly better approximation, the approximation problem becomes hard, and we prove it in several settings. Firstly, we show that a finite table that provides good approximations for Kolmogorov complexities of n-bit strings, necessarily has high complexity. Secondly, we show that there is no good computable approximation for Kolmogorov complexity of all strings. In particular, Kolmogorov complexity function is neither generically nor coarsely computable, as well as its approximations, and the time-bounded Kolmogorov complexity (for any computable time bound) deviates significantly from the unbounded complexity function. We also prove hardness of Kolmogorov complexity approximation in another setting: the mass problem whose solutions are good approximations for Kolmogorov complexity function is above the halting problem in the Medvedev lattice. Finally, we mention some proof-theoretic counterparts of these results. A preliminary version of this paper was presented at CiE 2019 conference (In Computing with Foresight and Industry – 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings (2019) 230–239 Springer).

Applicability of the Klein-Gordon equationfor pair production in vacuum and plasma.

In this paper, a phase-space description of electron-positron pair creation will be applied, based on a Wigner transformation of the Klein-Gordon equation. The resulting theory is similar in many respects to the equationsfrom the Dirac-Heisenberg-Wigner formalism. However, in the former case, all physics related to particle spin is neglected. In the present paper we compare the pair-production rate in vacuum and plasmas, with and without spin effects, in order to evaluate the accuracy and applicability of the spinless approximation. It is found that for modest frequencies of the electromagnetic field the pair production rate of the Klein-Gordon theory is a good approximation to the Dirac theory, provided the matter density is small enough for Pauli blocking to be neglected, and a factor of 2 related to the difference in the vacuum energy density is compensated for.

Cramér–Lundberg asymptotics for spectrally positive Markov additive processes

This paper studies the Cramér–Lundberg asymptotics of the ruin probability for a model in which the reserve level process is described by a spectrally-positive light-tailed Markov additive process. By applying a change-of-measure technique in combination with elements from Wiener-Hopf theory, the exact asymptotics of the ruin probability are expressed in terms of the model primitives. In addition a simulation algorithm of generalized Siegmund type is presented, under which the returned estimate of the ruin probability has bounded relative error. Numerical experiments show that, compared to direct estimation, this algorithm greatly reduces the number of runs required to achieve an estimate with a given accuracy. The experiments also reveal that our asymptotic results provide a good approximation of the ruin probability even for relatively small initial surplus levels.

Win/Loss Data and Consumer Switching Costs: Measuring Diversion Ratios and the Impact of Mergers*

We propose an identification strategy for diversion based on win/loss data. First, we show that win/loss data from the merging firms and market shares in two periods for all firms are sufficient to identify the diversion ratios between the merging partners. Second, we show that win/loss data from the merging firms alone are sufficient for partial identification, and we construct a lower bound that provides a good approximation to the diversion ratio when switching costs are high. We demonstrate the performance of our method with numerical simulations and with an application to the Anthem/Cigna merger.

Numerical homogenization of spatial network models

We present and analyze a methodology for numerical homogenization of spatial networks models, e.g. heat conduction and linear deformation in large networks of slender objects, such as paper fibers. The aim is to construct a coarse model of the problem that maintains high accuracy also on the micro-scale. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data, at a low computational cost. We prove optimal convergence to the micro-scale solution of the proposed method under mild assumptions on the homogeneity, connectivity, and locality of the network on the coarse scale. The theoretical findings are numerically confirmed for both scalar-valued (heat conduction) and vector-valued (linear deformation) models.

A Practical Approach to Estimate the Min-Entropy in PUFs

AbstractHelper data algorithms reliably extract secrets from physical unclonable functions. The necessary helper data can leak information, though. One state-of-the-art approach to assess the remaining min-entropy is limited to homogeneous bias or correlation, not both. Another one extends this to only local bias without correlation but is limited to short code lengths. This work presents a new approach for determining the min-entropy based on convolving histograms. It provides a better bound and good approximation given arbitrary bias, more realistic correlation effects, and practically relevant code sizes. Experiments on real-world and synthetic data show the benefit of the new method compared with state-of-the-art ones. This work also facilitates a better understanding of how the error correction as post-processing impacts the min-entropy.

Privacy-preserving and lossless distributed estimation of high-dimensional generalized additive mixed models

Various privacy-preserving frameworks that respect the individual’s privacy in the analysis of data have been developed in recent years. However, available model classes such as simple statistics or generalized linear models lack the flexibility required for a good approximation of the underlying data-generating process in practice. In this paper, we propose an algorithm for a distributed, privacy-preserving, and lossless estimation of generalized additive mixed models (GAMM) using component-wise gradient boosting (CWB). Making use of CWB allows us to reframe the GAMM estimation as a distributed fitting of base learners using the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-loss. In order to account for the heterogeneity of different data location sites, we propose a distributed version of a row-wise tensor product that allows the computation of site-specific (smooth) effects. Our adaption of CWB preserves all the important properties of the original algorithm, such as an unbiased feature selection and the feasibility to fit models in high-dimensional feature spaces, and yields equivalent model estimates as CWB on pooled data. Next to a derivation of the equivalence of both algorithms, we also showcase the efficacy of our algorithm on a distributed heart disease data set and compare it with state-of-the-art methods.

Kinematic response of pile groups and piled rafts to a distant stationary or moving harmonic load via the FEM

Using the finite-element method (FEM), this study presents results on piled foundations excited by a stationary or moving harmonic vertical point load acting on the soil surface at a given distance. For the stationary loading case, typical configurations of pile groups and corresponding piled rafts are examined in the frequency domain. The contribution of the raft alone in the overall response is also explored. The soil is modelled as a linear-elastic continuum with hysteretic damping. A wide range of frequencies from 8 to 64 Hz is considered. Of interest is the resulting vertical oscillation mode. The influence of the number of pile rows in the direction of wave propagation on the resulting vibration reduction is assessed by monitoring a reference pile at the furthest back pile row and an observation point at the free-field behind. The associated wave-passage effect is quantified through appropriate transfer functions for the piles. Subsequently, the case of a harmonic load travelling with a constant speed parallel to the pile group is investigated. Most importantly, it is revealed that a good approximation for this case is a stationary load located on the moving load path at the shortest distance from the pile group. This has important implications in the vibration protection practice, as detailed modelling of moving loads is then not required. The methodology presented can be extended to arbitrary foundation geometries and layered soil profiles.

On the problem of the classical limit for statistical characteristics of a quantum Coulomb plasma

The general structure of the perturbation theory series for the thermodynamic potential of a system of electrons and point ions is discussed. In the classical limit, when Planck’s constant ℏ → 0, the particular terms of the perturbation theory series diverge due to the impossibility of maintaining of stability of a purely classical system of the opposite sign charges attracted to each other. At the same time, it is known that in a certain range of parameters, a thermodynamic potential that does not contain Planck’s constant is a good approximation. Such thermodynamic functions describes the experimental data well and corresponds to the molecular dynamics calculations of model quasi-Coulomb systems. It is shown that the transition to a purely classical description, which implies the limit ℏ → 0, is impossible, but the terms of the series containing ℏ are small for certain plasma parameters and can be discarded. This is the rationale explanation of using classical models and approximations to describe plasma in certain parameter ranges. On this basis, a hypothesis is formulated about the absence of a classical limit for all measurable physical properties (including kinetic characteristics) of the Coulomb model of plasma, in which point particles interact according to the Coulomb law and, therefore, there are no adjustable parameters.

A Novel Low-Cost, Open-Source, Three-Dimensionally Printed Thyroplasty Simulator

Training of surgical procedures on awake patients, such as medialization thyroplasty, poses challenges to educators and trainees. Three-dimensionally (3D)-printed simulators provide opportunity to practice in low-stakes settings. We present the first 3D-printed thyroplasty simulator incorporating a cartilaginous framework, endolaryngeal soft tissue housed in a 3D-printed manikin with endoscopic visualization. Male and female laryngeal cartilages and endolarynx molds were 3D printed from an existing open-source design. Cartilage models were made of heat-treated polylactic acid (HTPLA), a material chosen for its thermal stability, allowing drilling. They were combined with molded silicone endolarynges modeling glottic insufficiency. Larynges were set in a 3D-printed head-and-neck manikin with an attached borescope for internal visualization similar to distal chip laryngoscopy. Eight laryngologists evaluated the simulator by drilling a thyroplasty window, inserting an implant for medialization, and rating the model using a modified Michigan Standard Simulation Experience Scale (1=strongly disagree, 5=strongly agree). The model was well rated in educational value (mean 4.7, standard deviation [SD] 0.3), fidelity (mean 3.8, SD 0.2), and overall value (mean 4.8, SD 0.5). Qualitative assessment concluded the model was anatomically realistic and that HTPLA was a good approximation of the density and texture of thyroid cartilage. The materials for one larynx cost $4.09. This high-fidelity 3D-printed simulator demonstrates educational value for thyroplasty training. The low-cost, open-source design has broad implications for universal access to this simulator platform.

Excess of high-z galaxies as a test for bumpy power spectrum of density perturbations

ABSTRACT Modified matter power spectra with approximately Gaussian bump on sub-Mpc scales can be a result of a complex inflation. We consider five spectra with different Gaussian amplitudes A and locations k0 and run N-body simulations in a cube (5 Mpc h−1)3 at z > 8 to reveal the halo mass functions and their evolution with redshift. We have found that the Sheth–Tormen formula provides a good approximation to a such kind of halo mass functions. In the considered models, the dark matter halo formation starts much more earlier than in Lambda cold dark matter (ΛCDM), which in turn can result in an earlier star formation and a nuclear activity in galaxies and can be detected and tested by, e.g. JWST. At z = 0, the halo mass functions are hardly distinguishable from the standard ΛCDM, therefore the models with the bumpy spectra can be identified in observations by their excess in number of bright sources at high redshift only.

The Maximum Clique Problem and Integer Programming Models, Their Modifications, Complexity and Implementation

The maximum clique problem is a problem that takes many forms in optimization and related graph theory problems, and also has many applications. Because of its NP-completeness (nondeterministic polynomial time), the question arises of its solvability for larger instances. Instead of the traditional approaches based on the use of approximate or stochastic heuristic methods, we focus here on the use of integer programming models in the GAMS (General Algebraic Modelling System) environment, which is based on exact methods and sophisticated deterministic heuristics incorporated in it. We propose modifications of integer models, derive their time complexities and show their direct use in GAMS. GAMS makes it possible to find optimal solutions to the maximum clique problem for instances with hundreds of vertices and thousands of edges within minutes at most. For extremely large instances, good approximations of the optimum are given in a reasonable amount of time. A great advantage of this approach over all the mentioned algorithms is that even if GAMS does not find the best known solution within the chosen time limit, it displays its value at the end of the calculation as a reachable bound.