Numerical Minimization Research Articles

Nonadditivity in Many-Body Interactions between Membrane-Deforming Spheres Increases Disorder.

Membrane-induced interactions play an important role in organizing membrane proteins. Measurements of the interactions between two and three membrane deforming objects have revealed their nonadditive nature. They are thought to lead to complex many-body effects, however, experimental evidence is lacking. We here present an experimental method to measure many-body effects in membrane-mediated interactions using colloidal spheres placed between a deflated giant unilamellar vesicles and a planar substrate. The confined colloidal particles cause a large deformation of the membrane while not being physicochemically attached to it and interact through it. Two particles attract with a maximum force of 0.2 pN. For three particles, compact equilateral triangles were preferred over linear arrangements. We use numerical energy minimization to establish that the attraction stems from a reduction in the membrane-deformation energy caused by the particles. Confining up to 36 particles, we find a preference for hexagonally close packed clusters. However, with increasing number of particles the order of the confined particles decreases, at the same time, diffusivity of the particles increases. Our experiments show that the nonadditive nature of membrane-mediated interactions affects the interactions and arrangements and ultimately leads to spherical aggregates with liquid-like order of potential importance for cellular processes.

Frequency-Dependent Fatigue Properties of Additively Manufactured PLA.

Vibration-fatigue failure occurs when a structure is dynamically excited within its natural frequency range. Unlike metals, which have constant fatigue parameters, polymers can exhibit frequency-dependent fatigue parameters, significantly affecting the vibration resilience of 3D-printed polymer structures. This manuscript presents a study utilizing a novel vibration-fatigue testing methodology to characterize the frequency dependence of polymer material fatigue parameters under constant temperature conditions. In this investigation, 3D-printed PLA samples with frequency-tunable geometry were experimentally tested on an electro-dynamical shaker with a random vibration profile. Using the validated numerical models, the estimation of vibration-fatigue life was obtained and compared to the experimental results. Performing the numerical minimization of estimated and actual fatigue lives, the frequency-dependent fatigue parameters were assessed. In particular, the results indicate that the tested samples exhibit varying fatigue parameters within the loading frequency range of 250-750 Hz. Specifically, as the loading frequency increases, the fatigue exponent increases and fatigue strength decreases. These findings confirm the frequency dependence of fatigue parameters for 3D-printed polymer structures, underscoring the necessity of experimental characterization to reliably estimate the vibration-fatigue life of 3D-printed polymer structures. The utilization of the introduced approach therefore enhances the vibration resilience of the 3D-printed polymer mechanical component.

Generalized Approach to Optimal Polylinearization for Smart Sensors and Internet of Things Devices

This study introduces an innovative numerical approach for polylinear approximation (polylinearization) of non-self-intersecting compact sensor characteristics (transfer functions) specified either pointwise or analytically. The goal is to partition the sensor characteristic optimally, i.e., to select the vertices of the approximating polyline (approximant) along with their positions, on the sensor characteristics so that the distance (i.e., the separation) between the approximant and the characteristic is rendered below a certain problem-specific tolerance. To achieve this goal, two alternative nonlinear optimization problems are solved, which differ in the adopted quantitative measure of the separation between the transfer function and the approximant. In the first problem, which relates to absolutely integrable sensor characteristics (their energy is not necessarily finite, but they can be represented in terms of convergent Fourier series), the polylinearization is constructed by the numerical minimization of the L1-metric (a distance-based separation measure), concerning the number of polyline vertices and their locations. In the second problem, which covers the quadratically integrable sensor characteristics (whose energy is finite, but they do not necessarily admit a representation in terms of convergent Fourier series), the polylinearization is constructed by numerically minimizing the L2-metric (area- or energy-based separation measure) for the same set of optimization variables—the locations and the number of polyline vertices.

Calculating Global Minimum Points to Binary Polynomial Optimization Problem: Optimizing the Optimal PMU Localization Problem as a Case-Study

State estimation (SE) is an algorithmic function of an energy management system (EMS). SE provides an actual-time monitoring and control of modern electrical power grids. State Estimation can be worked with sufficiency using Phasor Measurement Units optimally placed within a power grid. This paper concerns the implementation of proper algorithms embedded in optimization solvers to the optimal PMU localization problem solving globally. The optimization model is formulated as a 0 - 1 nonlinear minimization problem. The problem is transformed to a polyhedron using linearization methods and B&B tree. In this model, we use a linear cost function under polynomial constraints and binary restrictions on the design variables in a symbolic format. This mathematical model is programmed in the YALMIP environment which is fully compatible with MATLAB. The 0 - 1 Nonlinear Programming (NLP) model is suitable for getting concisely global optimal solutions. The optimal solution is given by a wrapped optimization engine including a local optimizer routine performing together with a mixed-Integer-Linear Programming routine. The solution is achieved within a zero-gap precisely encountered during the iterative process. This tolerance criterion is a necessity for a successful implementation of the B&B tree because it ensures global optimality with an acceptance relative gap. The minimization model is implemented in a YALMIP code fully compatible with MATLAB in two stages. Initially, an objective function with one term is minimized to discover a number of sensors for wide-area monitoring, control and state estimator applications. Then, an extra product is considered in the objective to suffice maximum reliability for observing the network buses. The numerical minimization models are applied to standard power networks in the direction to be solved globally.

Collective variable model for the dynamics of liquid crystal skyrmions

Liquid crystal skyrmions are topologically protected spatially-localized distortions of the director field which exhibit particle-like properties including translational motion in oscillating electric fields. Here, we develop a collective variable model of the skyrmion dynamics, extending the approach of Long and Selinger proposed earlier for one dimensional systems. The model relates the skyrmion motion to a complex dynamics of the width of the twist wall around the skyrmion core. The width evolves in a non-reciprocal way, quantifying squirming deformations of the high twist region within on and off states of the field. We analyze in details the average skyrmion velocity as a function of the frequency and strength of the field as well as its duty cycle. The model predictions agrees qualitatively with experiments and results of numerical minimization of the Frank-Oseen model. Our results provide insights into the conditions necessary to observe velocity reversal as a function of the field parameters.

Конструирование гладких выпуклых продолжений булевых функций

Systems of Boolean equations are widely used in mathematics, computer science, and applied sciences. In this regard, on the one hand, new research methods and algorithms are being developed for such systems, and on the other hand, existing methods and algorithms for solving such systems are being improved. One of these methods is that, firstly, the system of Boolean equations given over the ring of Boolean polynomials is transformed into a system of equations over the field of real numbers, and secondly, the transformed system is reduced either to the problem of numerical minimization of the corresponding objective function, to a MILP or QUBO problem, to a system of polynomial equations solved on the set of integers, or to an equivalent system of polynomial equations solved by symbolic methods. There are many ways to transform a system of Boolean equations into a continuous minimization problem, since the fundamental difference between such methods and “brute force” local search algorithms is that at each iteration of the algorithm, the shift along the antigradient is performed on all variables simultaneously. But one of the main problems that arise when applying these methods is that the objective function to be minimized in the desired area can have many local minima, which greatly complicates their practical use. In this paper, a non-negative convex and continuously differentiable extension of any Boolean function is constructed, which is applied to solving an arbitrary system of Boolean equations. It is argued that the problem of solving an arbitrary system of Boolean equations can be constructively reduced to the problem of minimizing a function, any local minimum of which in the desired domain is a global minimum.

Numerical minimization of neoclassical poloidal viscosity for supersonic equilibria in tokamak geometry

Extensive experimental evidence has shown that the presence of poloidal flow in tokamaks can dramatically improve transport properties. However, theory indicates that poloidal flows are damped by poloidal viscosity, thus necessitating external drivers, such as neutral beam injection or radio frequency heating. In this work, ideal magnetohydrodynamic equilibria are calculated via the FORTRAN code FLOW [Guazzotto et al., Phys. Plasmas 11, 604 (2004)] and a postprocessor is used to estimate the neoclassical poloidal viscosity. The equilibrium inputs, which correspond to intuitive physical quantities, are then numerically optimized to reduce a viscosity figure of merit. We present supersonic equilibria in tokamak geometry with minimized neoclassical poloidal viscosities for various velocity free function inputs, plasma aspect ratios, and collisionality regimes. Benchmarks are made against an analytic theory as well as a classical expression of poloidal viscosity. Numerical confirmation of the analytic theory is obtained in the high aspect ratio and high collisionality limit. Good agreement is also seen near the plasma core and edge, with discrepancies arising in the intermediate region. Outside of these limits, rotation input function profiles are found that provide ∼order of magnitude improvements over the analytic theory, with additional progress being made toward predictions for tokamak-relevant equilibria.

Efficient Heart Disease Prediction Using PSO Optimized LSTM Classifier

Accurate diagnosis and prognosis of cardiovascular disease are essential in the medical field, enabling cardiologists to administer appropriate treatments. Machine learning has gained prominence in the healthcare sector due to its ability to identify patterns in data. Leveraging machine learning for classifying cardiovascular disease occurrences, significantly minimize misdiagnosis. This research focuses on developing a robust model capable of accurately predicting cardiovascular diseases, ultimately aiming to decrease the mortality rates associated with these conditions. To find clusters of data objects in a dataset, K-means clustering algorithm is implemented. The technique known as particle swarm optimization (PSO) is used to estimate solutions to extremely challenging or unsolvable numerical minimization and maximization problems. LSTM classifier, a deep learning technology is employed to categorize the attacks under various scenarios. In addition to greatly enhancing the healthcare system, this research gives medical practitioners access to an invaluable diagnostic tool. The outcomes of proposed work is examined using programs written in Python Jupyter software tool.

On the numerical minimisation of the objective function applied to spherical harmonics fitting

On the numerical minimisation of the objective function applied to spherical harmonics fitting

Retrieving space-dependent polarization transformations via near-optimal quantum process tomography.

An optical waveplate rotating light polarization can be modeled as a single-qubit unitary operator. This analogy can be exploited to experimentally retrieve a polarization transformation within the paradigm of quantum process tomography. Standard approaches to tomographic problems rely on the maximum-likelihood estimation, providing the most likely transformation to yield the same outcomes as a set of experimental projective measurements. The performances of this method strongly depend on the number of input measurements and the numerical minimization routine that is adopted. Here we investigate the application of genetic and machine learning approaches to this problem, finding that both allow for accurate reconstructions and fast operations when processing a set of projective measurements very close to the minimal one. We apply these techniques to the case of space-dependent polarization transformations, providing an experimental characterization of the optical action of spin-orbit metasurfaces having patterned birefringence. Our efforts thus expand the toolbox of methodologies for optical process tomography. In particular, we find that the neural network-based scheme provides a significant speed-up, that may be critical in applications requiring a characterization in real-time. We expect these results to lay the groundwork for the optimization of tomographic approaches in more general quantum processes, including non-unitary gates and operations in higher-dimensional Hilbert spaces.

De-wetting of evaporating drops on regular patterns of triangular posts.

Directional wicking and spreading of liquids can be achieved by regular micro-patterns of specifically designed topographic features that break the reflection symmetry of the underlying pattern. The present study aims to understand the formation and stability of wetting films during the evaporation of volatile liquid drops on surfaces with a micro-pattern of triangular posts arranged in a rectangular lattice. Depending on the density and aspect ratio of the posts, we observe either spherical-cap shaped drops with a mobile three-phase contact line or the formation of circular or angular drops with a pinned three-phase contact line. Drops of the latter class eventually evolve into a liquid film extending to the initial footprint of the drop and a shrinking cap-shaped drop sitting on the film. The drop evolution is controlled by the density and aspect ratio of the posts, while no influence of the orientation of the triangular posts on the contact line mobility becomes evident. Our experiments corroborate previous results of systematic numerical energy minimization, predicting that conditions for a spontaneous retraction of a wicking liquid film depend weakly on the orientation of the film edge relative to the micro-pattern.

A complete characterization of minima of the spectral abscissa and rightmost roots of second-order systems with input delay

Abstract The numerical minimization of the spectral abscissa function of linear time-invariant time-delay systems, an established approach to compute stabilizing controllers with a fixed structure, often gives rise to minima characterized by active characteristic roots with multiplicity higher than one. At the same time, recent theoretical results reveal situations where the so-called multiplicity induced dominancy property holds, i.e., a sufficiently high multiplicity implies that the root is dominant. Using an integrative approach, combining analytical characterizations, computation of characteristic roots and numerical optimization, a complete characterization of the stabilizability of second-order systems with input delays is provided, for both state feedback and delayed output feedback. The level sets of the minimal achievable spectral abscissa are also characterized. These results shed light on the complex relations between (configurations involving) multiple roots, the property of being dominant roots and the property of corresponding to (local/global) minimizers of the spectral abscissa function.

High-Efficient Nonlinear Control for Brushless Doubly-Fed Induction Machines

The brushless doubly-fed induction machine (BDFIM) is suitable for applications that need adjustable speed within limited ranges. It is a promising machine if its control strategies, such as the maximum torque per total copper losses (MTPCL) control, are well designed. A BDFIM has two stator windings, a power winding (not controllable) and a control winding (CW) (controllable through a partially rated back-to-back converter). Unlike singly-fed electrical machines, which are fully controllable and the MTPCL can be easily implemented to them, implementing the MTPCL control in a BDFIM is challenging. In this paper, the model-based MTPCL control strategy in the presence of a nonlinear controller in the BDFIM drive is developed. To realize this strategy, we first determine an expression in terms of the angle of the CW's current as a controllable variable. The optimal angle of the CW's current, which guarantees the realization of the MTPCL strategy, is then mathematically obtained using the numerical minimization approach. Next, the proposed passivity-based nonlinear controller regulates both the MTPCL criterion and the electromagnetic torque directly as the output variables. The proposed strategy is validated by a TMS320F2833 microcontroller synchronized with a personal computer for a 3 kW prototype D132s-BDFIM.

Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of mathbb {R}^d. For restrictions to the Euclidean ball in odd dimensions, to the rotation group textrm{SO}(3), and to the Grassmannian manifold mathcal {G}_{2,4}, we compute the kernels’ Fourier coefficients and determine their asymptotics. The L_2-discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For textrm{SO}(3), the nonequispaced fast Fourier transform is publicly available, and, for mathcal {G}_{2,4}, the transform is derived here. We also provide numerical experiments for textrm{SO}(3) and mathcal {G}_{2,4}.

A continuous constraint satisfaction problem for the rigidity transition in confluent tissues

Models of confluent tissues are built out of tessellations of the space (both in two and three dimensions) in which the cost function is constructed in such a way that individual cells try to optimize their volume and surface in order to reach a target shape. At zero temperature, many of these models exhibit a rigidity transition that separates two phases: a liquid phase and a solid (glassy) phase. This phenomenology is now well established but the theoretical understanding is still not complete. In this work we consider an exactly soluble mean field model for the rigidity transition which is based on an abstract mapping. We replace volume and surface functions by random non-linear functions of a large number of degrees of freedom forced to be on a compact phase space. We then seek for a configuration of the degrees of freedom such that these random non-linear functions all attain the same value. This target value is a control parameter and plays the role of the target cell shape in biological tissue models. Therefore we map the microscopic models of cells to a random continuous constraint satisfaction problem with equality constraints. We argue that at zero temperature, the rigidity transition corresponds to the satisfiability transition of the problem. We also characterize both the satisfiable (SAT) and unsatisfiable (UNSAT) phase. In the SAT phase, before reaching the rigidity transition, the zero temperature SAT landscape undergoes an replica symmetry breaking (RSB)/ergodicity breaking transition of the same type as the Gardner transition in amorphous solids. By solving the RSB equations we compute the SAT/UNSAT threshold and the critical behavior around it. In the UNSAT phase we also compute the average shape index as a function of the target one and we compare the thermodynamical solution of the model with the results of the numerical greedy minimization of the corresponding cost function.

Two-stage game-theoretic model of inspection and embezzlement in a three-level control system

The subhierarchical two-stage game-theoretic model of embezzlement and bribery is designed, the particular example based on the scheme with six officials on three levels is solved. Theoretic conditions and numerical corruption minimization settings are suggested. The cooperative extension is proposed and the computer simulation is carried out for two different approaches to the isolated components, the results analyzed for stability.

Effect of weld geometry on the limit load of a cracked specimen under tension

The accuracy of the defect assessment methods strongly depends on the accuracy of limit load solutions. Usually, many parameters classify welded structures. Therefore, parametric analysis of such structures may be time-consuming. The upper bound theorem supplies an efficient method for reducing lengthy calculations. The mathematical features of this theorem allow for accurate estimations of the limit load to be found using rather simple kinematically admissible velocity fields. In the case of highly undermatched welded joints, the known singular velocity behavior near the bi-material interface can be taken into account to choose a kinematically admissible velocity field satisfying some features of the real velocity field. The present article provides such a solution for V-bevel butt welds containing a crack of quite an arbitrary shape. The welded joint is subject to tension. The solution consists of two steps. The first step determines the limit load of the specimen with no crack. This step requires the numerical minimization of a function of one variable. The second step accounts for the crack, assuming that the solution for the uncracked specimen is known. This step is purely analytical. The accuracy of the solution is confirmed by comparison with a solution for a particular geometry of the weld found using Riemann’s method. The effect of the weld geometry on the limit load is revealed.

Local magnetic moments due to loop currents in metals

We present Hartree-Fock calculations on a simple model to obtain the conditions of formation of local magnetic moments due to loop currents ${L}_{o}$ and spin-loop currents ${L}_{s}$ and compare them to the conditions of formation of local spin moments $M$ which were given long ago in a similar approximation by Anderson. A model with three degenerate orbitals sitting on an equilateral triangle, with on-site and nearest-neighbor repulsions $U$ and $V$, respectively, and intersite kinetic energy, hybridizing with conduction electrons with a parameter $\mathrm{\ensuremath{\Delta}}$ is investigated. ${L}_{o}$ and ${L}_{s}$ are promoted by large $V/\mathrm{\ensuremath{\Delta}}$ and their magnitude is relatively unaffected by $U/\mathrm{\ensuremath{\Delta}}$. Spin-magnetic moments $M$ promoted by large $U/\mathrm{\ensuremath{\Delta}}$ on the other hand are adversely affected by $V/\mathrm{\ensuremath{\Delta}}$. In this model, ${L}_{o}$ for $V$ multiplied by the number of neighbors is approximately the same as the $M$ promoted by $U$ in Anderson's local model for $M$. ${L}_{o}$ and ${L}_{s}$ are degenerate if exchange interactions and Hund's rule are neglected but ${L}_{o}$ is favored when they are included. Many of the qualitative results are visible in an expression for the Hartree-Fock ground state energy derived as a function of small ${L}_{o}, {L}_{s}$, and $M$. Numerical minimization of the Hartree-Fock energy is presented for larger values. We also briefly discuss the connection and differences of the interaction-generated orbital currents and spin currents discussed here and generalized to a lattice with the topological states in metals and semiconductors.

A Neural Network Based Approach to Inverse Kinematics Problem for General Six-Axis Robots.

Inverse kinematics problems (IKP) are ubiquitous in robotics for improved robot control in widespread applications. However, the high non-linearity, complexity, and equation coupling of a general six-axis robotic manipulator pose substantial challenges in solving the IKP precisely and efficiently. To address this issue, we propose a novel approach based on neural network (NN) with numerical error minimization in this paper. Within our framework, the complexity of IKP is first simplified by a strategy called joint space segmentation, with respective training data generated by forward kinematics. Afterwards, a set of multilayer perception networks (MLP) are established to learn from the foregoing data in order to fit the goal function piecewise. To reduce the computational cost of the inference process, a set of classification models is trained to determine the appropriate forgoing MLPs for predictions given a specific input. After the initial solution is sought, being improved with a prediction error minimized, the refined solution is finally achieved. The proposed methodology is validated via simulations on Xarm6-a general 6 degrees of freedom manipulator. Results further verify the feasibility of NN for IKP in general cases, even with a high-precision requirement. The proposed algorithm has showcased enhanced efficiency and accuracy compared to NN-based approaches reported in the literature.

GMM Model Averaging Using Higher Order Approximations

Moment conditions model averaging (MA) estimators in the GMM framework are considered. Under finite sample considerations, MA estimators with optimal weights are proposed, in the sense that weights minimize the corresponding higher-order asymptotic mean squared error (AMSE). It is shown that the higher-order AMSE objective function has a closed-form expression, which makes this procedure applicable in practice. In addition, and as an alternative, different averaging schemes based on moment selection criteria are considered, in which weights for averaging across GMM estimates can be obtained by direct smoothing or by numerical minimization of a specific criterion. Asymptotic properties assuming correctly specified models are derived and the performance of the proposed averaging approaches is contrasted with existing model selection alternatives i) analytically, for a simple IV example, and ii) by means of Monte Carlo experiments in a nonlinear setting, showing that MA compares favourably in many relevant setups. The usefulness of MA methods is illustrated by studying the effect of institutions on economic performance.