Minimization Of Function Research Articles

A second-order projection neurodynamic approach with exponential convergence for sparse signal reconstruction

Recently, a class of second-order neurodynamic approaches with convergence rates of O(1t) or O(1t2) has been developed to address the sparse signal reconstruction problem. In this paper, we propose a second-order projection neurodynamic approach (SOPNA) with exponential convergence to reconstruct a sparse signal by solving a modified inverted Gaussian function (MIGF) minimization problem. The existence, uniqueness, and feasibility of the solution to SOPNA are detailedly investigated, and the exponential convergence rate of O(exp(−μt)),μ>0 is proved. This implies that our proposed SOPNA can achieve a significantly superior convergence performance than several existing second-order neurodynamic approaches. Numerical experiments also confirm its effectiveness and superiority. Finally, the applications of the proposed SOPNA in real signal and real image reconstructions validate its practical feasibility.

The Role of Mathematics in Artificial Intelligence and Machine Learning

Mathematics serves as the foundational backbone of artificial intelligence (AI) and machine learning (ML), providing the essential tools and frameworks for developing sophisticated algorithms and models. the pivotal role of various mathematical disciplines, including linear algebra, calculus, probability theory, and optimization, in advancing AI and ML technologies. We begin by examining how linear algebra facilitates the manipulation and transformation of high-dimensional data, which is crucial for techniques such as principal component analysis (PCA) and singular value decomposition (SVD). Next, we delve into the applications of calculus in training neural networks through gradient-based optimization methods, highlighting the importance of differentiation and integration in backpropagation and loss function minimization. the role of probability theory in handling uncertainty and making predictions, emphasizing its application in Bayesian networks, Markov decision processes, and probabilistic graphical models. Additionally, we discuss optimization techniques, both convex and non-convex, that are fundamental to finding optimal solutions in machine learning tasks, including support vector machines (SVMs) and deep learning architectures.

Two quantum algorithms for solving the one-dimensional advection–diffusion equation

Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.

Practical one-shot data-driven design of fractional-order PID controller: Fictitious reference signal approach

This study proposes a one-shot data-driven tuning method for a fractional-order proportional-integral-derivative (FOPID) controller. The proposed method tunes the FOPID controller in the model-reference control formulation. A loss function is defined to evaluate the match between a given reference model and the closed-loop response while explicitly considering the closed-loop stability. A loss function value is based on the fictitious reference signal computed using the input/output data. Model matching is achieved via loss function minimization. The proposed method is simple and practical: it needs only one-shot input/output data of a plant (no plant model required), considers the bounded-input bounded-output stability of the closed-loop system from a bounded reference input to a bounded output, and automatically determines the appropriate parameter value via optimization. Numerical simulations show that the proposed approach facilitates good control performance, and destabilization can be avoided even if perfect model matching is unachievable.

Comparative study of PI-controller and neurocontroller performances in optimal by settling time control problems

When developing control systems, an important issue arises of choosing an operator, which forms the control function. The standard approach is to use a PI- or PID-controller, a more advanced approach involves ANNs training for this purpose. A comparative analysis of the PI- and neurocontroller performances makes it possible to establish the disadvantages and advantages of each of the compared controllers, which is an important scientific and applied problem. The purpose of the work was to conduct a comparative analysis of the performance of the PI-controller and the neurocontroller based on a set of evaluation indicators for plants of the second and third orders. Such a comparison was carried out by using an approach to the synthesis of both controllers, which involved the minimization of a complex objective function. The latter is obtained as a result of reducing the problem of optimal control with constraints to the problem of unconstrained optimization. The analysis showed that according to the settling time indicator (optimization criterion), the neurocontroller has an advantage of 6.1...96.2% for the modelled plants. At the same time, according to other indicators of the control quality, the PI-controller has an advantage. In addition, the synthesis of a neurocontroller in terms of finding the minimum of the objective function is a more difficult problem. For its solution, a bigger number of iterations of the VCT-PSO optimization algorithm is required. It is rationally to set more than 1000 iterations and swarm population in the range 30…50 particles. A comparative analysis by the settling time of the neurocontroller and PI-controller, which is tuned according to engineering methods, showed significant reserves for improving this indicator. Thus, if the requirements for settling time minimization are quite strict, then it is advisable to use a neurocontroller. The obtained results will make it possible to develop recommendations for the rational choice of the control operator when solving practical problems of the control systems synthesis

Global Minimization of Polynomial Integral Functionals

Global Minimization of Polynomial Integral Functionals

Disjoint sections of positive semidefinite matrices and their applications in linear statistical models

Abstract Given matrices A A and B B of the same order, A A is called a section of B B if R ( A ) ∩ R ( B − A ) = { 0 } {\mathscr{R}}\left(A)\cap {\mathscr{R}}\left(B-A)=\left\{0\right\} and R ( A T ) ∩ R ( ( B − A ) T ) = { 0 } {\mathscr{R}}\left({A}^{T})\cap {\mathscr{R}}\left({\left(B-A)}^{T})=\left\{0\right\} , where R ( . ) {\mathscr{R}}\left(.) denotes the range (column space) of a matrix argument and the superscript T T stands for the transpose of a matrix. A matrix G G is called positive semidefinite (p.s.d.) if x T G x {x}^{T}Gx is nonnegative for all real vector x x . However, we refer to a symmetric p.s.d. matrix simply as a p.s.d. matrix as the applications discussed in this article are concerned with only symmetric p.s.d. matrices. An n × n n\times n p.s.d. matrix G G admits of a symmetric section G X {G}_{X} of G G with regard to an n × k n\times k matrix X X such that R ( G X ) = R ( G ) ∩ R ( X ) {\mathscr{R}}\left({G}_{X})={\mathscr{R}}\left(G)\cap {\mathscr{R}}\left(X) . In this article, sections of the type G X {G}_{X} are used in minimization of quadratic functions under linear constraints and splitting of vector random variables into uncorrelated vector random variables. In the general Gauss-Markoff model, y = X β + ε y=X\beta +\varepsilon , with design matrix X X and a singular covariance matrix σ 2 G {\sigma }^{2}G of ε \varepsilon , y y is decomposed into four uncorrelated vector random variables as y = M 1 y + M 2 y + M 3 y + M 4 y y={M}_{1}y+{M}_{2}y+{M}_{3}y+{M}_{4}y , where M i , i = 1 , 2 , 3 {M}_{i},i=1,2,3 consist of sections of G G and X X T X{X}^{T} , and M 4 {M}_{4} is a matrix whose row space is the null space of G + X X T G+X{X}^{T} .

Damage detection of truss structures using meta-heuristic algorithms and optimized group method of data handling surrogate model

Optimization-based methods are increasingly being implemented for structural damage detection (SDD) problems through the minimization of the objective functions based on vibration data. Meanwhile, the challenge of high computational cost hinders the applied use of SDD methods, especially upon addressing large-scale structures. In this study, a two-stage damage detection approach is proposed for damage localization as well as extent quantification which reduces the computational time of SDD problems. In the first stage of the proposed framework, suspected locations of damage are identified by employing a residual force vector. Then in the second stage, damage extent of already identified elements will be assessed by model updating method through three different optimization algorithms, namely particle swarm optimization (PSO), differential evolution (DE), and enhanced colliding bodies optimization (ECBO). To spare time-consuming modal analysis in each iteration of optimization algorithms, a PSO-based group method of data handling (GMDH) polynomial neural network (PNN) surrogate model is proposed. Performance of GMDH PNN strongly depends on the input variables, the number of input variables, and the order of the polynomial. The PSO-based GMDH PNN, developed in this study, employs PSO for the optimum selection of these parameters. Furthermore, the efficiency of Elemental Energy Quotient Indicator (EEQI) as a new damage index proposed in this paper is compared with the approach based on the natural frequency-based index. This paper also deals with the incomplete modal data measured from limited number of sensors. This problem is addressed by employing the model reduction technique and solving the SDD problem with modal information obtained from the reduced model. Numerical examples show that the suggested PSO-based GMDH surrogate model is an accurate tool for various types of SDD problems, and it can provide promising results with a significant reduction in computational time.

Second-order Sobolev gradient flows for computing ground state of ultracold Fermi gases

Based on density functional theory, the ground state for ultracold Fermi gases with dipole–dipole interaction is a functional minimization problem with orthonormality constraints. Many methods such as direct minimization, self-consistent iteration method and first-order gradient flow had been proposed to solve such problem. In this paper, we introduce the second-order Sobolev gradient flows, solve them to the steady state and get the ground state of ultracold Fermi gases numerically. Two contributions have been made: firstly we extend the usual first-order gradient flows to the second-order gradient flows; secondly, we extend the usual L2 gradient to the Sobolev gradient and get the Sobolev gradient flows. We prove that the second-order Sobolev gradient flows have the properties of orthonormality preserving and ‘modified’ energy diminishing, which are desirable for the computation of the ground state solution. Several numerical techniques have been used to discretize the time-dependent second-order Sobolev gradient flows. These include explicit leap-frog method and stabilized semi-implicit method for temporal discretization and Fourier pseudo-spectral method for spatial variables. Extensive numerical examples for the ground state are reported to demonstrate the power of the numerical methods.

Model parameter estimation with imprecise information.

Model parameter estimation is a well-known inverse problem, as long as single-value point data are available as observations of system performance measurement. However, classical statistical methods, such as the minimization of an objective function or maximum likelihood, are no longer straightforward, when measurements are imprecise in nature. Typical examples of the latter include censored data and binary information. Here, we explore Approximate Bayesian Computation as a simple method to perform model parameter estimation with such imprecise information. We demonstrate the method for the example of a plain rainfall-runoff model and illustrate the advantages and shortcomings. Last, we outline the value of Shapley values to determine which type of observation contributes to the parameter estimation and which are of minor importance.

Reduced complexity model predictive direct power control for unbalanced grid

The model predictive control (MPC) group is frequently used as a controller in PWM rectifiers because of its accuracy. Yet, the requirement of a processor with high computational efficiency for its real-time implementation has been a significant drawback. Although a few low-complexity MPCs have been reported in the past, none of such methods are validated in an unbalanced grid. Hence, this work aims to propose a reduced-complexity MPC algorithm that can also handle an unbalanced grid. The conventional MPC evaluates the cost function for all the switching vectors and then chooses the switching vector that produces the minimum cost. In contrast, by equating its first-order derivative of the cost function to 0, the proposed analytical approach directly evaluates switching vectors, resulting in a minimum value of the cost function. Thus, the controller estimates only the closest possible solutions of the cost function minimization process and abandons the rest. Furthermore, the proposed method also endorses the use of additional non-linear constraints in the cost function, which eradicates one of the major drawbacks of the previous low-complexity techniques. To ascertain the practicability of the proposed method, it is examined in the MATLAB/Simulink environment and the real-time system. The obtained results look promising as the proposed controller is able to imitate the performance of conventional MPC with less computational burden.

High‐throughput in silico workflow for optimization and characterization of multimodal chromatographic processes

AbstractWhile high‐throughput (HT) experimentation and mechanistic modeling have long been employed in chromatographic process development, it remains unclear how these techniques should be used in concert within development workflows. In this work, a process development workflow based on HT experiments and mechanistic modeling was constructed. The integration of HT and modeling approaches offers improved workflow efficiency and speed. This high‐throughput in silico (HT‐IS) workflow was employed to develop a Capto MMC polishing step for mAb aggregate removal. High‐throughput batch isotherm data was first generated over a range of mobile phase conditions and a suite of analytics were employed. Parameters for the extended steric mass action (SMA) isotherm were regressed for the multicomponent system. Model validation was performed using the extended SMA isotherm in concert with the general rate model of chromatography using the CADET modeling software. Here, step elution profiles were predicted for eight RoboColumn runs across a range of ionic strength, pH, and load density. Optimized processes were generated through minimization of a complex objective function based on key process metrics. Processes were evaluated at lab‐scale using two feedstocks, differing in composition. The results confirmed that both processes obtained high monomer yield (>85%) and removed of aggregate species. Column simulations were then carried out to determine sensitivity to a wide range of process inputs. Elution buffer pH was found to be the most critical process parameter, followed by resin ionic capacity. Overall, this study demonstrated the utility of the HT‐IS workflow for rapid process development and characterization.

Multi-objective optimal power flow with wind–solar–tidal systems including UPFC using Adaptive Improved Flower Pollination Algorithm(AIFPA)

ABSTRACT Among the most significant non-linear challenges for power network design and smooth functioning of current modern updated power system networks is the optimum power flow (OPF) problem. Importance of the electrical system modeling has recently come to light due to the incremental use of energy from renewable sources in power systems networks. The goal is to use wind, solar and tidal sources to recreate the issue of OPF. In this work, Weibull, Lognormal, and also Gumbel probability distribution functions were applied to simulate the uncertainties of wind, photovoltaic, and tidal energy system. Additionally, by adding test scenarios of unpredictable wind, solar and tidal energy systems involving minimization of the cost function, loss of active power, voltage deviation, and increase stability of voltage. In accordance with the chosen thermal producing units, the solutions were evaluated using different locations using IEEE 30-bus systems of testing that incorporate renewable energy sources. The proposed power system planning problem was solved by multi-objective function where unified power flow controller are utilized as flexible AC transmission systems controllers via recently introduced optimization algorithms and the simulation outcomes of the aforementioned technique have been compared with Multi Objective Adaptive Guided Differential Evolution algorithms. The adaptive improved flower pollination algorithm (AIFPA) is a strong and reliable algorithm that is presented in this work. The AIFPA can efficiently deal with many kinds of high-complexity objective regions in multi-objective optimization situations. Utilizing an IEEE 30-bus test system, the suggested approaches’ performance was examined and evaluated for a range of objective functions. The simulation results obtained from the proposed algorithm were effective in finding the optimal solution compared to the results of the meta-heuristic algorithm and were reported in the literature.

Resolution of multiple semi-infinite delaminations using lock-in infrared thermography

Delaminations are flat subsurface defects parallel to the sample surface. Recently we have demonstrated that lock-in infrared thermography, with optical excitation, allows sizing the geometrical parameters (length, depth and thickness) of a semi-infinite delamination. Here, we analyse the ability of this technique to resolve several parallel and semi-infinite delaminations. First, we develop an analytical method (based on the thermal quadrupoles) together with a numerical formulation to calculate the surface temperature of a sample containing several semi-infinite parallel delaminations. We verify that both methods provide the same temperature values, indicating their consistency. Then, we study the ability of lock-in infrared thermography to resolve two close delaminations. In particular we focus on two main configurations: two non-overshadowed delaminations and two superimposed delaminations. Next, after analysing the inverse problem in terms of residual function minimization, we develop a dedicated parametric estimation procedure able to retrieve the geometry of the studied defects. Finally, we test this procedure with synthetic temperature amplitude and phase data to retrieve the geometrical parameters of both delaminations.

Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution

In a Hilbertian framework, for the minimization of a general convex differentiable function f, we introduce new inertial dynamics and algorithms that generate trajectories and iterates that converge fastly towards the minimizer of f with minimum norm. Our study is based on the non-autonomous version of the Polyak heavy ball method, which, at time t, is associated with the strongly convex function obtained by adding to f a Tikhonov regularization term with vanishing coefficient ε(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon (t)$$\\end{document}. In this dynamic, the damping coefficient is proportional to the square root of the Tikhonov regularization parameter ε(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon (t)$$\\end{document}. By adjusting the speed of convergence of ε(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon (t)$$\\end{document} towards zero, we will obtain both rapid convergence towards the infimal value of f, and the strong convergence of the trajectories towards the element of minimum norm of the set of minimizers of f. In particular, we obtain an improved version of the dynamic of Su-Boyd-Candès for the accelerated gradient method of Nesterov. This study naturally leads to corresponding first-order algorithms obtained by temporal discretization. In the case of a proper lower semicontinuous and convex function f, we study the proximal algorithms in detail, and show that they benefit from similar properties.

Constructal design for H-shaped compound heat transfer path in a rectangular heat generation body

An H-shaped compound heat transfer path (HSCHTP) model in a rectangular heat generation body (RHGB) is built in present article. With specified total volumes of HSCHTP and RHGB, constructal design for HSCHTP model is conducted by varying path length ratios and radius. The minimum complex functions composed of thermal and flow performances are acquired, and the multi-objective optimizations are performed furtherly. Three decision-making approaches are introduced to obtain the optimal design scheme (ODS) from Pareto-optimal frontiers. It indicates that complex function after triple-optimization is reduced by 14.7 % compared with that of initial design. Compared with performance of RHGB with no heat conduction path, complex function after optimization is reduced by 77.5 %. Thus, performance of the HSCHTP is apparently superior to that with no heat conduction path in this case. The complex function minimization is a special case of the multi-objective optimization (MOO). The fifth order path radius, adjacent order and internal order length ratios acquired by MOO are distributed within the ranges of 1.50mm-1.60 mm, 1.39–1.47 and 0.066–0.90, respectively. By chasing for the smallest deviation-index, optimal structure of HSCHTP acquired by LINMAP decision-making method is taken for the ODS when multiple requirements are considered in the heat dissipation designs.

Modified Memoryless Spectral-Scaling Broyden Family on Riemannian Manifolds

This paper presents modified memoryless quasi-Newton methods based on the spectral-scaling Broyden family on Riemannian manifolds. The method involves adding one parameter to the search direction of the memoryless self-scaling Broyden family on the manifold. Moreover, it uses a general map instead of vector transport. This idea has already been proposed within a general framework of Riemannian conjugate gradient methods where one can use vector transport, scaled vector transport, or an inverse retraction. We show that the search direction satisfies the sufficient descent condition under some assumptions on the parameters. In addition, we show global convergence of the proposed method under the Wolfe conditions. We numerically compare it with existing methods, including Riemannian conjugate gradient methods and the memoryless spectral-scaling Broyden family. The numerical results indicate that the proposed method with the BFGS formula is suitable for solving an off-diagonal cost function minimization problem on an oblique manifold.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

This paper introduces a formulation of the robust topology optimization problem that is tailored for designing fiber-reinforced composite structures with spatially varying principal mechanical properties. Specifically, a methodology is developed that incorporates the spatial variability in the engineering constants of the composite lamina into the concurrent topology (i.e., material distribution) and morphology (i.e., fiber orientation distribution) optimization problem for the minimization of the robust compliance function. The spatial variability in the mechanical properties of the lamina is modeled as a homogeneous random field within the design domain by means of the Karhunen-Loe´ve series expansion, and is thereafter intrusively propagated into the stochastic finite element analysis of the composite structure. To carry out the stochastic finite element analysis per iteration of the optimization cycle, the first-order perturbation method is utilized for approximating the current state variables of the physical system. The resulting robust topology and fiber orientation optimization problem is formulated step-by-step for the minimization of the robust compliance function. With the view of solving the optimization problem at hand by means of gradient-based solution algorithms, the first-order derivatives of the involved design functions w.r.t. the associated design variables are analytically derived. The present work concludes with a series of numerical examples, focusing on the benchmark academic case studies of the 2D cantilever and the half part of the Messerschmitt-Bölkow-Blohm beam, aiming to demonstrate the developed methodology as well as to explore the effect that different parameterization instances of the random field bear on the predicted topology and morphology of the beams.

A new chromatographic response function with automatically adapting weight factor for automated method development

A critical factor for automated method development in chromatography is the maximization or minimization of an objective function describing the quality (and speed) of the separation. In chromatography, this function is commonly referred to as a chromatographic response function (CRF). Many CRFs have previously been introduced, but many have unfavourable properties such as featuring multiple optima, insufficient discriminatory power, and a too strong dependence on the weight factors needed to balance resolution and time penalty components. To overcome these problems, the present study introduces a new type of CRF wherein the relative weight of the time penalty term is a self-adaptive function of the separation quality. The ability to unambiguously identify the optimal gradient settings of this newly proposed CRF is compared to that of some of the most frequently used CRFs in a study covering 100 randomly composed in silico samples. Doing so, the new CRF is found to flawlessly lead to the correct solution (=linear gradient parameters providing the highest resolution in the shortest potential time) in 100 % of the cases, while the most frequently used literature CRFs were off-target for about 50 to 60 % of the samples, even when considering the availability of spectral peak shape data. Some slight alterations to the proposed CRF are introduced and discussed as well.

Securing Internet-of-Medical-Things networks using cancellable ECG recognition

Reinforcement of the Internet of Medical Things (IoMT) network security has become extremely significant as these networks enable both patients and healthcare providers to communicate with each other by exchanging medical signals, data, and vital reports in a safe way. To ensure the safe transmission of sensitive information, robust and secure access mechanisms are paramount. Vulnerabilities in these networks, particularly at the access points, could expose patients to significant risks. Among the possible security measures, biometric authentication is becoming a more feasible choice, with a focus on leveraging regularly-monitored biomedical signals like Electrocardiogram (ECG) signals due to their unique characteristics. A notable challenge within all biometric authentication systems is the risk of losing original biometric traits, if hackers successfully compromise the biometric template storage space. Current research endorses replacement of the original biometrics used in access control with cancellable templates. These are produced using encryption or non-invertible transformation, which improves security by enabling the biometric templates to be changed in case an unwanted access is detected. This study presents a comprehensive framework for ECG-based recognition with cancellable templates. This framework may be used for accessing IoMT networks. An innovative methodology is introduced through non-invertible modification of ECG signals using blind signal separation and lightweight encryption. The basic idea here depends on the assumption that if the ECG signal and an auxiliary audio signal for the same person are subjected to a separation algorithm, the algorithm will yield two uncorrelated components through the minimization of a correlation cost function. Hence, the obtained outputs from the separation algorithm will be distorted versions of the ECG as well as the audio signals. The distorted versions of the ECG signals can be treated with a lightweight encryption stage and used as cancellable templates. Security enhancement is achieved through the utilization of the lightweight encryption stage based on a user-specific pattern and XOR operation, thereby reducing the processing burden associated with conventional encryption methods. The proposed framework efficacy is demonstrated through its application on the ECG-ID and MIT-BIH datasets, yielding promising results. The experimental evaluation reveals an Equal Error Rate (EER) of 0.134 on the ECG-ID dataset and 0.4 on the MIT-BIH dataset, alongside an exceptionally large Area under the Receiver Operating Characteristic curve (AROC) of 99.96% for both datasets. These results underscore the framework potential in securing IoMT networks through cancellable biometrics, offering a hybrid security model that combines the strengths of non-invertible transformations and lightweight encryption.