Function Approximation Research Articles

Self-organizing surrogate-assisted non-dominated sorting differential evolution

Multi-objective optimization problems (MOPs) involve optimizing multiple conflicting objectives simultaneously, resulting in a set of Pareto optimal solutions. Due to the high computational or financial cost associated with evaluating fitness, expensive multi-objective optimization problems (EMOPs) further complicate the optimization process. Surrogate-assisted evolutionary algorithms (SAEAs) have emerged as a promising approach to address EMOPs by substituting costly evaluations with computationally efficient surrogate models. This paper introduces the self-organizing surrogate-assisted non-dominated sorting differential evolution (SSDE), which uses surrogate model based on a self-organizing map (SOM) to approximate the fitness function. SSDE offers advantages such as reduced computational cost, improved accuracy, and the speed of enhanced convergence. The SOM-based surrogate models effectively capture the underlying structure of the Pareto optimal set and Pareto optimal front, leading to superior approximations of the fitness function. Experimental results on benchmark functions and real-world problems, including Model-Free Adaptive Control (MFAC) and the Yagi-Uda Antenna design, demonstrate the competitiveness and efficiency of SSDE compared to other algorithms.

Robust iterative value conversion: Deep reinforcement learning for neurochip-driven edge robots

A neurochip is a device that reproduces the signal processing mechanisms of brain neurons and calculates Spiking Neural Networks (SNNs) with low power consumption and at high speed. Thus, neurochips are attracting attention from edge robot applications, which suffer from limited battery capacity. This paper aims to achieve deep reinforcement learning (DRL) that acquires SNN policies suitable for neurochip implementation. Since DRL requires a complex function approximation, we focus on conversion techniques from Floating Point NN (FPNN) because it is one of the most feasible SNN techniques. However, DRL requires conversions to SNNs for every policy update to collect the learning samples for a DRL-learning cycle, which updates the FPNN policy and collects the SNN policy samples. Accumulative conversion errors can significantly degrade the performance of the SNN policies. We propose Robust Iterative Value Conversion (RIVC) as a DRL that incorporates conversion error reduction and robustness to conversion errors. To reduce them, FPNN is optimized with the same number of quantization bits as an SNN. The FPNN output is not significantly changed by quantization. To robustify the conversion error, an FPNN policy that is applied with quantization is updated to increase the gap between the probability of selecting the optimal action and other actions. This step prevents unexpected replacements of the policy’s optimal actions. We verified RIVC’s effectiveness on a neurochip-driven robot. The results showed that RIVC consumed 1/15 times less power and increased the calculation speed by five times more than an edge CPU (quad-core ARM Cortex-A72). The previous framework with no countermeasures against conversion errors failed to train the policies. Videos from our experiments are available: https://youtu.be/Q5Z0-BvK1Tc.

New global minimum conformers for the Pt19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{19}$$\\end{document} and Pt20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{20}$$\\end{document} clusters: low symmetric species featuring different active sites

ContextThe study of platinum (Pt) clusters and nanoparticles is essential due to their extensive range of potential technological applications, particularly in catalysis. The electronic properties that yield optimal catalytic performance at the nanoscale are significantly influenced by the size and structure of Pt clusters. This research aimed to identify the lowest-energy conformers for Pt18\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{18}$$\\end{document}, Pt19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{19}$$\\end{document}, and Pt20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{20}$$\\end{document} species using Density Functional Theory (DFT). We discovered new low-symmetry conformers for Pt19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{19}$$\\end{document} and Pt20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{20}$$\\end{document}, which are 3.0 and 1.0 kcal/mol more stable, respectively, than previously reported structures. Our study highlights the importance of using density functional approximations that incorporate moderate levels of exact Hartree-Fock exchange, alongside basis sets of at least quadruple-zeta quality. The resulting structures are asymmetric with varying active sites, as evidenced by sigma hole analysis on the electrostatic potential surface. This suggests a potential correlation between electronic structure and catalytic properties, warranting further investigation.MethodsAn equivariant graph neural network interatomic potential (NequIP) within the Atomic Simulation Environment suite (ASE) was used to provide initial geometries of the aggregates under study. DFT calculations were performed with the ORCA 5 package, using functional approximations that included Generalized Gradient Approximation (PBE), meta-GGA (TPSS, M06-L), hybrid (PBE0, PBEh), meta-GGA hybrid (TPSSh), and range-separated hybrid (ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}B97x) functionals. Def2-TZVP and Def2-QZVP as well as members of the cc-pwCVXZ-PP family to check basis set convergence were used. QTAIM calculations were performed using the AIMAll suite. Structures were visualized with the AVOGADRO code.

Statistical inference on nonparametric regression model with approximation of Fourier series function: Estimation and hypothesis testing

Nonparametric regression is an approximation method in regression analysis that is not constrained by the assumption of knowing the regression curve. One of the functions to approximate the curve is a Fourier series function. The nonparametric regression model with approximation of a Fourier series function has been widely discussed by several researchers. However, discussions on statistical inference, particularly in partial hypothesis testing, has not been carried out previously. Therefore, the purpose of this research is to discuss the statistical inference on nonparametric regression model with approximation of a Fourier series function. The discussion includes parameter and model estimations, simultaneous and partial hypotheses testing. In the application, we use life expectancy data from East Java Province during 2022. Based on data analysis, we obtain a model estimation with an R-square value of 96.24 %. At a 5 % significance level, the parameters simultaneously have a significant influence on the model. Partially, four parameters are not significant. However, overall, the predictor variables significantly influence the life expectancy data.•The Fourier series function used is a Fourier series function introduced by Bilodeau (1992).•The model estimation is obtained by selecting the optimal number of oscillation parameters.•The statistical test is obtained using the LRT method.

Padé Approximations and Irrationality Measures on Values of Confluent Hypergeometric Functions

Padé approximations are approximations of holomorphic functions by rational functions. The application of Padé approximations to Diophantine approximations has a long history dating back to Hermite. In this paper, we use the Maier–Chudnovsky construction of Padé-type approximation to study irrationality properties about values of functions with the form f(x)=∑k=0∞xkk!(bk+s)(bk+s+1)⋯(bk+t), where b,t,s are positive integers and obtain upper bounds for irrationality measures of their values at nonzero rational points. Important examples includes exponential integral, Gauss error function and Kummer’s confluent hypergeometric functions.

A low-complexity fixed-time prescribed performance control for uncertain full-vehicle active suspensions with hydraulic actuators

This paper addresses the low-complexity fixed-time prescribed performance control problem for uncertain full-vehicle active suspension systems. Different from existing results that ignore the dynamics of the actuator, this paper considers a hydraulic actuator in the controller design. To address the nonlinearities of hydraulic active suspension systems, a new low-complexity fixed-time prescribed performance control method is proposed. In this method, the function approximators (e.g. neural networks (NNs) and fuzzy systems (FSs)) are not needed, which means that the heavy computational costs are avoided. Furthermore, by developing a fixed-time performance function, the proposed method can ensure that the suspension motions converge to the prescribed range within a finite time. Based on the Lyapunov theorem and Extreme Value Theorem, the stability of the closed-loop suspension system is strictly proved. Finally, the comparative simulation results show that the proposed method effectively improves the attitude stability and ride comfort of the suspension system.

Ensemble Generalization of the Perdew-Zunger Self-Interaction Correction: A Way Out of Multiple Minima and Symmetry Breaking.

The Perdew-Zunger (PZ) self-interaction correction (SIC) is an established tool to correct unphysical behavior in density functional approximations. Yet, the PZ-SIC is well-known to sometimes break molecular symmetries. An example of this is the benzene molecule, for which the PZ-SIC predicts a symmetry-broken electron density and molecular geometry, since the method does not describe the two possible Kekulé structures on an even footing, leading to local minima [Lehtola et al. J. Chem. Theory Comput. 2016, 12, 3195]. The PZ-SIC is often implemented with Fermi-Löwdin orbitals (FLOs), yielding the FLO-SIC method, which likewise has issues with symmetry breaking and local minima [Trepte et al. J. Chem. Phys. 2021, 155, 224109]. In this work, we propose a generalization of the PZ-SIC─the ensemble PZ-SIC (E-PZ-SIC) method─which shares the asymptotic computational scaling of the PZ-SIC (albeit with an additional prefactor). The E-PZ-SIC is straightforwardly applicable to various molecules, merely requiring one to average the self-interaction correction over all possible Kekulé structures, in line with chemical intuition. We showcase the implementation of the E-PZ-SIC with FLOs, as the resulting E-FLO-SIC method is easy to realize on top of an existing implementation of the FLO-SIC. We show that the E-FLO-SIC indeed eliminates symmetry breaking, reproducing a symmetric electron density and molecular geometry for benzene. The ensemble approach suggested herein could also be employed within approximate or locally scaled variants of the PZ-SIC and its FLO-SIC versions.

Geometric and electronic structures of Cs2BB′X6 double perovskites: The importance of exact exchange

A widely adopted computational protocol in contemporary materials research is to first relax materials' geometries using semilocal density functional approximations (DFA), and then determining their electronic band structures using the more expensive hybrid functionals. This procedure often works well, as the popular semilocal DFAs, such as the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation, yield rather good geometries for a wide range of materials. However, here we show that, for some of the lead-free halide double perovskites (HDPs) Cs2BB′X6 (B=Ag+, Na+; B′=In3+, Bi3+; X=Cl−, Br−), the validity of this common practice is questionable. We find that, for these HDPs, the geometrical structures, in particular, the B(B′)−X bond lengths predicted by PBE show large deviations from the experimental values. Additionally, the band gaps of some of these materials (specifically, the In-based HDPs) are sensitive to the B(B′)−X bond lengths. As a consequence, the band gaps obtained using the hybrid functionals (such as the Heyd-Scuseria-Ernzerhof functional) based on the PBE geometries can still be quite off, in particular, for HDPs with B′=In3+. The situation is significantly improved by using hybrid functionals with tuned portion of exact exchange, based on the geometries determined consistently under the same level of theory. The successes and failures of several popular exchange-correlation (XC) functionals are traced back to the so-called delocalization error, and can be quantitatively analyzed and understood via a three-atom linear-chain B−X−B′ molecular model. Finally, our findings provide a practical guide for choosing appropriate XC functionals for describing HDPs and point to a promising path for band structure engineering via doping and alloying. Published by the American Physical Society 2024

Extended benchmark set for lattice parameters of inorganic solids.

The development of novel methods in solid-state quantum chemistry necessitates reliable reference data sets for their assessment. The most fundamental solid-state property of interest is the crystal structure, quantified by the lattice parameters. In the last decade, several studies were conducted to assess theoretical approaches based on the agreement of calculated lattice parameters with respect to experiment as a measure. However, most of these studies used a limited number of reference systems with high symmetry. The present work offers a more comprehensive reference benchmark denoted as Sol337LC, which consists of 337 inorganic compounds with 553 symmetry-inequivalent lattice parameters, representing every element of the periodic table for atomic numbers between 1 and 86, except noble gases, the radioactive elements and lanthanoids. The reference values were taken from earlier benchmarks and from measurements at very low temperature or extrapolation to 0 K. The experimental low-temperature lattice parameters were then corrected for zero-point energy effects via the quasi-harmonic approximation for direct comparison with quantum-chemical optimized structures. A selection of standard density functional approximations was assessed for their deviations from the experimental reference data. The calculations were performed with the crystal orbital program CRYSTAL23, applying optimized atom-centered basis sets of triple-zeta plus polarization quality. The SCAN functional family and the global hybrid functional PW1PW, augmented with the D3 dispersion correction, were found to provide closest agreement with the Sol337LC reference data.

Towards circular economy of wasted printed circuit boards of mobile phones fuelled by machine learning and robust mathematical optimization framework

Estimating the operating conditions using conventional process analysis techniques for the maximum metal extraction from the wasted printed circuit boards (WPCB) can provide sub-optimal solutions leading to the low yield of the process. In this paper, we present a closed-loop methodological framework built on machine learning and robust mathematical optimization technique, that offers the mathematical rigour, to determine the optimum operating conditions for the maximum Cu and Ni recovery from the WPCB. Alkali leaching based novel metals recovery process from the WPCB is designed, and the experiments are conducted to collect the data on the percentage recovery of Cu and Ni against the operating levels of the process input variables (ammonia concentration (NH3 conc. (g/L)), ammonium sulfate concentration ((NH4)2SO4 conc. (g/L)), H2O2 concentration (H2O2 conc. (M)), time (h), liquid to solid ratio (L/S ratio, (mL/g)), temperature (Temp. (°C)), and stirring speed (rpm)). The experimental data is deployed to construct the functional mapping between the nonlinear output variables of metals recovery process with the hyperdimensional input space through artificial neural network (ANN) based modelling algorithm – a powerful universal function approximator. Well-predictive ANN models for Cu and Ni recovery are developed having co-efficient of determination (R2) value more than 0.90. Partial derivative-based sensitivity analysis is then carried out to establish the order of the significance of the input variables that is backed by the domain knowledge, thus promotes the interpretability of the trained ANN models. The hybridization of ANN with NLP (nonlinear programming) framework is implemented for the determination of optimized operating conditions to extract maximum Cu and Ni under separate and combined model of metal extraction. The robustness of the determined solutions is verified, the determined optimized solutions for the metal recovery are validated in the lab, and the maximum metal recovery, i.e., 100 % Cu and 90 % Ni is extracted from the WPCB. This research demonstrates the effective utilization of ANN model-based robust optimization approach for the metal recovery from the WPCB that supports the circular economy for the metal extraction industry.

Nonsingular fixed-time K-filter output-feedback control for high-order nonlinear multi-agent systems

This paper investigates the problem of nonsingular fixed-time control for high-order nonlinear multi-agent systems with unmeasurable states and disturbances. To address the unmeasurable states, a K-filter is introduced to reconstruct the system with the input and output only and to facilitate the construction of the observer. To provide an approximation for the unknown nonlinear functions, the fuzzy logic system technique is applied. A novel tracking controller is proposed to ensure that all signals remain bounded for the closed-loop system and that the tracking error converges into a small and adjustable set within a fixed time by utilising the adaptive backstepping recursive technology and the adding power integrator technique. Meanwhile, the singular problems arising from the fixed-time stability criterion can be also avoided. Finally, a simulation experiment verifies the effectiveness of the designed control scheme.

Exchange-correlation kernel for perturbation dependent auxiliary functions in auxiliary density perturbation theory

ContextAnalytic exchange-correlation kernel formulations are of the outermost importance for density functional theory (DFT) perturbation calculations. In this paper, the working equation for the exchange-correlation kernel of the generalized gradient approximation (GGA) for perturbation dependent auxiliary functions is derived and discussed in the framework of auxiliary density functional theory (ADFT). The presented new formulation is extended to the unrestricted approach, too. A comprehensive discussion of the implementation of the GGA ADFT kernel, using either the native exchange-correlation functional implementations in deMon2k or the ones from the LibXC library, is given. Calculations with analytic exchange-correlation kernels are compared to their finite difference counterparts. The obtained results are in quantitative agreement. Nevertheless, analytic GGA ADFT kernel implementations show substantial improvement in the computational performance. Similar results are reported for analytic second derivatives of effective core potential (ECP) and model core potential (MCP) matrix elements when compared to their finite difference counterparts in molecular frequency analyses.MethodAll calculations are performed in the framework of ADFT as implemented in deMon2k. In the ADFT analytic frequency calculations, auxiliary density perturbation theory was used. The underlying two-center exchange-correlation kernel matrix elements are calculated by numerical integration either with analytic or finite difference kernel expressions. Validation calculations are performed with the VWN and PBE functionals employing DFT-optimized DZVP basis sets in conjunction with automatically generated GEN-A2 auxiliary density function sets. In the (Pt3Cu)n cluster benchmark calculations, the RPBE functional was used. For Pt atoms, the quasi-relativistic LANL2DZ effective core potential with the corresponding valence basis set was employed, whereas for Cu atoms, the all-electron DFT-optimized TZVP basis was applied. The auxiliary density was expanded by the automatically generated GEN-A2* auxiliary function set. We run all benchmark calculations in parallel on 24 cores.

Geodesic metrics for RBF approximation of some physical quantities measured on sphere

The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula.We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general.

Experimental molecular structures in the gas phase at the upper size limit: The case of Si6Tip6.

Currently, the largest (ramax = 19.9Å) and by far the most complicated (234 atoms, C1 symmetry, 696 independent geometrical parameters, and 27 261 interatomic terms) experimental molecular structure of a cage-type Si6Tip6 (Tip = 2,4,6-iPr3C6H2) isomer has been investigated in the gas phase by the electron diffraction method (Tav = 645K) supplemented with theoretical simulations. A detailed analysis of the current possibilities for experimentally investigating large molecular structures is performed. A series of density functional theory approximations and the role of dispersion interactions have been benchmarked using the obtained data. Based on the refined geometry of Si6Tip6, various quantum-chemical methods have been applied for the investigation of the electronic structure of its Si6 core. In particular, natural bond orbital, quantum theory of atoms in molecules, interacting quantum atoms, fractional occupation number weighted density, and complete active space self-consistent field (CASSCF) methods were utilized. The diradical character of the molecule has been assessed by the UHF and CASSCF approximations. The problem of bonding between the hemispheroidal silicon atoms has been investigated.

Degree of approximation of functions conjugate to the functions belonging to Lip((𝜉1,𝜉2)𝑟)-class through double Nörlund means

Abstract In this paper, we determine the error of approximation of functions, conjugate to the functions of two variables ( 2 ⁢ π 2\pi -periodic in both variables) belonging to the Lipschitz class Lip ( ( ξ 1 , ξ 2 ) ; r \operatorname{Lip}((\xi_{1},\xi_{2});r ) ( r ≥ 1 r\geq 1 ), through the double Nörlund means of its conjugate double Fourier series. Additionally, in the form of corollaries, we present some particular cases of the theorem proved here.

Decentralized Adaptive temporal-difference learning over time-varying networks and its finite-time analysis

In reinforcement learning, centralized temporal-difference (TD) learning is commonly used to solve the policy evaluation problem. However, the decentralized adaptive variant of the TD learning algorithm has rarely been investigated in multi-agent reinforcement learning. To fill this gap, based on linear function approximation, we propose a decentralized adaptive TD learning algorithm over time-varying networks, referred to as D-AdaTD. We rigorously analyze the convergence performance of D-AdaTD, i.e., the explicit finite-time analysis is established under different step-sizes. Specifically, under constant step-sizes, the average estimated value function can converge to a neighborhood of the optimal value at rate O(1/(k+1)) and the estimated parameter of each agent can converge to a neighborhood of the optimal parameter at rate O(ξk), where k is the number of iterations and ξ∈(0,1). Under diminishing step-sizes, the average estimated value function can converge to the optimal value and the average estimated parameter can converge to the optimal parameter at rate O((1+log(k+1))/k) and O((1+log(k+1))/(k+1)), respectively. In addition, we evaluate the performance of D-AdaTD via simulation experiments, which are commonly insufficient in the existing decentralized temporal-difference learning. The experimental results also validate the effectiveness of D-AdaTD.

Uniform momentum zones in turbulent channel flow

The characteristics of uniform momentum zones (UMZs) and UMZ interfaces in turbulent channel flow (TCF) at friction Reynolds number ranging from 600 to 2000 are investigated. The focus is on the Reynolds number effect and the difference among the turbulent channel flow, turbulent pipe flow (TPF), and turbulent boundary layer (TBL). The UMZ is identified using the approach of probability density histograms, a discrete approximation of the probability density function. We find that the average number of UMZs increases with the Reynolds number, similar to the findings in the TBL. The first group of UMZ in the TCF, namely the quiescent core, has a thicker thickness at a higher Reynolds number. Therefore, the UMZ closer to the channel centre significantly influences high Reynolds number wall turbulence by occupying a large portion of the channel. With increasing Reynolds number, the intensity of spanwise vorticity at the UMZ interface increases, while the thickness of the local maximum decreases. The dominance of prograde vorticity at the interface also increases with the Reynolds number. Moreover, the contortion of the UMZ interface is significantly enhanced at high Reynolds numbers than at low Reynolds numbers in both the streamwise and spanwise directions. Furthermore, the number of UMZ for a given snapshot is related to the overall velocity fluctuations in the subregion, increasing with large-scale Q2 events and decreasing with large-scale Q4 events. Comparing the three flow configurations, UMZs show similarities in their trends, along with two notable differences. First, the TPF has more UMZs compared to the TCF and TBL. Second, the innermost group in the TPF is considerably thicker than that of the higher ranks, a pattern absent in the TCF and TBL. The present findings advance the understanding of UMZs in TCF and demonstrate the similarities and differences between the cases of TCF and TBL.

ANALYSIS OF OPTIMAL PORTFOLIO ON FINITE AND SMALL-TIME HORIZONS FOR A STOCHASTIC VOLATILITY MODEL WITH MULTIPLE CORRELATED ASSETS

In this paper, we consider the portfolio optimization problem in a financial market where the underlying stochastic volatility model is driven by n-dimensional Brownian motions. At first, we derive a Hamilton–Jacobi–Bellman equation including the correlations among the standard Brownian motions. We use an approximation method for the optimization of portfolios. With such approximation, the value function is analyzed using the first-order terms of expansion of the utility function in the powers of time to the horizon. The error of this approximation is controlled using the second-order terms of expansion of the utility function. It is also shown that the one-dimensional version of this analysis corresponds to a known result in the literature. We also generate a close-to-optimal portfolio near the time to horizon using the first-order approximation of the utility function. It is shown that the error is controlled by the square of the time to the horizon. Finally, we provide an approximation scheme to the value function for all times and generate a close-to-optimal portfolio.

Efficient opportunistic maintenance strategies via pruning in parallel–series systems with economic dependence

Opportunistic maintenance stands out as an effective strategy for minimizing maintenance costs and enhancing system efficiency. Despite its importance, current opportunistic maintenance models are often tailored to specific system types, particularly those with series structures, or they offer heuristic approaches for multi-component redundant systems. This paper shifts focus to the most common of such systems, the parallel–series system. We approach the optimization of opportunistic maintenance in a parallel–series system through a Markov Decision Process framework, utilizing phase-type approximations of individual component hazard functions. Addressing the challenge of the curse of dimensionality in the action space, this study conducts a thorough structural analysis of the optimal opportunistic maintenance policy, deriving a pruning procedure, which can reduce the extensive combinatorial action space into a more manageable linear complexity without sacrificing the strategy’s effectiveness in parallel–series systems. Building on this more manageable action space, we introduce a more efficient algorithm leveraging the proposed pruning procedure for determining the optimal opportunistic maintenance policy. The effectiveness and practical applicability of this approach are rigorously demonstrated through comprehensive numerical experiments.

Piecewise nonlinear approximation for non-smooth functions

Piecewise affine or linear approximation has garnered significant attention as a technique for approximating piecewise-smooth functions. In this study, we propose a novel approach: piecewise non-linear approximation based on rational approximation, aimed at approximating non-smooth functions. We introduce a method termed piecewise Padé Chebyshev (PiPC) tailored for approximating univariate piecewise smooth functions. Our investigation focuses on assessing the effectiveness of PiPC in mitigating the Gibbs phenomenon during the approximation of piecewise smooth functions. Additionally, we provide error estimates and convergence results of PiPC for non-smooth functions. Notably, our technique excels in capturing singularities, if present, within the function with minimal Gibbs oscillations, without necessitating the explicit specification of singularity locations. To the best of our knowledge, prior research has not explored the use of piecewise non-linear approximation for approximating non-smooth functions. Finally, we validate the efficacy of our methods through numerical experiments, employing PiPC to reconstruct a non-trivial non-smooth function, thus demonstrating its capability to significantly alleviate the Gibbs phenomenon.