Low-dimensional Problems Research Articles

Sparse adaptive basis set methods for solution of the time dependent Schrodinger equation

Scalable numerical solutions to the time dependent Schrodinger equation remain an outstanding goal in theoretical chemistry. Here we present a method which utilises recent breakthroughs in signal processing to consistently adapt a dictionary of basis functions to the dynamics of the system. We show that for two low-dimensional model problems the size of the basis set does not grow quickly with time and appears only weakly dependent on dimensionality. The generality of this finding remains to be seen. The method primarily uses energies and gradients of the potential, opening the possibility for its use in on-the-fly ab initio quantum wavepacket dynamics.

An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition

A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection–diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods.

Relaxation-based importance sampling for structural reliability analysis

This study presents an importance sampling formulation based on adaptively relaxing parameters from the indicator function and/or the probability density function. The formulation embodies the prevalent mathematical concept of relaxing a complex problem into a sequence of progressively easier sub-problems. Due to the flexibility in constructing relaxation parameters, relaxation-based importance sampling provides a unified framework for various existing variance reduction techniques, such as subset simulation, sequential importance sampling, and annealed importance sampling. More crucially, the framework lays the foundation for creating new importance sampling strategies, tailoring to specific applications. To demonstrate this potential, two importance sampling strategies are proposed. The first strategy couples annealed importance sampling with subset simulation, focusing on low-dimensional problems. The second strategy aims to solve high-dimensional problems by leveraging spherical sampling and scaling techniques. Both methods are desirable for fragility analysis in performance-based engineering, as they can produce the entire fragility surface in a single run of the sampling algorithm. Three numerical examples, including a 1000-dimensional stochastic dynamic problem, are studied to demonstrate the proposed methods.

Extreme value analysis of high-dimensional Gaussian vector processes

The analysis of the extreme value pertaining to first-passage failure is a key consideration for engineering systems undergoing random vibration. Most of the literature focus on a univariate stochastic process, however for system reliability analysis, the extreme value of multiple dependent processes is required. Due to the problem complexity, analytical methods are scarce, and they are limited to low-dimensional problems (dimension refers to the number of component processes). This paper presents an analytical method for extreme analysis of multivariate stationary Gaussian processes. The method can efficiently solve high-dimensional problems, thus it has diverse applications such as analysing dynamic systems with many degrees-of-freedom, or evaluating space-time extremes of random fields by discretizing the spatial domain. A maximum process is defined, representing the maxima of the components at each time instant. The exact upcrossing rate of the maximum process is derived, and the Poisson approximation is invoked to estimate the extreme value distribution. The method is fast as it involves multiple one-dimensional integrals, which can be computed using Gaussian quadrature. Moreover, an accurate approximation is proposed that obviates the need for numerical integration. The proposed method is implemented on two examples, specifically a dynamic system and spreading ocean waves. Monte Carlo simulations, which are computationally demanding, are performed to verify the accuracy and correctness of the proposed method.

Optimal control of mineral processing plants using constrained model predictive static programming

The model predictive static programming (MPSP) technique, which is extended recently to incorporate applicable state and control constraints, operates on the philosophy of nonlinear model predictive control (NMPC). However, it reduces the problem into a lower-dimensional problem of control variables alone, thereby enhancing computational efficiency significantly. Because of this, problems with larger dimensions and/or increased complexity can be solved using MPSP without changing the computational infrastructure. In this paper, the MPSP technique with applicable constraints is applied to two challenging control problems in the mineral processing industry: (i) a single-stage grinding mill circuit model, and (ii) a four-cell flotation circuit model. The results are compared with a conventional nonlinear MPC approach. Comparison studies show that constrained MPSP executes much faster than constrained MPC with similar/improved performance. Therefore, it can be considered a potential optimal control candidate for mineral processing plants.

A decision-making design approach to optimize the hole arrangement of effusion cooling plates under non-uniform heat load conditions

The harsh operating conditions of aero-engines contain non-uniform temperatures and variable hotspots, which require adaptive cooling techniques to accommodate the heat loads on the external surfaces of metal. Among the existing technologies, effusion cooling is an important way to fulfill such quests due to its flexibility in controlling coolant locally. As an effusively cooled component generally contains hundreds of holes, hole arrangement optimization is a high-dimensional problem that is difficult to solve using existing evolutionary algorithms. Recently matured deep reinforcement learning algorithms showed success in many research fields by incorporating data modeling with sequential optimization, which demonstrated the ability to convert high dimensional optimization problems into very low dimensional iterative optimization problems. The present study proposed a decision-making design approach to optimize the hole arrangement of a series of effusion cooling plates. Hole arrangements were transformed into Markov decision chains of opening or closing each hole. The decision chains were solved by a reinforcement learning framework integrated with a deep learning-based environment simulator. Results indicated that the proposed approach could yield an elevation of cooling performance by 20% as compared with the results from genetic algorithms. Moreover, an analytical correlation was extracted from the clear decision boundaries of reinforcement learning. Upon further validation, the computational cost to apply the correlation to other non-uniform heat loads was within seconds, while the cooling effect could reach the same optimization level as reinforcement learning. The idea proposed in this study could be applied to the arrangement optimization of multiple cooling technologies, such as film/effusion cooling, impingement cooling, and pin–fin array cooling.

Deterministic bridge regression for compressive classification

Pattern classification using a compact representation is a crucial component of machine intelligence. Specifically, it is essential to learn a model with well-regulated parameters to achieve good generalization. Bridge regression provides a mechanism for regulating parameters through a penalized ℓp-norm. However, due to the nonlinear nature of the formulation, an iterative numerical search is typically used to solve the optimization problem. In this work, we propose an analytic solution for bridge regression based on solving a penalized error formulation using an approximated ℓp-norm. The solution is presented in primal form for over-determined systems and in dual form for under-determined systems. The primal form is suitable for low-dimensional problems with a large number of data samples, while the dual form is suitable for high-dimensional problems with a small number of data samples. We also extend the solution to problems with multiple classification outputs. Numerical studies using simulated and real-world data demonstrate the effectiveness of our proposed solution.

Classification model-based and assisted environment selection for evolutionary algorithms to solve high-dimensional expensive problems

Surrogate-assisted evolutionary algorithms (SAEAs) have been proven to be very effective in tackling low-dimensional expensive problems. However, it remains a challenge to solve high-dimensional expensive problems (HEPs) with the curse of dimensionality. Therefore, this paper proposes a surrogate-assisted improved multioperator differential evolution (SA-IMODE) algorithm to address HEPs up to 2000 dimensions. Specifically, this paper proposes a novel relationship classification model-based environment selection strategy (RCES), in which a classification model is used to distinguish between “good” and “bad” solutions to assist in environment selection. By doing this, the “unpromising” solutions are thrown away directly without evaluation to reduce the number of expensive fitness evaluations. Moreover, the solution-improvement data point and error samples are added to construct the training dataset to improve the model's prediction accuracy. Two coordinate systems are utilized to produce five types of DE operators to suit different fitness landscapes. An adaptive strategy is also used to select suitable DE operators to generate promising offspring. Furthermore, a local search mechanism is applied to refine the best solution in the current population to accelerate the algorithm's convergence. The systematic experiment results show SA-IMODE has a significant advantage over thirteen state-of-the-art algorithms on benchmark problems.

Stochastic dynamic analysis of composite plates in thermal environments using nonlinear autoregressive model with exogenous input in polynomial chaos expansion surrogate

The application of composite materials has increased manifolds in aerospace and high-speed vehicle industries, where structures experience dynamic loads along with temperature variation. To ensure safety of these structures, the stochastic dynamic analysis in varying thermal environments is essential. The generalized polynomial chaos (gPC) expansion method is a well-known metamodel used to quantify uncertainties in engineering systems instead of using the direct Monte Carlo simulation. Though the gPC expansion method is used extensively for engineering systems, its classical version has some limitations while implementing in dynamical systems. The accuracy of the gPC expansion method reduces in delayed time domain and can be improved to some extent by increasing the order of polynomial, which though is computationally intensive. Furthermore, the classical gPC expansion method is suitable for low-dimensional problems having independent random parameters. To overcome these limitations in the classical gPC expansion method, a system identifier, i.e., nonlinear autoregressive model with exogenous input (NARX) is coupled with it to avoid response degeneration in delayed time domain. The temperature-dependent material properties of composite lamina can be random and correlated, however the classical gPC expansion method is unable to accommodate the correlated random parameters. To address these challenges, the NARX-gPC expansion method is modified to conduct stochastic dynamic analysis of composite plates using correlated random material properties of the composite lamina in varying thermal environments. The study indicates that the correlation has a significant effect on the stochastic dynamic response at low and high temperatures for cross-ply laminates in comparison with angle-ply laminates, which is not captured through uncorrelated random material properties. The prediction efficiency of the developed surrogate by using the adaptive NARX-gPC expansion method is established even though it is trained using data in a shorter time domain, thereby computational performance is enhanced. Additionally, this surrogate is implemented effectively for higher order randomness in the input parameters.

Combined approximation algorithms for interactive design of road routes in CAD

Objectives. The aim of the work is to create algorithms for approximating a sequence of points on a plane by arcs of clothoids and circles. Such a problem typically arises in the design of railroad and highway routes. The plan (projection onto a horizontal plane) of the road route is a curve (spline) consisting of a repeating bundle of elements “straight line + clothoid arc + circle arc + clothoid arc + ...”. Such a combination of elements provides continuity not only for the curve and its tangent, but also for the curvature. Since the number of spline elements is not known in advance, and their parameters are subject to restrictions, there is no mathematically consistent algorithm for this problem. The two-stage scheme for solving the problem is developed at RTU MIREA only for a spline with lines and circles (i.e., without clothoid elements). At the first stage, the scheme uses dynamic programming to determine the number of spline elements. At the second stage, the scheme optimizes parameters of the spline using nonlinear programming. This scheme has yet to be implemented for a spline with clothoids due to a significantly more complicated nature of this problem. Therefore, the design of route plans in existing computer aided design (CAD) systems is carried out in interactive mode using iterative selection of elements. In this regard, it makes sense to develop mathematically consistent algorithms for element-by-element approximation.Methods. The problem of element-by-element approximation by a circle and a clothoid is formalized as a lowdimensional non-linear programming problem. The objective function is the sum of squared deviations from the original points. Since a clothoid can only be represented in Cartesian coordinates by power series, there are difficulties in calculating the derivatives of the objective function with respect to the desired parameters of the spline elements. The proposed mathematically consistent algorithm for calculating these derivatives is based on the integral representation of the Cartesian coordinates of the points of the clothoid as functions of its length.Results. A mathematical model and algorithms have been developed for approximating a sequence of points on a plane by clothoids and circles using the method of nonlinear programming. A second-order algorithm is implemented with the calculation and inversion of the matrix of second derivatives (Hesse matrix).Conclusions. For approximation by circles and clothoids using nonlinear programming, it is not necessary to have an analytical expression of the objective function in terms of the required variables. The proposed algorithms make it possible to calculate not only the first, but also the second derivatives in the absence of such expressions.

Finite-range effect in the two-dimensional density-induced BCS–BEC crossover

Abstract We theoretically investigate the Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensation (BEC) crossover in a two-dimensional Fermi gas with the finite-range interaction by using the Hartree–Fock–Bogoliubov theory. Expanding the scattering phase shift in terms of the scattering length and effective range, we discuss the effect of the finite-range interaction on the pairing and thermodynamic properties. By solving the gap equation and the number equation self-consistently, we numerically calculate the effective-range dependence of the pairing gap, chemical potential, and pair size throughout the BCS–BEC crossover. Our results would be useful for further understanding of low-dimensional many-body problems.

Physics-Informed Deep Learning for Solving Coupled Electron and Phonon Boltzmann Transport Equations

Electron-phonon (e-ph) coupling and transport are ubiquitous in modern electronic devices. The coupled electron and phonon Boltzmann transport equations (BTEs) hold great potential for the simulation of thermal transport in metal and semiconductor systems. However, solving the BTEs is often computationally challenging owing to their high dimensional complexity and a wide span of heat carrier properties, which hinder large-scale thermal modeling at the device level. In this work, we present a physics-informed neural network framework for solving the coupled electron and phonon BTEs. Instead of relying on labeled data, the proposed framework directly learns the spatiotemporal solutions (i.e., the electron and phonon distribution functions) within a parameterized space by enforcing physical laws. The efficacy of this framework is demonstrated through its ability to accurately resolve temperature profiles in low-dimensional thermal transport problems and visualize the ultrafast electron and phonon dynamics in laser heating experiments on thin metal films. The results indicate that our approach can accurately describe nonequilibrium e-ph energy transfer with improved efficiency, opening alternative avenues for the predictive design and optimization of micro- and nanostructures.

Implicit function characterization of the curvilinear mask to realize parametric optical proximity correction with a neighborhood parallel tabu search.

The curvilinear mask structures provide significant benefits in improving lithographic resolution. Curvilinear masks, as opposed to rectilinear masks, have a wider range of structure types that can be used precisely to correct the contour of diffraction at sharp technological nodes. However, the curvilinear structure also makes the inverse design of mask in optical proximity correction (OPC) flow difficult. The current OPC of curvilinear masks uses pixel-based inverse optimization, which is extremely computationally intensive and takes up a lot of design data storage space. This paper proposes an implicit function to represent a large number of curve types with a small number of parameters to reduce computational complexity and the R&D cycle. Therefore, the ultra-high dimensional pixel-based OPC problem is transformed into a low-dimensional parameter search problem in the critical diffraction area of the mask pattern. The tabu search algorithm and neighborhood parallel computing strategy are then used to quickly search for optimal characterized parameters. The results of the simulation show that the parametric curvilinear OPC method achieves a higher image fidelity than that of rectilinear OPC. At the same time, it addresses the shortcomings of the traditional pixelated curvilinear mask OPC method, including high computational complexity, low manufacturability, and storage space occupancy.

Cooperative Coevolutionary CMA-ES With Landscape-Aware Grouping in Noisy Environments

Many real-world optimization tasks suffer from noise. So far, the research on noise-tolerant optimization algorithms is still restricted to low-dimensional problems with less than 100 decision variables. In reality, many problems are high-dimensional. Cooperative coevolutionary (CC) algorithms based on a divide-and-conquer strategy are promising in solving complex high-dimensional problems. However, noisy fitness evaluations pose a challenge in problem decomposition for CC. The state-of-the-art grouping methods, such as differential grouping and recursive differential grouping, are unable to work properly in noisy environments. Because it is impossible to distinguish whether the change of one variable’s difference value is caused by noise or the perturbation of its interacting variables. As a result, every pair of variables will be identified as nonseparable in these methods. In this paper, we study how to group decision variables with covariance matrix adaptation evolution strategy (CMA-ES) in noisy environments and subsequently propose a landscape-aware grouping (LAG) method. Instead of detecting pairwise interacting variables, we directly identify a nonseparable subcomponent. To this end, we propose to use two convergence features, variable convergence time and accumulative path, to describe variables’ fitness landscapes; then, variables are clustered according to these two features. Numerical experiments show that LAG can more effectively identify interactive decision variables in the presence of multiplicative noise than the differential grouping and some of its variants. Up to 500 dimensions, the performance of cooperative coevolutionary CMA-ES with landscape-aware grouping (CC-CMAES-LAG) is competitive compared with existing CC algorithms and uncertainty-handling CMA-ES (UH-CMA-ES).

Semi-Parametric Functional Calibration Using Uncertainty Quantification Based Decision Support

Abstract While most calibration methods focus on inferring a set of model parameters that are unknown but assumed to be constant, many models have parameters that have a functional relation with the controllable input variables. Formulating a low-dimensional approximation of these calibration functions allows modelers to use low-fidelity models to explore phenomena at lengths and time scales unattainable with their high-fidelity sources. While functional calibration methods are available for low-dimensional problems (e.g., one to three unknown calibration functions), exploring high-dimensional spaces of unknown calibration functions (e.g., more than ten) is still a challenging task due to its computational cost and the risk for identifiability issues. To address this challenge, we introduce a semiparametric calibration method that uses an approximate Bayesian computation scheme to quantify the uncertainty in the unknown calibration functions and uses this insight to identify what functions can be replaced with low-dimensional approximations. Through a test problem and a coarse-grained model of an epoxy resin, we demonstrate that the introduced method enables the identification of a low-dimensional set of calibration functions with a limited compromise in calibration accuracy. The novelty of the presented method is the ability to synthesize domain knowledge from various sources (i.e., physical experiments, simulation models, and expert insight) to enable high-dimensional functional calibration without the need for prior knowledge on the class of unknown calibration functions.

Bayesian Co-evolutionary Optimization based entropy search for high-dimensional many-objective optimization

Bayesian evolutionary optimization algorithms have been widely employed to solve expensive many-objective optimization problems. However, the existing approaches are generally designed for low-dimensional problems. In high-dimensional problems, the accuracy of the prediction decreases. And the acquisition function becomes ineffective. The combination of these challenges renders existing approaches unsuitable for selecting potential individual solutions for high-dimensional many-objective optimization problems. To address these limitations, we propose a novel Entropy Search-based Bayesian Co-Evolutionary Optimization approach (ESB-CEO). With the co-evolutionary algorithm as the basic optimizer, it executes an adaptive acquisition function combining the Lp-norm and information entropy to efficiently solve computationally expensive many-objective optimization problems. Individual solutions that have a significant effect on different search stages can be effectively identified, which improves the convergence and diversity of the algorithm. Extensive experimental results based on a set of expensive multi/many-objective test problems demonstrate that the proposed approach significantly outperforms five state-of-the-art surrogate-assisted evolutionary algorithms.

Simultaneous perturbation stochastic approximation: towards one-measurement per iteration

When measuring the value of a function to be minimized is not only expensive but also with noise, the popular simultaneous perturbation stochastic approximation (SPSA) algorithm requires only two function values in each iteration. In this paper, we present a method requiring only one function measurement value per iteration in the average sense. We prove the strong convergence and asymptotic normality of the new algorithm. Limited experimental results demonstrate the effectiveness and potential of our algorithm for solving low-dimensional problems.

Application of Natural Language Processing-based Emotional Semantic Analysis in the “One Core, Three Integrations” Vocal Music Teaching Model

Abstract To be able to better improve musical literacy and artistic aesthetics, this paper presents a study on the application of sentiment analysis for vocal music teaching under natural language processing. Firstly, for the training text that has been labeled with categories, a suitable classification model is built for the unlabeled test text using the model category prediction. The sum of the identified sentiment scores is calculated, and the different results are compared to set a reasonable threshold to determine the sentiment polarity. According to the mutual information statistic value of each feature item, the feature items with a mutual information statistic value higher than the threshold are retained. The weights of each feature term are normalized to the interval to which the range of weights is normalized, and the feature weights are obtained. The optimal classification surface is calculated by converting the Lagrangian optimization method into a problem of finding the pair, which makes the low-dimensional problem transformed into a high-dimensional space and combines the advantages of Bayes’ ability to automatically adjust the weight coefficients in training and the huge training capacity of neural networks to improve the flexibility and robustness of the training model. Feature extraction is performed by convolutional layers, and the dimensionality of the feature vector is reduced by the maximum pooling method to realize an emotional-semantic vocal teaching model. The results show that the proposed method can improve the flexibility and accuracy of the semantic analysis model, and also enhance the training robustness of the network, with the F1 value reaching 91.65%, which can accurately mine the application data of “one core and three integrations” vocal teaching model and improve the teaching model in universities.

Beating a Benchmark: Dynamic Programming May Not Be the Right Numerical Approach

We analyze dynamic investment strategies for benchmark outperformance using two widely used objectives of practical interest to investors: (i) maximizing the information ratio (IR), and (ii) obtaining a favorable tracking difference (cumulative outperformance) relative to the benchmark. In the case of the tracking difference, we propose a simple and intuitive objective function based on the quadratic deviation (QD) from an elevated benchmark. In order to gain some intuition about these strategies, we provide closed-form solutions for the controls under idealized assumptions. For more realistic cases, we represent the control using a neural network (NN) and directly solve a sampled optimization problem, which approximates the original optimal stochastic control formulation. Unlike the typical approach based on dynamic programming (DP), e.g., reinforcement learning, solving the sampled optimization with an NN as a control avoids computing conditional expectations and leads to an optimization problem with a small number of variables. In addition, our NN parameter size is independent of the number of portfolio rebalancing times. Under some assumptions, we prove that a traditional DP approach results in a high-dimensional problem, whereas directly solving for the control without using DP yields a low-dimensional problem. Our analytical and numerical results illustrate that, compared with IR-optimal strategies with the same expected value of terminal wealth, the QD-optimal investment strategies result in comparatively more diversified asset allocations during certain periods of the investment time horizon.

Multi-surrogate-assisted stochastic fractal search algorithm for high-dimensional expensive problems

Surrogate models have been radically used in metaheuristic algorithms owing to their capacity in solving computationally expensive problems. However, despite the promising performance of surrogate-assisted metaheuristic algorithms in coping with low-dimensional problems, they failed to tackle high-dimensional problems efficiently. Thus, a multi-surrogate-assisted stochastic fractal search algorithm (MSASFS) is proposed in this paper. Several improvements are integrated into the algorithm design: (1) By combining the original coordinate system with the eigencoordinate system, an improved surrogate-assisted differential evolution (SDE) updating mechanism is proposed to ameliorate the generalization ability of the algorithm and extend the scope of exploration. (2) A new expected improvement (EI) pre-screening strategy based on the Gaussian process (GP) model is employed to select promising candidate solutions. (3) Two different surrogate models are applied to enhance the robustness of the proposed algorithm. The effectiveness of MSASFS is further demonstrated by numerical experiments on some widely used benchmark problems with dimensions ranging from 30 to 200 and parameter estimation problem of fractional-order chaotic systems. The results reveal that, compared with state-of-the-art surrogate-assisted evolutionary algorithms (SAEAs), the proposed algorithm can effectively solve high-dimensional expensive problems. Furthermore, MSASFS shows a more significant efficiency when the dimension of problems becomes higher.