Numerical Experiments Research Articles

A fast local search based memetic algorithm for the parallel row ordering problem

The parallel row ordering problem (PROP) is concerned with arranging two groups of facilities along two parallel lines with the goal of minimizing the sum of the flow cost-weighted distances between the pairs of facilities. As the main result of this paper, we show that the insertion neighborhood for the PROP can be explored in optimal time Θ(n2) by providing an O(n2)-time procedure for performing this task, where n is the number of facilities. As a case study, we incorporate this procedure in a memetic algorithm (MA) for solving the PROP. We report on numerical experiments that we conducted with MA on PROP instances with up to 500 facilities. The experimental results demonstrate that the MA is superior to the adaptive iterated local search algorithm and the parallel hyper heuristic method, which are state-of-the-art for the PROP. Remarkably, our algorithm improved best known solutions for six largest instances in the literature. We conjecture that the time complexity of exploring the interchange neighborhood for the PROP is Θ(n2), exactly as in the case of insertion operation.

Hybrid Hu-Storey type methods for large-scale nonlinear monotone systems and signal recovery

We propose two hybrid methods for solving large-scale monotone systems, which are based on derivative-free conjugate gradient approach and hyperplane projection technique. The conjugate gradient approach is efficient for large-scale systems due to low memory, while projection strategy is suitable for monotone equations because it enables simply globalization. The derivative-free function-value-based line search is combined with Hu-Storey type search directions and projection procedure, in order to construct globally convergent methods. Furthermore, the proposed methods are applied into solving a number of large-scale monotone nonlinear systems and reconstruction of sparse signals. Numerical experiments indicate the robustness of the proposed methods.

Is maximum tolerated dose (MTD) chemotherapy scheduling optimal for glioblastoma multiforme?

In this study, we investigate a control problem involving a reaction–diffusion partial differential equation (PDE). Specifically, the focus is on optimizing the chemotherapy scheduling for brain tumor treatment to minimize the remaining tumor cells post-chemotherapy. Our findings establish that a bang-bang increasing function is the unique solution, affirming the MTD scheduling as the optimal chemotherapy profile. Several numerical experiments on a real brain image with parameters from clinics are conducted for tumors located in the frontal lobe, temporal lobe, or occipital lobe. They confirm our theoretical results and suggest a correlation between the proliferation rate of the tumor and the effectiveness of the optimal treatment.

Construction of mathematical models of thermal conductivity for modern electronic devices with elements of a layered structure

This paper considers a heat conduction process for isotropic layered media with internal thermal heating. As a result of the heterogeneity of environments, significant temperature gradients arise as a result of the thermal load. In order to establish the temperature regimes for the effective operation of electronic devices, linear and non-linear mathematical models for determining the temperature field have been constructed, which would make it possible to further analyze the temperature regimes in these heat-active environments. The coefficient of thermal conductivity for the above structures was represented as a whole using asymmetric unit functions. As a result, the conditions of ideal thermal contact were automatically fulfilled on the surfaces of the conjugation of the layers. This leads to solving one heat conduction equation with discontinuous and singular coefficients and boundary conditions at the boundary surfaces of the medium. For linearization of nonlinear boundary value problems, linearizing functions were introduced. Analytical solutions to both linear and nonlinear boundary value problems were derived in a closed form. For heat-sensitive environments, as an example, the linear dependence of the coefficient of thermal conductivity of structural materials on temperature, which is often observed when solving many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in these environments were obtained. Based on this, a numerical experiment was performed, and it was geometrically represented depending on the spatial coordinates. This proves that the constructed linear and nonlinear mathematical models testify to their adequacy to the real physical process. They make it possible to analyze heat-active media regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements but also the entire structure

Asymptotically optimal energy consumption and inventory control in a make-to-stock manufacturing system

We study a make-to-stock manufacturing system in which a single server makes the production. The server consumes energy, and its power consumption depends on the server state: a busy server consumes more power than an idle server, and an idle server consumes more power than a turned-off server. When a server is turned on, it completes a costly set-up process that lasts a while. We jointly control the finished goods inventory and the server’s energy consumption. The objective is to minimize the long-run average inventory holding, backorder, and energy consumption costs by deciding when to produce, when to idle or turn off the server, and when to turn on a turned-off server. Because the exact analysis of the problem is challenging, we consider the asymptotic regime in which the server is in the conventional heavy-traffic regime. We formulate a Brownian control problem (BCP) with impulse and singular controls. In the BCP, the impulse control appears due to server shutdowns, and the singular control appears due to server idling. Depending on the system parameters, the optimal BCP solution is either a control-band or barrier policy. We propose a simple heuristic control policy from the optimal BCP solution that can easily be implemented in the original (non-asymptotic) system. Furthermore, we prove the asymptotic optimality of the proposed control policy in a Markovian setting. Finally, we show that our proposed policy performs close to optimal in numerical experiments.

Rosen–Zener transition in a nonlinear nonreciprocal two-level system

The dynamics of the Rosen–Zener transition (RZT) in interacting two-level systems within nonreciprocal systems is examined, where nonreciprocity stems from asymmetric couplings between energy levels. For the linear case, an analytical expression for the RZT probability is derived. In the nonlinear scenario, numerical experiments are utilized to investigate the impact of various initial states and scanning periods on transition probabilities under adiabatic conditions. Additionally, an equivalent classical Hamiltonian is formulated to elucidate the origin of transition asymmetry under different initial state conditions. The influence of energy level detuning on the RZT is explored, leading to the suggestion of new applications for studying parameters of two-level systems related to nonlinearity and nonreciprocity. This opens avenues for potential Rosen–Zener–Stückelberg interferometry implementations.

Numerical prediction of beam capacity using the new constitutive model based on concrete biaxial behavior

The biaxial strength of concrete exhibits intriguing nonlinearity and scattering, posing challenges to the assessment of building structural safety. Experiments have confirmed that biaxial strength envelopes can manifest concave, quasi-linear, and convex shapes. Despite various empirical formulas being proposed to describe this behavior, they cannot be applied to finite element simulations due to the absence of a constitutive perspective. To study the mechanisms forming the shape of strength envelopes, a novel three-invariant model constitutive model is proposed, which reveals the characteristics of the envelopes resulting from the coupling effect of hydrostatic pressure and Lode angle. By adjusting the model parameters, it effectively describes concave, quasi-linear, and convex envelopes. The model's accuracy and compatibility were validated against experimental data spanning various concrete types, with compression strengths ranging from 19.29 to 96.5 MPa and tension-compression ratios form 0.056 to 0.095. To investigate how biaxial strength impacts the capacity of beams, numerical experiments were conducted on notched and intact beams based on the proposed model. The results underscored that beams prepared using concrete with convex envelopes exhibited higher capacity compared to those with quasi-linear envelopes. Therefore, conducting biaxial tension-compression experiments is recommended to accurately evaluate beam bearing capacity. The proposed model is expected to have further applications in the safety evaluation of building structures.

The Stress State in an Elastic Disk Due to a Temperature Variation in One Sector

Abstract Consider a linear elastic infinite disk, a sector of which, of arbitrary opening angle 2β, is subjected to a uniform temperature increase ΔT with respect to the complementary portion. An analytical solution is sought, imagining that the disk is first cut along the interface with the heated sector, now free to expand; then the two parts are re-joined and the thermal mismatch is annihilated by arrays of glide dislocations, distributed along the interfaces. A sequence of approximate solutions is found as the length of the arrays of reconciling dislocations is increased, characterized by a logarithmic stress singularity at the sector tip. However, modulo the particular case 2β=π, the stress grows unboundedly when the length of the dislocation arrays tends to infinity. This is in agreement with the predictions from dimensional analysis, because for the infinite disk problem there is no internal length scale. If the disk is finite in size, its radius R represents an additional length scale enriching the class of solutions, but the analytical treatment results much more complicated. Therefore, we propose to correlate the solution of this problem with that of an infinite disk for which the length of the arrays of reconciling dislocations is finite and depends upon R. An excellent agreement with numerical experiments in abaqus is thus found. An approach of this type can be useful in many engineering problems for which the limit condition of infinite body, though leading to analytic simplifications, could imply spurious results.

Expanding the reach of quantum optimization with fermionic embeddings

Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the special orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.

Nonlinear steering control law under input magnitude and rate constraints with exponential convergence

AbstractA ship steering control law is designed for a nonlinear maneuvering model whose rudder manipulation is constrained in both magnitude and rate. In our method, the tracking problem of the target heading angle with input constraints is converted into the tracking problem for a strict-feedback system without any input constraints. To derive this system, hyperbolic tangent ($$\tanh$$ tanh ) function and auxiliary variables are introduced to deal with the input constraints. Furthermore, using the feature of the derivative of $$\tanh$$ tanh function, auxiliary systems are successfully derived in the strict-feedback form. The backstepping method is utilized to construct the control input for the resulting cascade system. The proposed steering control law is verified in numerical experiments, and the result shows that the tracking of the target heading angle is successful using the proposed control law.

A Method for Real-Time Measurement of the Vertical Vortex at Flood Discharge Outlets Using Ultrasonic Sensors.

In this study, ultrasonic sensors were used to measure the vertical vortex at flood discharge outlets in real time, and numerical simulations and model experiments were conducted. When a sound signal passes through a vortex, its propagation characteristics will change, which helps to determine the existence of the vortex. Moreover, its characteristic parameters can be obtained through inversion. In this paper, first, the theories of acoustic measurement methods were introduced and their feasibility was verified through a comparison between Particle Image Velocimetry (PIV) measurement and numerical simulation results. Then, the Computational Fluid Dynamics (CFD) method was used to simulate the vertical vortex at the flood discharge outlets of hydraulic structures and the simulation data were restored to the actual size at scale. Finally, acoustic numerical simulations of actual vortex data were conducted, and ultrasonic sensors were used to measure the velocity of a simplified vertical vortex model under laboratory conditions. The research results indicate that the acoustic measurement method proposed in this article is effective in the measurement of the characteristic parameters of vertical vortex with a core radius of 0.03~0.05 m and a maximum tangential velocity of 0.5 m/s, the measurement error of the maximum tangential velocity is within 10%.

Comparative analysis of different spacing policies for longitudinal control in vehicle platooning

Most existing automated driving vehicles in the platoon equipped with an Adaptive Cruise Control (ACC) systems and a Cooperative Adaptive Cruise Control (CACC) systems have mainly focused on enhancing safety, improving traffic efficiency problems, and reducing workload. The spacing strategy is the core of all platoon designs, and the performance of an ACC/CACC systems hinges on the select of the spacing strategy. Although, in the literature, there are many papers dealing with platoon control, detailed explanations of the operating mechanisms of two types of spacing policies including the Headway Spacing Strategy (CTHS), and Constant Spacing Strategy (CSS), and comparative studies on them are still lacking. This work presents the studies of the longitudinal control strategy of a platoon of vehicles equipped with the existing ACC systems and CACC systems under two different spacing policies to evaluate the performances. The contributions in this work are carefully reviewed and the operating mechanisms and characteristics of two different spacing policies: the CTHS and CSS, the general evaluation criteria for spacing strategies are provided and their advantages and disadvantages are based on the numerical results. Both numerical simulations and experiments with a platoon of smart cars in real-time have demonstrated the effectiveness and practicability of the presented methodology.

A three-directional stress-strain model-based physics-embedded prediction framework for metal tube full-bent cross-sectional characteristics

A metal tube system is known as the industrial blood vessel, among which the bent section is the most vulnerable part. The cross-sectional defects (CSDs) of the bent tube cause the flow fluctuation of the fluid inside the tube. The existing defect characterization methods are roughly presented by describing CSDs in some specific cross-sections, which results in the lack of the tube full-bent section (FBS) characteristic information. To comprehensively describe and predict the tube FBS characteristics, an advanced physics-embedded CSDs prediction framework is proposed. This framework includes an FBS-neutral layer displacement angle (NLDA) prediction module and an FBS-CSDs prediction module, which uses the method that integrates the analytical model and BiLSTM-based deep learning (DL) models to predict the CSDs in the FBS of the tube. A novel analytical model of CSDs that considers both three-directional stresses and strains during tube bending is embedded in the FBS-CSDs prediction module. The analytical model provides the initial predicted values of CSDs through the NLDA sequence obtained from the FBS-NLDA module. The inaccurate CSDs are then treated as physical information to be fed into DL models for further correction and prediction. The prediction performance of this framework is validated through numerical simulations and experiments. The results prove that the framework can accurately predict the CSDs in the tube FBS. The integration of DL models with the analytical model not only overcomes the limitations of the analytical model, but also improves the prediction accuracy and convergence speed of DL models.

On quadrature of highly oscillatory Bessel function via asymptotic analysis of simplex integrals

In this article, two methods for evaluating highly oscillatory Bessel integrals are explored. Firstly, a polynomial is analyzed as an effective approximation of the simplex integral of a highly oscillatory Bessel function based on Laplace transform, and its error rapidly decreases as the frequency increases. Furthermore, the inner product of f and highly oscillatory Bessel function can be approximated by two other forms of inner product by which one depends on a polynomial and the higher derivatives of f, another depends on Bessel function and the interpolation polynomial of f. In addition, three issues related to highly oscillatory Bessel integrals have also been discussed: inequalities for the convergence rate of Filon-type methods, evaluation of Cauchy principal values, and simplified evaluation on infinite intervals. Through some preliminary numerical experiments, our theoretical analysis has been preliminarily confirmed, and the proposed numerical method is accurate and effective.

Analysis and optimal control of a hierarchical age-structured population model in a polluted environment

For individual biological populations, examining the impacts of competition within the species and the presence of toxic substances in the environment can effectively facilitate control measures. However research in this area remains limited. This article introduces a toxicant-population model incorporating a hierarchical age-structured and examines the minimum deviation problem within this model. The aim is to explore effective control strategies that strike a balance between improving the environment and population benefits. Through a novel approach, we reformulate the problem as an abstract Cauchy problem on a non-dense domain and then apply integrated semigroup theory to establish the existence, uniqueness, and positivity of solutions to the model. The existence of optimal control is derived using Ekeland's variational principle in functional analysis. The expression for optimal control is given by the Pontryagin's minimum principle. Lastly, we conducted numerical experiments on the model, applied optimality conditions to delineate optimal control, and illustrated the comparison between crucial population parameters under optimal control and those without any control.

Optimization and evaluation of thermo-hydraulic performance of solar air heaters equipped with different roughness geometries

Solar air heaters have poor thermal efficiency. Applying artificial roughness elements is a common approach to enhance thermal performance of a solar air heater. In this study, fluid flow and heat transfer characteristics of solar air heaters equipped with six different roughness geometries, including inline cubical elements, staggered cubical elements, rectangular baffles, square ribs, V-shaped baffles, and V-shaped ribs, were assessed. First, parameters of each roughness geometry were optimized and then thermo-hydraulic performance of solar air heaters having optimum roughness geometries was compared. For each roughness geometry, two design parameters were selected and 25 numerical experiments were designed based on the full factorial design of experiment approach. The thermo-hydraulic performance parameter (THPP) was set out as the objective function and its relationship with roughness parameters was defined using the non-parametric regression response surface method. Finally, the optimum values of roughness parameters were determined using the nonlinear programming by quadratic Lagrangian (NPQL) technique. Comparing the performance of solar air heaters having optimum roughness geometries showed that rectangular baffles led to the largest and inline cubical elements brought about the smallest THPP at almost all Reynolds numbers. The maximum THPP of rectangular baffles was 2.28 at the Reynolds number of 25,000.

Higher order numerical approximations for non-linear time-fractional reaction–diffusion equations exhibiting weak initial singularity

In the present study, we introduce a high-order non-polynomial spline method designed for non-linear time-fractional reaction–diffusion equations with an initial singularity. The method utilizes the L2-1σ scheme on a graded mesh to approximate the Caputo fractional derivative and employs a parametric quintic spline for discretizing the spatial variable. Our approach successfully tackles the impact of the singularity. The obtained non-linear system of equations is solved using an iterative algorithm. We provide the solvability of the novel non-polynomial scheme and prove its stability utilizing the discrete energy method. Moreover, the convergence of the proposed scheme has been established using the discrete energy method in the L2 norm. It is proven that the method is convergent of order min{θα,2} in the temporal direction and 4.5 in the spatial direction, where α∈(0,1) denotes the order of the fractional derivative and the parameter θ is utilized in the construction of the graded mesh. Finally, we conduct numerical experiments to validate our theoretical findings and to illustrate how the mesh grading influences the convergence order when dealing with a non-smooth solution to the problem.

The fast committor machine: Interpretable prediction with kernels.

In the study of stochastic systems, the committor function describes the probability that a system starting from an initial configuration x will reach a set B before a set A. This paper introduces an efficient and interpretable algorithm for approximating the committor, called the "fast committor machine" (FCM). The FCM uses simulated trajectory data to build a kernel-based model of the committor. The kernel function is constructed to emphasize low-dimensional subspaces that optimally describe the A to B transitions. The coefficients in the kernel model are determined using randomized linear algebra, leading to a runtime that scales linearly with the number of data points. In numerical experiments involving a triple-well potential and alanine dipeptide, the FCM yields higher accuracy and trains more quickly than a neural network with the same number of parameters. The FCM is also more interpretable than the neural net.

On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields

Deep Learning is having a remarkable impact on the design of Reduced Order Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited as a powerful tool for tackling complex problems for which classical methods might fail. In this respect, deep autoencoders play a fundamental role, as they provide an extremely flexible tool for reducing the dimensionality of a given problem by leveraging on the nonlinear capabilities of neural networks. Indeed, starting from this paradigm, several successful approaches have already been developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs). Nevertheless, when it comes to stochastic problems parameterized by random fields, the current understanding of DL-ROMs is mostly based on empirical evidence: in fact, their theoretical analysis is currently limited to the case of PDEs depending on a finite number of (deterministic) parameters. The purpose of this work is to extend the existing literature by providing some theoretical insights about the use of DL-ROMs in the presence of stochasticity generated by random fields. In particular, we derive explicit error bounds that can guide domain practitioners when choosing the latent dimension of deep autoencoders. We evaluate the practical usefulness of our theory by means of numerical experiments, showing how our analysis can significantly impact the performance of DL-ROMs.

A new filled function method for solving constrained global optimization problems

ABSTRACT Filled function methods have been considered as effective algorithms for solving global optimization problems. However, their effectiveness is greatly affected by the selection of parameters, the noncontinuous or non-differentiable properties of the constructed filled function. In addition, many of the constructed filled functions are only for unconstrained optimization problems, and they are unable to solve constrained optimization problems. In this paper, a new filled function is constructed for solving constrained global optimization problems. The new filled function has only one parameter which needs to be adjusted, and, when the objective functions and constrained functions are all continuously differentiable functions, the constructed filled function is also a continuously differentiable function. Then, the classical local optimization methods can be used to find a better minimum of the proposed filled function and a few parameter adjustments are needed. At last, a new filled function algorithm for constrained global optimization is developed based on the proposed filled function. The new algorithm is applied to several test examples. The results of the numerical experiments show that the new filled function algorithm is effective and efficient.