Auxiliary Problem Research Articles

A discrete sine–cosine based method for the elasticity of heterogeneous materials with arbitrary boundary conditions

The aim of this article is to extend Moulinec and Suquet (1998)’s FFT-based method for heterogeneous elasticity to non-periodic Dirichlet/Neumann boundary conditions. The method is based on a decomposition of the displacement into a known term verifying the boundary conditions and a fluctuation term, with no contribution on the boundary, and described by appropriate sine–cosine series. A modified auxiliary problem involving a polarization tensor is solved within a Galerkin-based method, using an approximation space spanned by sine–cosine series. The elementary integrals emerging from the weak formulation of the equilibrium are approximated by discrete sine–cosine transforms, which makes the method relying on the numerical complexity of Fourier transforms. The method is finally assessed in several problems including kinematic uniform, static uniform and arbitrary Dirichlet/Neumann boundary conditions.

High-order schemes of exponential time differencing for stiff systems with nondiagonal linear part

Exponential time differencing methods are a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models often possess fast oscillating or decaying modes—in other words, are stiff systems. Practical implementation of these methods for the systems with nondiagonal linear part of equations is exacerbated by infeasibility of an analytical calculation of the exponential of a nondiagonal linear operator; in this case, the coefficients of the exponential time differencing scheme cannot be calculated analytically. We suggest an approach, where these coefficients are numerically calculated with auxiliary problems. We rewrite the high-order Runge–Kutta type schemes in terms of the solutions to these auxiliary problems and practically examine the accuracy and computational performance of these methods for a heterogeneous Cahn–Hilliard equation, a sixth-order spatial derivative equation governing pattern formation in the presence of an additional conservation law, and a Fokker–Planck equation governing macroscopic dynamics of a network of neurons.

Existence and stability of standing waves for a weakly coupled nonlinear fractional Schrödinger system

We study the system of two weakly coupled nonlinear fractional Schrödinger equations{(−Δ)su+ωu=|u|2p−2u+β|u|p−2u|v|pin RN,(−Δ)sv+ωv=|v|2p−2v+β|u|p|v|p−2vin RN, where N≥2,s∈(0,1],2<2p<2s⁎=2N/(N−2s),ω>0 and β>0. For s∈(0,1] and 1<p<2, by finding a new kind of solution to the above system and comparing its energy level with those of other kinds of solutions, we show that each component of the least energy solution is nontrivial for any β>0, which forms a striking contrast with case p≥2. We emphasize that this result can be viewed as a complement to that of Guo and He (2016) [13] and Maia et al. (2006) [28]. When s∈(0,1),1<p<1+2s/N and ω>0 is fixed, we obtain the orbital stability of standing waves associated with the synchronized solution to the above system. To this end, we introduce an auxiliary mass constraint problem and establish the connection between the orbital stability of standing waves and the stability of ground state set to the auxiliary problem.

Two Numerical Schemes Via an Auxiliary Function for Nonlocal Problems With Integrable Kernels

ABSTRACTIn this study, we explore volume constrained nonlocal problem with integrable kernels. A central challenge addressed is the discontinuity of true solutions across the boundary, posing difficulties for designing efficient numerical methods. To address this issue, we introduce an auxiliary function which serves as the solution of an auxiliary problem. This approach effectively transforms the original problem to a new nonlocal problem with a true solution which is continuous across the boundary. We propose two numerical schemes to solve the new nonlocal problem: one is exclusively applicable to one‐dimensional case, while the other is suitable for general dimensional cases. Experimental results show that both schemes exhibit optimal convergence order under mild condition.

Existence of Positive Solutions for Singular Difference Equations with Nonlinear Boundary Conditions

In this paper, we delve into a discrete nonlinear singular semipositone problem, characterized by a nonlinear boundary condition. The nonlinearity, given by f(u)−auα with α&gt;0, exhibits a singularity at u=0 and tends towards −∞ as u approaches 0+. By constructing some suitable auxiliary problems, the difficulty that arises from the singularity and semipositone of nonlinearity and the lack of a maximum principle is overcome. Subsequently, employing the Krasnosel’skii fixed-point theorem, we determine the parameter range that ensures the existence of at least one positive solution and the emergence of at least two positive solutions. Furthermore, based on our existence results, one can obtain the symmetry of the solutions after adding some symmetric conditions on the given functions by using a standard argument.

Stochastic Control Problems with State Reflections Arising from Relaxed Benchmark Tracking

This paper studies stochastic control problems motivated by optimal consumption with wealth benchmark tracking. The benchmark process is modeled by a combination of a geometric Brownian motion and a running maximum process, indicating its increasing trend in the long run. We consider a relaxed tracking formulation such that the wealth compensated by the injected capital always dominates the benchmark process. The stochastic control problem is to maximize the expected utility of consumption deducted by the cost of the capital injection under the dynamic floor constraint. By introducing two auxiliary state processes with reflections, an equivalent auxiliary control problem is formulated and studied, which leads to the Hamilton-Jacobi-Bellman equation with two Neumann boundary conditions. We establish the existence of a unique classical solution to the dual partial differential equation using some novel probabilistic representations involving the local time of some dual processes together with a tailor-made decomposition-homogenization technique. The proof of the verification theorem on the optimal feedback control can be carried out by some stochastic flow analysis and technical estimations of the optimal control. Funding: L. Bo and Y. Huang are supported by the National Natural Science Foundation of China [Grant 12471451], the Natural Science Basic Research Program of Shaanxi [Grant 2023-JC-JQ-05], the Shaanxi Fundamental Science Research Project for Mathematics and Physics [Grant 23JSZ010], and Fundamental Research Funds for the Central Universities [Grant 20199235177]. X. Yu is supported by the Hong Kong RGC General Research Fund (GRF) [Grant 15304122] and the Research Centre for Quantitative Finance at the Hong Kong Polytechnic University [Grant P0042708].

Modification of the dimensional doma method for linear elliptic equations

Currently, one of the most common methods for the numerical solution of boundary value problems are difference methods (grid method, finite difference method). These methods are well suited for solving problems in regular domains. In areas of complex structure, their use leads to a number of difficulties both in the construction of difference schemes, and in the implementation on a computer. One of the methods to overcome these difficulties is the method of fictitious regions. There are four interrelated directions in the development of the method of fictitious regions: study of various ways to continue the original problems in a fictitious area; obtaining unimprovable estimates in stronger metrics; extension of the class of problems on the application of the method of fictitious areas; construction of effective difference schemes for solving auxiliary problems constructed by the method of fictitious areas. For the first time, the fictitious domain method for the Navier-Stokes equation was studied in the work of A.N. Bugrova, Sh. Smagulov, Smagulov Sh. This article discusses the method of additional areas, which is an analogue of the method of fictitious areas for solving linear boundary value problems, but in the auxiliary problem of this method there is no small perimeter. This method is based on the variational principle of solving linear boundary value problems of elliptic type in an arbitrary region. The existence of a solution is proved by the Galerkin uniqueness method.

Does the type of transportation mode matter in the effect of freight transportation on the environmental pollution of OECD countries?

This study aims forward to explain the effect of freight transportation on environmental pollution and to investigate the role played by the usage rates of road and railway transportation modes in this relationship. As an auxiliary problem of the research, it was tried to determine whether there is a threshold value for these types of transportation and what consequences will arise in terms of environmental pollution below and above this threshold. For this purpose, the data of OECD countries for the period 2000-2018 were analyzed with the PSTR method. Analysis results showed a non-linear relationship between freight transportation and environmental pollution and that the road/rail transportation ratio plays an important role in this relationship. The determined threshold value is 68.18% for road freight transport and 29.55% for rail freight transport. Below the determined threshold values, as the amount of cargo transported increases, the rate of carbon emission increase gradually increases in both types of transport. Above the threshold values, road transport increases the rate of increase even more, while rail transport decreases this rate of increase. Within the scope of these results, countries should develop policies to increase the use of railways in freight transport while reducing the use of roads.

A Modified Robust Homotopic Method for the Low-Thrust Fuel-Optimal Orbital Control Problem

This study focuses on improving the application of the quadratic penalty smoothing technique for solving the fuel-optimal problem of low-thrust trajectory. Some practical techniques, including the Particle Swarm Optimization algorithm (PSO), the co-state normalization technique, and the single shooting method, are employed. These approaches can enhance convergence to the globally optimal solution. The application of PSO is improved through the introduction of the diversity termination term. Furthermore, a new parameter, the maximum thrust to initial mass ratio (CTm), is introduced and analyzed. The relationships among Lagrange multipliers for different CTm in the energy-optimal auxiliary problem are also derived. These relationships can be used in scenarios with high CTm to reduce the variable search space, leading to a significant decrease in computation time. Nonetheless, few researchers have proposed a well-defined strategy for the homotopic process. We propose a robust iteration strategy, involving the determination of the maximum step size and the distinction of sub-optimal solutions. This iterative strategy could ensure a robust convergence toward the optimal solutions. An interplanetary rendezvous example is provided to demonstrate the effectiveness of the presented techniques and strategy.

Joint multi-feature extraction and transfer learning in motor imagery brain computer interface

Motor imagery brain computer interface (BCI) systems are considered one of the most crucial paradigms and have received extensive attention from researchers worldwide. However, the non-stationary from subject-to-subject transfer is a substantial challenge for robust BCI operations. To address this issue, this paper proposes a novel approach that integrates joint multi-feature extraction, specifically combining common spatial patterns (CSP) and wavelet packet transforms (WPT), along with transfer learning (TL) in motor imagery BCI systems. This approach leverages the time-frequency characteristics of WPT and the spatial characteristics of CSP while utilizing transfer learning to facilitate EEG identification for target subjects based on knowledge acquired from non-target subjects. Using dataset IVa from BCI Competition III, our proposed approach achieves an impressive average classification accuracy of 93.4%, outperforming five kinds of state-of-the-art approaches. Furthermore, it offers the advantage of enabling the design of various auxiliary problems to learn different aspects of the target problem from unlabeled data through transfer learning, thereby facilitating the implementation of innovative ideas within our proposed approach. Simultaneously, it demonstrates that integrating CSP and WPT while transferring knowledge from other subjects is highly effective in enhancing the average classification accuracy of EEG signals and it provides a novel solution to address subject-to-subject transfer challenges in motor imagery BCI systems.

Improved a posteriori error bounds for reduced port-Hamiltonian systems

Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem, to the case of port-Hamiltonian systems. The approaches rely on a secondary approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are adapted to port-Hamiltonian systems. The mathematical relationship between the two methods is discussed both theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid–structure interaction.

A REFINED CLASSIFICATION OF THE SUBSET SUM PROBLEM IN POLYNOMIAL TIME COMPLEXITY

The problem considered is the selection of at least one subset from a set (array) of distinct positive integers, such that the sum of the subset's elements exactly matches a given target sum (target certificate). According to R. M. Karp, this problem belongs to the class of NP-complete problems. Diophantine equations and an auxiliary problem, which facilitates the solution of the original problem and has independent scientific interest, have been introduced. A novel method has been developed, which includes proven lemmas and theorems. These results enable the development of efficient and straightforward algorithms for solving the subset sum problem. The time and space complexity for selecting the required subsets do not exceed the square of the length of the original set. An analytical framework has been proposed for managing indices within the original set. These algorithms are applicable to solving problems related to the independent set of cardinality k and the k-vertex cover problem. Additionally, we present examples to confirm claimed results. It should be noted that the time complexity of sorting an array of integers is proportional to the square of the array's size, and this problem belongs to class P. Therefore, based on the newly developed method, it can be inferred that the subset sum problem, originally classified as NP-complete within the NP class, also belongs to P.

Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method

Abstract In this study, a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is considered. The addition of a new function converts the system of Volterra-Fredholm integro-differential equations to a system of Fredholm integro-differential equations. In contrast to the original problem, the dimension of a Fredholm integro-differential equation is determined by the number of matrices in the degenerate kernel of the Volterra integral. A numerical algorithm of Dzhumabaev parameterization method for addressing a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is proposed. The main advantage of the proposed method is splitting the problem into auxiliary Cauchy problems for ordinary differential equations and a system of algebraic equations with respect to the parameters. The conditions for the unique solvability of the multipoint boundary value problem for Fredholm integro-differential equations are established. Finally, various numerical examples are provided to demonstrate the efficiency and correctness of the suggested technique.

Multitasking optimization for the imaging problem in electrical capacitance tomography

Electrical capacitance tomography excels in measuring multiphase flows, but it faces challenges in reconstructing high-accuracy images. To tap into the potential of this measurement technique, we model the image reconstruction problem as a bilevel multitasking optimization problem, aiming to enhance imaging quality via effective knowledge transfer and integration across various tasks. This new imaging model consists of a target optimization problem and an auxiliary optimization problem, facilitating knowledge transfer and fusion between tasks. We design a new algorithm to solve the target optimization problem and integrate it with an autoencoder used for the cross-task knowledge transfer, forming a new multitasking optimizer to solve this proposed bilevel multitasking optimization imaging model. The extreme learning machine is developed for the prediction of learnable prior images by introducing a novel bilevel optimization framework that enables concurrent parameter selection and model training. A novel nested algorithm, designed to efficiently address optimization problems structured hierarchically, is developed to solve the training model. Evaluation results show that the new imaging algorithm outperforms popular imaging methods in terms of reconstruction accuracy and robustness, while also demonstrating stable performance. This proposed imaging algorithm achieves cross-task knowledge transfer and fusion, integrates measurement physics and machine learning, mines and integrates effective image priors, adaptively learns model parameters, alleviates the ill-posed property of the image reconstruction problem, increase the automation of the model, and improves imaging quality, robustness and performance stability, which provides a comprehensive solution to mitigate challenges in imaging.

The maximum norm error estimate and Richardson extrapolation methods of a second-order box scheme for a hyperbolic-difference equation with shifts

The study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ 2 + h 2) in L ∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ 4 + h 4) in L ∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings.

A parameter uniform numerical method on a Bakhvalov type mesh for singularly perturbed degenerate parabolic convection–diffusion problems

AbstractWe are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper.

Innovative air supply management in fuel cells: An economic predictive control approach without pre-measured OER

Existing fuel cell air supply control methods improve the economic performance of Proton Exchange Membrane Fuel Cells (PEMFC) under steady-state conditions through oxygen excess ratio curve, while ignoring the transient process of the air supply system under varying operating conditions. Furthermore, the oxygen excess ratio curve may exhibit deviations due to fuel cell aging and measurement inaccuracies, which would further diminish the economic performance derived from the control process. Not relying on pre-measuring the oxygen excess ratio curve, an EMPC (Economic Model Predictive Control) method is proposed for fuel cell air supply systems. It incorporates an economic cost function reflecting the economic objectives of the air supply system. By dynamically optimizing the parasitic power of the air compressor, this method enhances the overall economic performance by achieving performance optimization for both transient and steady-state processes of the fuel cell system. The asymptotic stability of PEMFC is achieved through the utilization of terminal penalty functions and terminal domain constraints. Stability and convergence analysis is carried out by employing auxiliary optimization problems and the Lyapunov method. In transient processes, EMPC method can achieve a maximum 4.97 % improvement in the economic performance of PEMFC. A hardware in loop (HIL) experiment was finally conducted to show the real-time capacity of the proposed EMPC method.

Indirect stabilization of locally weakly coupled second‐order evolution systems of hyperbolic type

The purpose of this work is to investigate the stabilization of locally weakly coupled second‐order evolution equations of hyperbolic type, where only one of the two equations is directly damped. As such system cannot be exponentially stable, we are interested in polynomial energy decay rates. Our main contributions concern strong abstract and polynomial stability properties based on stability properties of two auxiliary problems: the sole damped equation and the second equation with a damping related to the coupling operator. The main novelty is that the polynomial energy decay rates are obtained in different important situations not covered before; let us mention the case of a coupling operator not partially coercive and not necessarily bounded. The main tools are the use of the frequency domain approach combined with new multipliers technique based on the solutions of the resolvent equations of the two auxiliary problems mentioned before. Finally, our abstract framework is illustrated by several concrete examples not treated before.

Discrete inverse problem for a hyperbolic equation, properties of the solution to a discrete direct and auxiliary discrete problems

This paper considers the formulation of a discrete inverse problem for a hyperbolic equation. First, the continuous inverse problem is reduced to a convenient form for research. In the inverse problem, the required function is considered even. Since the Dirac delta function is present in the problem data, the structure of a generalized solution to the Cauchy problem for a hyperbolic equation is determined. The solution to the Cauchy problem for a hyperbolic equation is determined only for positive values in time, therefore the solution to the Cauchy problem for negative values in time is determined using odd continuation. After some transformations, the formulation of the continuous inverse problem is reduced to a form convenient for research. A grid domain is introduced, and for all functions in the problem statement the corresponding grid functions and a discrete analogue of the Dirac delta function are determined. Differential operators, initial conditions and additional data of the inverse problem are approximated by finite differences. Assuming that a solution to the discrete inverse problem exists, we prove the data lemma of the discrete inverse problem. In order to study the discrete inverse problem for a hyperbolic equation, a theorem on the existence and uniqueness of the discrete direct problem is proved, as well as on the properties of the solution to this discrete problem. In the course of proving the theorem, a discrete analogue of d'Alembert's formula for solving the Cauchy problem for a hyperbolic equation was obtained. The theorem on the existence of a unique solution to the auxiliary discrete problem and its properties is proved.

Distributed economic model predictive control for the joint energy dispatch of wind farms and run-of-the-river hydropower plants

This study addresses the energy dispatch problem of a virtual power plant (VPP) acting as a price-tacker in the day-ahead electricity market. The VPP comprises wind farms and a cascade of run-of-the-river hydropower plants. Even if the storage capacity of the cascade is limited, it can still be exploited to compensate the variability of wind. This implies dispatching the water reservoirs near to real-time, while accounting for complex constraints and various sources of uncertainty. To this aim, we present a control strategy based on economic model predictive control (MPC), which is then decomposed using the auxiliary problem principle. As a distinctive feature, the proposed algorithm is fully-distributed, i.e. no central coordinator is required. Compared to centralized MPC, the distributed algorithm brings a ∼10% reduction in the average execution time of the controller. Moreover, the joint operation of hydropower and wind is shown to enhance the economic value of both assets.