Subspace Of Solutions Research Articles

An incremental method-based machine learning approach for max–min knapsack with multiple scenarios

In this paper, the max–min knapsack problem with multiple scenarios is addressed using an incremental method-based machine learning approach. The method is structured around three key phases: a learning phase, an exploitation phase, and an exploration phase. During the learning phase, the algorithm initiates by generating a sequential set of initial elite solutions through a Q-table scoring procedure. Moving on to the exploitation phase, the goal is to improve the quality of the current solution through an intensification-based tabu search, emphasizing iterative solution optimization. The exploration phase introduces controlled randomness to diversify exploration, revealing promising sub-spaces and facilitating the discovery of new solutions. This iterative process results in a series of new elite solutions, some of which are combined to highlight the method’s efficiency. To maintain diversity, a new fusion operator is introduced, where couples related to elite solutions are combined, forming an incremental subset of elite solutions capable of exploring new solution subspaces. The proposed method is evaluated using benchmark instances from the literature, and its results are compared to those provided by recently published algorithms. The computational analysis underscores the algorithm’s competitiveness in producing high-quality solutions, often surpassing previously published methods with superior solution bounds.

A multi-scale model-order reduction strategy for vibration analysis of coupled structures with local inhomogeneities

Although the wave finite-element method (WFEM) is an efficient numerical technique to analyze wave propagation characteristics and forced response of waveguides at medium and high frequencies, its application to large-scale and complex structures is still a challenge. In this paper, we introduce a model-order reduction (MOR) strategy for the WFEM to rapidly investigate the dynamics of coupled elastic systems involving one-dimensional waveguides with local inhomogeneities, such as steel bridges, fuselages and pipeline systems. The proposed method combines the advantages of mode-based component mode synthesis (CMS) and wave-based WFEM. A reduced modal basis is first established to describe the internal degrees of freedom using CMS. Subsequently, a reduced solution subspace considering the contribution of propagating and evanescent waves is constructed by the duality of Bloch’s formulations for the waveguide part, while wave scattering characteristics of the coupled element are used to quantify the influence of fixed-interface modes determined by CMS. Structural dynamic response can be then efficiently calculated using a dynamic stiffness matrix method. The performance of the MOR method in dispersion analysis and forced-response calculation is elaborately illustrated via an example of a steel beam with a diaphragm. The influence of the diaphragm on structural dynamic response is explained from the perspective of wave-propagation characteristics. Finally, the method is applied to a real steel–concrete composite girder, and the effects of the number and thickness of diaphragms on the vibration of each plate of the girder are discussed to provide some suggestions on vibration control.

FastSVD-ML–ROM: A reduced-order modeling framework based on machine learning for real-time applications

Digital twins have emerged as a key technology for optimizing the performance of engineering products and systems. High-fidelity numerical simulations constitute the backbone of engineering design, providing insight into the performance of complex systems. However, large-scale, dynamic, non-linear models require significant computational resources and are prohibitive for real-time digital twin applications. To this end, reduced order models (ROMs) are employed, to approximate the high-fidelity solutions while accurately capturing the dominant aspects of the physical behavior. The present work proposes a new machine learning (ML) platform for the development of ROMs to handle large-scale numerical problems dealing with transient nonlinear partial differential equations. Our framework, named as FastSVD-ML-ROM, utilizes (i) a singular value decomposition (SVD) update methodology, to compute a linear subspace of the multi-fidelity solutions during the simulation process, (ii) convolutional autoencoders for nonlinear dimensionality reduction, (iii) feed-forward neural networks to map the input parameters to the latent spaces, and (iv) long–short term memory networks to predict and forecast the dynamics of parametric solutions. The efficiency of the FastSVD-ML-ROM framework is demonstrated for a 2D linear convection–diffusion benchmark, the problem of fluid flow around a cylinder, the 2D lid-driven cavity problem at high Reynolds numbers, and the 3D blood flow inside an arterial segment. The accuracy of the reconstructed results indicates the robustness of the proposed approach.

Feedback of Control on Mathematics: Bettering Iterative Methods by Observer System Design

The control approaches generally resort to the tools from the mathematics, but whether and how the mathematics can benefit from the control approaches is unclear. This article aims to bring the “control design” idea into the mathematics by providing an observer-based iterative method that focuses on solving linear algebraic equations (LAEs). An inherent relationship is revealed between the problem-solving of LAEs and the design of observer-based control systems, with which the iterative method for solving LAEs is exploited based on the design of the basic state observers. It is shown that all (least squares) solutions for any (un)solvable LAEs can be determined exponentially fast or monotonically with the different selections of initial conditions. Moreover, the general solution subspace and particular (least squares) solutions of LAEs are related closely with the unobservable subspace and observable states of their associated observer systems, respectively. Through incorporating the design idea of the deadbeat control, the solving of LAEs can be realized within only finite iterations. In particular, our proposed iterative method can be leveraged to develop a new observer-based design algorithm for traditional two-dimensional iterative learning control to realize the perfect tracking tasks.

A Majorization-Minimization Golub-Kahan bidiagonalization method for <inline-formula><tex-math id="M1">$ \ell_{2}-\ell_{q} $</tex-math></inline-formula> mimimization with applications in image restorization

An image restorization problem is often modelled as a discrete ill-posed problem. In general, its solution, even if it exists, is very sensitive to the perturbation in the data. Regularization methods reduces the sensitivity by replacing this problem with a minimization problem with a fidelity term and $ \ell_{q} $ regularization term. In order to improve the sparsity of the solution, we only consider the case of $ 0<q\leq1 $ in this paper. This paper presents a majorization-minimization Golub-Kahan bidiagonalization algorithm to solve this kind of minimization problems. The solution subspace is extended by the Golub-Kahan bidiagonalization process. The restarted case is also considered. The regularization parameter is determined by using the discrepancy principle. Several examples in image restorization are shown for the proposed methods.

Some numerical aspects of Arnoldi-Tikhonov regularization

Large linear discrete ill-posed problems are commonly solved by first reducing them to small size by application of a few steps of a Krylov subspace method, and then applying Tikhonov regularization to the reduced problem. A regularization parameter determines how much the given problem is regularized before solution. If the matrix of the linear discrete ill-posed problem is nonsymmetric, then the reduction often is carried out with the aid of the Arnoldi process, while when the matrix is symmetric, the Lanczos process is used. This paper discusses several numerical aspects of these solution methods. We illustrate that it may be beneficial to apply the Arnoldi process also when the matrix is symmetric and discuss how certain user-chosen basis vectors can be added to the solution subspace. Finally, we compare the application of the discrepancy principle for the determination of the regularization parameter to another approach that is based on the solution of a cubic equation and has been proposed in the literature.

An efficient data-driven optimal sizing framework for photovoltaics-battery-based electric vehicle charging microgrid

The rapid growth of electric vehicles (EV) in cities has led to the development of microgrids (MGs) combined with photovoltaics (PV) and the energy storage system (ESS) as charging stations. Traditional sizing methods cannot efficiently evaluate large-scale scenarios through nonlinear optimization models to ensure the economy and reliability of the design. This paper combines the physical-economical model (PEM) and data-driven model (DDM) for fast-speed and accurate determination of MG sizes, considering nonlinear battery degradation and optimal power dispatch under variant EV charging profiles. The DDM predicts a truncated initial solution subspace where the optimal result is likely to reside, which performs 90 % time savings while maintaining over 95 % optimality compared with the benchmark PEM. The artificial neural network (ANN) is adopted for DDM with the input of load features and economic parameters, and the accuracy can be continuously improved with more data input. The EV load training set is extracted from charging piles and enlarged two orders of magnitude larger than the original based on reversed Monte Carlo (RMC) and feature space expansion. Through sensitivity analysis of over 40 scenarios, we show that the optimal results are mainly related to load features, correlation with solar irradiation, PV and ESS prices. Most adequately designed stations will pay back around five years. Projects with poor revenue will become profitable when the ESS price and PV price are below 154 $/kWh and 615 $/kW, and all the studied cases are worth the investment with the ESS price lower than 77 $/kWh.

Improved Model Predictive Control for Single-Phase Grid-Tied Inverter With Virtual Vectors in the Compacted Solution-Space

In this letter, an improved model predictive control (MPC) has been proposed for single-phase grid-tied converter with virtual vectors in a reshaped and compacted solution space, while improving ac power quality, without increase of the virtual vectors number. Combining linear solution space reconstruction method, the global optimization solution space can be shrunken and reshaped in real-time, only relying on grid angle information. Therefore, MPC can select the optimal virtual-vector in a finer and closer solution subspace, then, much finer control effect can be obtained. Implemented by finite-control set MPC with an integrated difference modulator, equivalent fixed double switching frequency at the output can be achieved. Experimental results validate the effectiveness of the proposed methods.

Multi-spin-flip engineering in an Ising machine

A merge process is proposed to engineer a multi-spin flip in an Ising machine. The merge process deforms the Hamiltonian (energy function) of the Ising model. We prove a theorem for the merge process and show that a single-spin flip in the deformed Hamiltonian is equivalent to a multi-spin flip in the original Hamiltonian. A merge process induces a transition within the subspace of feasible solutions. We propose a hybrid simulated annealing (SA) algorithm with the merge process. The hybrid algorithm outperforms the conventional SA algorithm, genetic algorithm, and tabu search in the binary quadratic knapsack problems (QKP) and the quadratic assignment problems (QAP). Finally, the hybrid merge process is used in a real Ising machine. The performance is improved in QKP and QAP instances. The merge process is generally applicable to existing Ising machine hardware because the deformed Hamiltonian keeps the format of the Ising model.

Optimal station locations for en-route charging of electric vehicles in congested intercity networks: A new problem formulation and exact and approximate partitioning algorithms

This paper addresses a new electricity-charging station location problem for congested intercity or regional networks, in which electric vehicles with a limited driving range require recharges for their long-distance trips. When the distribution of charging stations does not sufficiently cover all used routes, some drivers may take a detour to find charging opportunities (or switch to another transportation mode). The goal of this station location problem is to find, under a limited construction budget, an optimal set of charging station locations such that all vehicles finish their trips by choosing a self-optimal route with necessary charging opportunities while the overall network congestion caused by possible detours is minimized. The problem is written as a bi-level mixed nonlinear integer programming model, where the upper level of the model is set to regulate the selection of station locations subject to the construction budget while the lower level is used to characterize the equilibrium flow pattern of electric vehicles with the charging requirement. Selected exact and approximate algorithms, namely, the branch-and-bound algorithm and the nested partitions algorithm are adopted to solve this station location problem. While both algorithms imply a divide-and-conquer strategy, the branch-and-bound algorithm poses a deterministic, exact procedure that utilizes the bounding mechanism to prune impossible solution subspaces, whereas the nested partitions algorithm performs a stochastic search and selection process in terms of the optimality probability for near-optimal solutions. To get numerical insights on the algorithmic performance and solution behavior, we test these algorithms through a couple of benchmark network instances. A performance comparison of the algorithms indicates that, the branch-and-bound algorithm can quickly obtain the global optimum when the driving range is relatively low, while the nested partitions algorithm can find optimal solutions or extremely near-optimal solutions in all cases and typically spend on average only one fourth of the computing time of the branch-and-bound algorithm. The network flow solutions clearly show, compared to gasoline vehicle drivers, how an insufficient driving range or number of charging stations may significantly reduce the number of feasible paths and force electric vehicle drivers to choose more costly paths, which thus reshapes flow patterns and possibly increases congestion levels.

Multi-resolution wavelet basis for solving steady forced Korteweg\u2013de Vries model

A steady forced Korteweg–de Vries (fKdV) model which includes gravity, capillary, and pressure distributions is solved numerically using the wavelet Galerkin method. The anti-derivatives of Daubechies wavelets are developed as the basis of the solution subspaces for the mixed boundary condition type. Accuracy of numerical solutions can be improved by increasing the number of wavelet levels in the multi-resolution analysis. The theoretical result of convergence rate is also shown. The problem can be viewed as gravity-capillary wave flows over an applied pressure distribution. The flow regime can be characterized by subcritical, supercritical, and critical flows depending on the value of the Froude number. Trapped depression and elevation waves are found over the pressure distribution. For a near-critical flow regime, a generalized solitary wave with ripples is presented. This shows a capillary effect in balance to gravity and the pressure force on the free surface.

Efficacy of the Method of Four Coefficients to Determine Charge-Carrier Scattering

The investigation of the electronic properties of semiconductors is inherently challenging due to the ensemble averaging of fundamentals to transport measurements (i.e., resistivity, Hall, and Seebeck coefficient measurements). Here, we investigate the incorporation of a fourth measurement of electronic transport, the Nernst coefficient, into the analysis, termed the method of four-coefficients. This approach yields the Fermi level, effective mass, scattering exponent, and relaxation time. We begin with a review of the underlying mathematics and investigate the mapping between the four-dimensional material property and transport coefficient spaces. We then investigate how the traditional single parabolic band method yields a single, potentially incorrect point on the solution sub-space. This uncertainty can be resolved through Nernst coefficient measurements and we map the span of the ensuing sub-space. We conclude with an investigation of how sensitive the analysis of transport coefficients is to experimental error for different sample types.

Invariant Scalar Product and Associated Structures for Tachyonic Klein–Gordon Equation and Helmholtz Equation

Although describing very different physical systems, both the Klein–Gordon equation for tachyons (m2<0) and the Helmholtz equation share a remarkable property: a unitary and irreducible representation of the corresponding invariance group on a suitable subspace of solutions is only achieved if a non-local scalar product is defined. Then, a subset of oscillatory solutions of the Helmholtz equation supports a unirrep of the Euclidean group, and a subset of oscillatory solutions of the Klein–Gordon equation with m2<0 supports the scalar tachyonic representation of the Poincaré group. As a consequence, these systems also share similar structures, such as certain singularized solutions and projectors on the representation spaces, but they must be treated carefully in each case. We analyze differences and analogies, compare both equations with the conventional m2>0 Klein–Gordon equation, and provide a unified framework for the scalar products of the three equations.

On the block Lanczos and block Golub–Kahan reduction methods applied to discrete ill‐posed problems

AbstractThe reduction of a large‐scale symmetric linear discrete ill‐posed problem with multiple right‐hand sides to a smaller problem with a symmetric block tridiagonal matrix can easily be carried out by the application of a small number of steps of the symmetric block Lanczos method. We show that the subdiagonal blocks of the reduced problem converge to zero fairly rapidly with increasing block number. This quick convergence indicates that there is little advantage in expressing the solutions of discrete ill‐posed problems in terms of eigenvectors of the coefficient matrix when compared with using a basis of block Lanczos vectors, which are simpler and cheaper to compute. Similarly, for nonsymmetric linear discrete ill‐posed problems with multiple right‐hand sides, we show that the solution subspace defined by a few steps of the block Golub–Kahan bidiagonalization method usually can be applied instead of the solution subspace determined by the singular value decomposition of the coefficient matrix without significant, if any, reduction of the quality of the computed solution.

A machine learning approach to optimal Tikhonov regularization I: Affine manifolds

Despite a variety of available techniques, such as discrepancy principle, generalized cross validation, and balancing principle, the issue of the proper regularization parameter choice for inverse problems still remains one of the relevant challenges in the field. The main difficulty lies in constructing an efficient rule, allowing to compute the parameter from given noisy data without relying either on any a priori knowledge of the solution, noise level or on the manual input. In this paper, we propose a novel method based on a statistical learning theory framework to approximate the high-dimensional function, which maps noisy data to the optimal Tikhonov regularization parameter. After an offline phase where we observe samples of the noisy data-to-optimal parameter mapping, an estimate of the optimal regularization parameter is computed directly from noisy data. Our assumptions are that ground truth solutions of the inverse problem are statistically distributed in a concentrated manner on (lower-dimensional) linear subspaces and the noise is sub-gaussian. We show that for our method to be efficient, the number of previously observed samples of the noisy data-to-optimal parameter mapping needs to scale at most linearly with the dimension of the solution subspace. We provide explicit error bounds on the approximation accuracy from noisy data of unobserved optimal regularization parameters and ground truth solutions. Even though the results are more of theoretical nature, we present a recipe for the practical implementation of the approach. We conclude with presenting numerical experiments verifying our theoretical results and illustrating the superiority of our method with respect to several state-of-the-art approaches in terms of accuracy or computational time for solving inverse problems of various types.

Ginzburg--Landau Patterns in Circular and Spherical Geometries: Vortices, Spirals, and Attractors

This paper consists of three results on pattern formation of Ginzburg-Landau $m$-armed vortex solutions and spiral waves in circular and spherical geometries. First, we completely describe the global bifurcation diagram of vortex equilibria. Second, we prove persistence of all bifurcation curves under perturbations of parameters, which yields the existence of spiral waves for the complex Ginzburg-Landau equation. Third, we explicitly construct the global attractor of $m$-armed vortex solutions. Our main tool is a new shooting method that allows us to prove hyperbolicity of vortex equilibria in the invariant subspace of vortex solutions.

ОСОБЛИВОСТІ ПОБУДОВИ МАТЕМАТИЧНОЇ МОДЕЛІ ВИЗНАЧЕННЯ ПАРАМЕТРІВ ДЛЯ ПЕРЕХВАТУ ПОВІТРЯНИХ ЦІЛЕЙ

The article proposes an approach to formalize knowledge about the process of determining the parameters of the planned interception using heuristic methods, which are the best in terms of practice, experience, intuition, knowledge of decision makers when aiming fighters at air targets, and looking for solutions within some subspace of possible solutions. The proposed mathematical model allows to formalize the factors that are taken into account when guiding fighters, in the form of linguistic and interval-estimated parameters for each option, which allow to take into account the uncertainty. The initial data of the method is a recommendation regarding the appropriate method of guidance, the hemisphere of the attack of the fighter during guidance. Decisions on the application of the appropriate method of guidance is possible only after the analysis of the conditions of hostilities, the tactical position of the fighter at the time of detection of air targets, taking into account the dynamic characteristics of each method of guidance. It is revealed that automation of information preparation, formation of various variants of application of parameters of the planned interception, is possible at the expense of realization of the corresponding system of support of decision-making. It is substantiated that a logical-linguistic production hierarchical model is expedient as a mathematical model for determining the parameters of interception..

Ingham type approach for uniform observability inequality of the semi-discrete coupled wave equations

This article concerns an observability inequality for a system of coupled wave equations for the continuous models as well as for the space semi-discrete finite difference approximations. For finite difference and standard finite elements methods on uniform numerical meshes it is known that a numerical pathology produces a blow-up of the constant on the observability inequality as the mesh-size tends to zero. We identify this numerical anomaly for coupled wave equations and we prove that there exists a uniform observability inequality in a subspace of solutions generated by low frequencies. We use the Ingham type approach for getting a uniform boundary observability.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/127/abstr.html

Proper scalar product for tachyon representations in configuration space

We propose a new inner product for scalar fields that are solutions of the Klein-Gordon equation with $m^2<0$. This inner product is non-local, bearing an integral kernel including Bessel functions of the second kind, and the associated norm proves to be positive definite in the subspace of oscillatory solutions, as opposed to the conventional one. Poincar\'e transformations are unitarily implemented on this subspace, which is the support of a unitary and irreducible representation of the proper orthochronous Poincar\'e group. We also provide a new Fourier Transform between configuration and momentum spaces which is unitary, and recover the projection onto the representation space. This new scenario suggests a revision of the corresponding quantum field theory.

Isogeometric Residual Minimization Method (iGRM) with direction splitting preconditioner for stationary advection-dominated diffusion problems

In this paper, we introduce the isoGeometric Residual Minimization (iGRM) method. The method solves stationary advection-dominated diffusion problems.We stabilize the method via residual minimization. We discretize the problem using B-spline basis functions. We then seek to minimize the isogeometric residual over a spline space built on a tensor product mesh. We construct the solution over a smooth subspace of the residual. We can specify the solution subspace by reducing the polynomial order, by increasing the continuity, or by a combination of these. The Gramm matrix for the residual minimization method is approximated by a weighted H1 norm, which we can express as Kronecker products, due to the tensor-product structure of the approximations. We use the Gramm matrix as a preconditional which can be applied in a computational cost proportional to the number of degrees of freedom in 2D and 3D. Building on these approximations, we construct an iterative algorithm. We test the residual minimization method on several numerical examples, and we compare it to the Discontinuous Petrov–Galerkin (DPG) and the Streamline Upwind Petrov–Galerkin (SUPG) stabilization methods. The iGRM method delivers similar quality solutions as the DPG method, it uses smaller grids, it does not require breaking of the spaces, but it is limited to tensor-product meshes. The computational cost of the iGRM is higher than for SUPG, but it does not require the determination of problem specific parameters.