Exact Decomposition Research Articles

Exact representations of many-body interactions with restricted-Boltzmann-machine neural networks.

Restricted Boltzmann machines (RBMs) are simple statistical models defined on a bipartite graph which have been successfully used in studying more complicated many-body systems, both classical and quantum. In this work, we exploit the representation power of RBMs to provide an exact decomposition of many-body contact interactions into one-body operators coupled to discrete auxiliary fields. This construction generalizes the well known Hirsch's transform used for the Hubbard model to more complicated theories such as pionless effective field theory in nuclear physics, which we analyze in detail. We also discuss possible applications of our mapping for quantum annealing applications and conclude with some implications for RBM parameter optimization through machine learning.

Exact Optimization and Decomposition Approaches for 2D Shelf Space Allocation

Shelf space is one of the scarcest resources, and its effective management to maximize profits has become essential to gain a competitive advantage for retailers. We consider the two-dimensional shelf space allocation problem (2DSSAP) with additional features motivated by literature and our interactions with a local bookstore. Two dimensions represent the width and height of rectangular arrangement space of a product. We determine optimal number of facings of all products in both dimensions and allocate them as contiguous rectangles to maximize profit. We first develop a mixed-integer linear mathematical programming model (MIP) for our problem and propose a solution method based on logic-based Benders decomposition (LBBD). Next, we construct an exact 2-stage algorithm (IP1/IP2), inspired by LBBD, which can handle larger and real-world size instances. To compare performances of our methods, we generate 100 test instances inspired by real-world applications and benchmarks from the literature. We observe that IP1/IP2 finds optimal solutions for real-world instances efficiently and can increase the local bookstore's profit up to 16.56\%. IP1/IP2 can provide optimal solutions for instances with 100 products in minutes and optimally solve up to 250 products (assigned to 8 rows x 160 columns) within a time limit of 1800 seconds. This exact 2-stage IP1/IP2 solution approach can be effective in solving similar problems such as display problem of webpage design, allocation of product families in grocery stores, and flyer advertising.

Normalising Flows and Nonlinear Normal Modes

Abstract In the context of dynamic decoupling problems, engineering dynamics has long held modal analysis as an exemplar. The method allows the exact decomposition of linear multi-degree-of-freedom (MDOF) systems into single-degree-of-freedom (SDOF) oscillators, thus simplifying complex dynamic problems considerably. However, modal analysis is very much a linear theory; if applied to nonlinear systems, the decoupling property (among others) is lost. This unfortunate situation has led to numerous attempts to formulate workable nonlinear versions of the theory. The current paper extends previous work by the authors in using machine learning methods to learn nonlinear modal transformations on measured data, based on the premise that any latent modal variables should be statistically independent. Unlike previous work, the transformation here exploits the recent development of normalising flows in constructing the required transformations. The new approach is shown to overcome a number of the problems in the original approach when demonstrated on a simulated nonlinear system.

Damping Energy Dissipation and Parameter Identification of the Bellows Structure Covered with Elastic‐Porous Metal Rubber

In order to improve damping energy dissipation of a U‐shaped bellows structure, elastic‐porous metal rubber as a cover layer was adopted and the corresponding vibration parameters were identified. First, the evolution of energy dissipation characteristics with respect to the changes of amplitude and frequency was investigated through a dynamic experimental test in the bending direction of the covered bellows structure. Second, the conspicuous hysteresis loop characteristics were described while the nonlinear constitutive relation was analytically modelled based on the exact decomposition method. Third, the corresponding parameters on dynamic properties of the covered bellows structure were determined by generalized least‐squares estimation. Finally, the prediction results were compared with the measured displacement‐restoring force curves to verify the accuracy of the developed dynamic model. The results indicate that the proposed dynamic model associated with the nonlinear constitutive relation for the covered bellows structure can well describe the evolution of the restoring force in terms of amplitude and frequency.

Integrated Production and Distribution Scheduling in Mobile Facilities

Many supply chains lack flexibility and adaptability in today's competitive market, resulting in customer dissatisfaction, backorders, and several extra costs for the business. Additionally, the inability to quickly meet the customer's demands and the unnecessary transportation costs is also one of the significant challenges faced by the fixed facilities' supply chain. To address these challenges, this study analyzed the mobile facilities supply chain and the production, distribution, and delivery of goods conducted by trucks based on customer preferences. This study proposes a bi-objective mixed-integer linear programming model to ensure the mobile facilities' routing and manufacturing schedules are optimized to meet the customer's needs. Furthermore, this model minimizes production and distribution costs in the shortest amount of time. An exact decomposition algorithm based on Benders decomposition is used to find high-quality solutions in a reasonable amount of time to tackle the problem efficiently. We present several acceleration strategies for increasing the convergence rate of Benders' decomposition algorithm, including Pareto optimality cut and warm-up start. The warm-up start acceleration strategy itself is a meta-heuristic based on particle swarm optimization (PSO). Using the Benders decomposition, we demonstrate the superior accuracy of our solution methodology for large-scale cases with 10 kinds of products ordered by 30 customers using 10 mobile facilities.

Fast algorithm with theoretical guarantees for constrained low-tubal-rank tensor recovery in hyperspectral images denoising

Hyperspectral images (HSIs) are unavoidably degraded by mixed noise, including Gaussian noise and sparse noise. In this paper, we consider a constrained tubal rank and sparsity model (CTSD) to tackle the HSIs mixed noise removal, which characterizes the clean HSIs via the low-tubal-rank constraint and the sparse noise via the l0 and l∞ norm constraints, respectively. Due to the strong non-convexity, the CTSD model is challenging to solve. To tackle the CTSD, we develop the proximal alternating minimization (PAM) algorithm via the exact tensor singular value decomposition (t-SVD) and establish the global convergence under mild assumptions. Since the t-SVD is computationally expensive, especially for large scale images, we further design an efficient inexact PAM algorithm via an inexact t-SVD. The inexact PAM enjoys two advantages: (1) The computational complexity for SVDs of the inexact PAM (O(rn1n2n3)) is about twofold faster than that of the exact PAM (O(min(n1,n2)n1n2n3)) for r≪min(n1,n2); (2) The accuracy of the inexact PAM is theoretically guaranteed. Extensive experiments on HSIs denoising demonstrate that the exact and inexact methods both outperform comparing methods in quantitative evaluation metrics and visual effects, and the inexact PAM can compromise between the accuracy and efficiency for large scale HSIs.

Simultaneously exploiting two formulations: An exact benders decomposition approach

When modelling a given problem using integer linear programming techniques several possibilities often exist, each resulting in a different mathematical formulation of the problem. Usually, advantages and disadvantages can be identified in any single formulation. In this paper we consider mixed integer linear programs and propose an approach based on Benders decomposition to exploit the advantages of two different formulations when solving a problem. We propose applying Benders decomposition to a combined formulation, comprised of two separate formulations, augmented with linking constraints to ensure consistency between the decision variables of the respective formulations. We demonstrate the applicability of the proposed methodology to situations in which one of the formulations models a relaxation of the problem and to cases where one formulation is the Dantzig-Wolfe reformulation of the other. The proposed methodology guarantees a lower bound that is as good as the tighter of the two formulations, and we show how branching can be performed on the decision variables of either formulation. Finally, we test and compare the performance of the proposed approach on publicly available instances of the Cutting Stock Problem and the Split Delivery Vehicle Routing Problem. Compared to the best approaches from the literature, the proposed method shows promising performance and appears to be an attractive alternative.

On extension of near-wall domain decomposition to turbulent compressible flows

For modeling turbulent flow, the near-wall domain decomposition (NDD) approach initially proposed by the second author and recently developed in a number of papers proved to be very efficient. It leads to a non-overlapping domain decomposition with a Robin-to-Dirichlet map between an inner (near-wall) and outer regions. The regions are linked with each other via interface boundary conditions of Robin type which equivalently replace both the boundary conditions at the wall and simplified governing equations in the inner region. As has been shown, this approach can reduce the computational time by one order of magnitude while retaining sufficiently high accuracy. In the current paper, for the first time the technique is extended to compressible gas flows. In addition, it is modified to include an exact domain decomposition applied to the original Reynolds-averaged Navier-Stokes equations (RANS) without any simplifications near the wall. The efficiency and accuracy of the algorithm are demonstrated on a number of test cases with the use of the Spalart-Allmaras turbulence model for compressible flows implemented in the in-house code “FlowModellium”. Apart from the approximate NDD (ANDD) based on the thin boundary layer model, for the first time an exact NDD (ENDD) is implemented. The interface boundary conditions in both ANDD and ENDD approaches are consistent. Thereby, the ENDD can effectively complete the ANDD approach when it is needed.

The Exact Form of the “Ockham Factor” in Model Selection

We explore the arguments for maximizing the “evidence” as an algorithm for model selection. We show, using a new definition of model complexity which we term “flexibility,” that maximizing the evidence should appeal to both Bayesian and frequentist statisticians. This is due to flexibility’s unique position in the exact decomposition of log-evidence into log-fit minus flexibility. In the Gaussian linear model, flexibility is asymptotically equal to the Bayesian information criterion (BIC) penalty, but we caution against using BIC in place of flexibility for model selection.

Developing a new model for simultaneous scheduling of two grand projects based on game theory and solving the model with Benders decomposition

Grand infrastructure projects, such as dam, power plant, petroleum, and gas industry projects, have several contractors working on them in several independent sub-projects. The concern of reducing the duration of these projects is one of the important issues among various aspects; thus, our aim is to fulfill the requirements by using the game theory approach. In this study, a mixed-integer programming model consisting of game theory and project scheduling is developed to reduce the duration of projects with a minimum increase in costs. In this model, two contractors in successive periods are entered into a step-by-step competition by the employer during dynamic games, considering an exchange in their limited resources. The optimum solution of the game in each stage are selected as the strategy, and the resources during the game are considered to be renewable and limited. The strategy of each contractor can be described as follows: 1) share their resources with the other contractor and 2) not share the resources with the other contractor. This model can act dynamically in all circumstances during project implementation. If a player chooses a non-optimum strategy, then this strategy can immediately update itself at the succeeding time period. The proposed model is solved using the exact Benders decomposition method, which is coded in GAMS software. The results suggest the implementation of four step-by-step games between the contractors. Then, the results of our model are compared with those of the conventional models. The projects’ duration in our model is reduced by 22.2%. The nominal revenue of both contractors has also reached a significant value of 46078 units compared with the relative value of zero units in the original model. Moreover, we observed in both projects the decreases of 19.5%, 20.9%, and 19.7% in the total stagnation of resources of types 1, 2, and 3, respectively.

The stochastic critical node problem over trees

AbstractWe tackle a stochastic version of the critical node problem (CNP) where the goal is to minimize the pairwise connectivity of a graph by attacking a subset of its nodes. In the stochastic setting considered, the outcome of attacks on nodes is uncertain. In our work, we focus on trees and demonstrate that over trees the stochastic CNP actually generalizes to the stochastic critical element detection problem where the outcome of attacks on edges is also uncertain. We prove the NP‐completeness of the decision version of the problem when connection costs are one, while its deterministic counterpart was proved to be polynomial. We then derive a nonlinear model for the considered CNP version over trees and provide a corresponding linearization based on the concept of probability chains. Moreover, given the features of the derived linear model, we devise an exact Benders decomposition (BD) approach where we solve the slave subproblems analytically. A strength of our approach is that it does not rely on any statistical approximation such as the sample average approximation, which is commonly employed in stochastic optimization. We also introduce an approximation algorithm for the problem variant with unit connection costs and unit attack costs, and a specific integer linear model for the case where all the survival probabilities of the nodes in case of an attack are equal. Our methods are capable of solving relevant instances of the problem with hundreds of nodes within 1 hour of computational time. With this work, we aim to foster research on stochastic versions of the CNP, a problem tackled mainly in deterministic contexts so far. Interestingly, we also show a successful application of the concept of probability chains for problem linearizations significantly improved by decomposition methods such as the BD.

Many‐body response of benzene at monolayer MoS2: Van der Waals interactions and spectral broadening

AbstractModels of surface enhancement of molecular electronic response properties are challenging for two reasons: (a) molecule‐surface interactions require a simultaneous solution of the molecular and the surface dynamic response (a daunting task), and (b) when solving for the electronic structure of the combined molecule + surface system, it is not trivial to single out the particular physical effects responsible for enhancement. To tackle this problem, in this work, we apply a formally exact decomposition of the system's response function into subsystem contributions by using subsystem density functional theory (DFT), which grants access to dynamic polarizabilities and optical spectra. In order to access information about the interactions between the subsystems, we extend a previously developed subsystem‐based adiabatic connection fluctuation‐dissipation theorem of DFT to separate the additive from the nonadditive correlation energy and identify the nonadditive correlation as the van der Waals interactions. As an example, we choose benzene adsorbed on monolayer MoS2. We isolate the contributions to benzene's dynamic response arising from the interaction with the surface, and for the first time, we evaluate the enhancements to the effectiveness of C6 coefficients as a function of benzene‐MoS2 distance and adsorption site. We also quantify the spectral broadening of the benzene's electronic excited states due to their interaction with the surface. We find that the broadening has a similar decay law with the molecule‐surface distance as the leading van der Waals interactions (ie, R−6) and that the surface enhancement of dispersion interactions between benzene molecules is less than 5% but is still large enough (0.5 kcal/mol) to likely play a role in the prediction of interface morphologies.

Response Spectra Method for Extended Systems with Proportional Damping

The application of the response spectra method for extended damped systems with a point support on the base is considered. The support excitations can be different. External and internal damping is taken into account, it being assumed proportional i.e. admitting an exact or approximate decomposition of motion according to the undamped system modes. It is noted that the presence of externaldamping leads to additional system perturbation, i.e. to additional members appearance in the right part of motion equations. It is shown that well-known formulas of the response spectra method for systems on a rigid platform are kept intact, but the vector of load projections on the directions of generalized coordinates is modified taking into account the influence of support displacements on generalized loads in the system.

Solving Highly Nonlinear Partial Differential Equations Using Homotopy Perturbation Method

An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linear and nonlinear ordinary and partial differential equations. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which can be solved easily. The method does not depend upon a small parameter in the equation. Using the initial conditions this method provides an analytical or exact solution. From the calculation and its graphical representation it is clear that how the solution of the original equation and its behavior depends on the initial conditions. Therefore there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Many problems in natural and engineering sciences are modeled by nonlinear partial differential equations (NPDEs). The theory of nonlinear problem has recently undergone much study. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. In this paper we have applied this method to Burger’s equation and an example of highly nonlinear partial differential equation to get the most accurate solutions. The final results tell us that the proposed method is more efficient and easier to handle when is compared with the exact solutions or Adomian Decomposition Method (ADM).

Application of the Partial Wave QP Decomposition to the Angular Scattering of the State-to-State F + H2 Reaction at Etrans = 0.04088 eV.

We analyze the physical content of structures present in the product differential cross sections (DCSs) of the benchmark F + H2(vi, ji, mi) → FH(vf, jf, mf) + H reaction, where v, j, and m are the vibrational, rotational, and helicity quantum numbers, respectively, for the initial and final states. We analyze three state-to-state transitions: 000 → 300, 000 → 310, and 000 → 320. Accurate quantum S matrix elements are employed at a translational energy of 0.04088 eV for the Fu-Xu-Zhang potential energy surface. Our analysis of the DCSs uses a new technique called the QP decomposition; it makes an exact decomposition of the scattering (S) matrix into a Q part and a P part. The P part consists of a partial wave (PW) sum of Regge poles (involving both positions and residues) together with a rapidly oscillating quadratic phase. The Q part of the decomposition is then constructed exactly by subtracting the rapidly oscillating phase and the PW Regge pole sum from the input PW S matrix. In practice, it is convenient to make a small modification, which we call the QmodPmod decomposition. All our calculations use only integer values of the total angular momentum quantum number, namely, J = 0, 1, 2,... We find that the QmodPmod decomposition is successful and physically meaningful, in that the properties of Qmod matrix are simpler than those of the input S matrix. We then carry out a QmodPmod analysis of the DCSs, which provides novel insights into interference structures present in the angular scattering. In particular, we find for all three reactions that Regge resonances contribute across the whole angular range of the DCSs, being particularly pronounced at small angles. The techniques of nearside-farside decomposition and local angular momentum analysis for resummed Legendre PW series are also employed to provide additional insights into the angular scattering.

Logic-based benders decomposition algorithm for contamination detection problem in water networks

To prevent the serious hazards caused by the intrusion of contaminants into the water distribution network, equipping the network with monitoring sensors is necessary. In this regard, the identification criterion is an important and recently addressed issue indicating that the sensors should be located so that in the case of intrusion of contamination, not only at least one sensor responds, but also it is possible to identify the source of contamination intrusion, as well. This paper addresses the sensor location problem with the identification criterion assuming that a limited budget is available for the sensor placement, and the aim is to minimize the number of vulnerable nodes having the same alarm pattern. First, the problem is formulated as a bi-objective mixed-integer linear programming model, assuming that the objective functions are ordered based on a given prioritization. Then, by utilizing the underlying problem structure, an exact logic-based Benders decomposition algorithm is presented. Computational results over moderate and large-sized instances confirm the efficiency of the proposed algorithm.

Density-Friendly Graph Decomposition

Decomposing a graph into a hierarchical structure via k -core analysis is a standard operation in any modern graph-mining toolkit. k -core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere degree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connectedness, and it allows to reveal the structural organization of the graph. Despite the fact that k -core analysis relies on vertex degrees, k -cores do not satisfy a certain, rather natural, density property. Simply put, the most central k -core is not necessarily the densest subgraph. This inconsistency between k -cores and graph density provides the basis of our study. We start by defining what it means for a subgraph to be locally dense , and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by k -cores, but in this case the components are arranged in order of increasing density. We show that such a locally dense decomposition for a graph G =( V , E ) can be computed in polynomial time. The running time of the exact decomposition algorithm is O (| V | 2 | E |) but is significantly faster in practice. In addition, we develop a linear-time algorithm that provides a factor-2 approximation to the optimal locally dense decomposition. Furthermore, we show that the k -core decomposition is also a factor-2 approximation, however, as demonstrated by our experimental evaluation, in practice k -cores have different structure than locally dense subgraphs, and as predicted by the theory, k -cores are not always well-aligned with graph density.

Multipole decompositions for directional light scattering

Applications of the multipole decomposition method for investigations of directional light scattering by a single nanoparticle and nanoparticle structures located in a finite spatial region are discussed. It is shown that, even in the case of relatively large scatterers, the multipole decomposition obtained in the long-wavelength approximation (LWA) may provide much better convergence than the multipole decomposition with the exact multipoles obtained from the spherical harmonics expansion. For an explanation of this seeming paradox, we derive in real space the exact multipole decomposition based on the spherical harmonics, presenting exact expressions for multipoles up to the electric 16-pole. Results obtained with the exact and approximate multipole expressions are discussed and compared. It is shown that for shape-anisotropic finite-size scatterers with different geometrical dimensions (like plates, rods, disks, rings, etc.), the required number of approximate multipoles providing accurate results may be much smaller than the required number of exact multipoles. For applicability of the LWA multipole decomposition, the only important parameter is the small ratio of the scatter size (its projection) in the scattering direction to the light wavelength. If this condition is fulfilled, the multipole decomposition with a small number of LWA multipoles is simpler than that based on the exact multipoles.

Perspectives on CUR decompositions

This note discusses an interesting matrix factorization called the CUR Decomposition. We illustrate various viewpoints of this method by comparing and contrasting them in different situations. Additionally, we offer a new characterization of CUR decompositions which synergizes these viewpoints and shows that they are indeed the same in the exact decomposition case.

Quantum algorithm for nonhomogeneous linear partial differential equations

We describe a quantum algorithm for preparing states that encode solutions of non-homogeneous linear partial differential equations. The algorithm is a continuous-variable version of matrix inversion: it efficiently inverts differential operators that are polynomials in the variables and their partial derivatives. The output is a quantum state whose wavefunction is proportional to a specific solution of the non-homogeneous differential equation, which can be measured to reveal features of the solution. The algorithm consists of three stages: preparing fixed resource states in ancillary systems, performing Hamiltonian simulation, and measuring the ancilla systems. The algorithm can be carried out using standard methods for gate decompositions, but we improve this in two ways. First, we show that for a wide class of differential operators, it is possible to derive exact decompositions for the gates employed in Hamiltonian simulation. This avoids the need for costly commutator approximations, reducing gate counts by orders of magnitude. Additionally, we employ methods from machine learning to find explicit circuits that prepare the required resource states. We conclude by studying an example application of the algorithm: solving Poisson's equation in electrostatics.