Linear Subproblems Research Articles

Post-processing of the planewave approximation of Schrödinger equations. Part I: linear operators

Abstract In this article we prove a priori error estimates for the perturbation-based post-processing of the plane-wave approximation of Schrödinger equations introduced and tested numerically in previous works (Cancès, Dusson, Maday, Stamm and Vohralík, (2014), A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations. C. R. Math., 352, 941--946; Cancès, Dusson, Maday, Stamm and Vohralík, (2016), A perturbation-method-based postprocessing for the planewave discretization of Kohn–Sham models. J. Comput. Phys., 307, 446--459.) We consider here a Schrödinger operator ${{\mathscr{H}} \,}= -\frac{1}{2}\varDelta +{\mathscr{V}}$ on $L^2(\varOmega )$, where $\varOmega $ is a cubic box with periodic boundary conditions and where ${\mathscr{V}}$ is a multiplicative operator by a regular-enough function ${\mathscr{V}}$. The quantities of interest are, on the one hand, the ground-state energy defined as the sum of the lowest $N$ eigenvalues of ${{\mathscr{H}} \,}$, and, on the other hand, the ground-state density matrix that is the spectral projector on the vector space spanned by the associated eigenvectors. Such a problem is central in first-principle molecular simulation, since it corresponds to the so-called linear subproblem in Kohn–Sham density functional theory. Interpreting the exact eigenpairs of ${{\mathscr{H}} \,}$ as perturbations of the numerical eigenpairs obtained by a variational approximation in a plane-wave (i.e., Fourier) basis we compute first-order corrections for the eigenfunctions, which are turned into corrections on the ground-state density matrix. This allows us to increase the accuracy by a factor proportional to the inverse of the kinetic energy cutoff ${E_{\textrm{c}}}^{-1}$ of both the ground-state energy and the ground-state density matrix in Hilbert–Schmidt norm at a low computational extra cost. Indeed, the computation of the corrections only requires the computation of the residual of the solution in a larger plane-wave basis and two fast Fourier transforms per eigenvalue.

Two-scale command shaping for arresting motion in nonlinear systems

This paper presents a feedforward technique for arresting motion in nonlinear systems based on two-scale command shaping (TSCS). The advantages of the proposed technique arise from its feedforward nature and ease of implementation in linear and nonlinear systems. Using the TSCS strategy, the control input required to arrest motion is decomposed into two scales—the first arrests dynamics associated with the linear subproblem, while the second eliminates response from the nonlinearities. Using direct numerical integration, the method is assessed using a traditional Duffing system and multi-degree-of-freedom nonlinear systems. Experiments are conducted on a compound pendulum attached to a servomotor, documenting effective arrest of the system in close agreement with theoretical predictions.

A Gradient Sampling Method Based on Ideal Direction for Solving Nonsmooth Optimization Problems

In this paper, a modification to the original gradient sampling method for minimizing nonsmooth nonconvex functions is presented. One computational component in the gradient sampling method is the need to solve a quadratic optimization problem at each iteration, which may result in a time-consuming process, especially for large-scale objectives. To resolve this difficulty, this study proposes a new descent direction, for which there is no need to consider any quadratic or linear subproblem. It is shown that this direction satisfies the Armijo step size condition. We also prove that under proposed modifications, the global convergence of the gradient sampling method is preserved. Moreover, under some moderate assumptions, an upper bound for the number of serious iterations is presented. Using this upper bound, we develop a different strategy to study the convergence of the method. We also demonstrate the efficiency of the proposed method using small-, medium- and large-scale problems in our numerical experiments.

A filter-type method for solving nonlinear semidefinite programming

Recently, there has been much interest in filter methods for solving linear and nonlinear problems. We describe extensions of such methods to nonlinear semidefinite programming problems. These extensions are based on linear subproblems that combine filter techniques and trust region methods. The results of convergence and the rate of convergence are given. Numerical examples on semidefinite programming problems are also reported.

English

The high shear rate thrombus formation was only recently recognized as another way of thrombosis. Models proposed in Weller (2008), (2010) take into account this type of thrombosis. This work uses the phase-field method to model these evolving interface problems. A loosely coupled iterative procedure is introduced to solve the coupled system of equations. Convergence behavior on two levels of refinement of perfusion chamber geometry and cylinder geometry is then studied. The perfusion chamber simulations show good agreement with the original results of Weller. The code is implemented in FEM-library deal.ii Alzeta et al. (2018), which enables distribution of computations to large number of processing units. A scalability and numerical performance study of the loosely coupled iterative procedure is performed, combined with several preconditioners for the linear subproblems.

Probabilistic decomposition‐based security constrained transmission expansion planning incorporating distributed series reactor

This study presents a probabilistic transmission expansion planning model incorporating distributed series reactors, which are aimed at improving network flexibility. Although the whole problem is a mixed-integer non-linear programming problem, this study proposes an approximation method to linearise it in the structure of the Benders decomposition (BD) algorithm. In the first stage of the BD algorithm, optimal number of new transmission lines and distributed series reactors are determined. In the second stage, the developed optimal power flow problem, as a linear sub-problem, is performed for different scenarios of uncertainties and a set of probable contingencies. The Benders cuts are iteratively added to the first stage problem to decrease the optimality gap below a given threshold. The proposed model utilises the Monte Carlo simulation method to take into account uncertainty of wind generations and demands. Several case studies on three test systems are presented to validate the efficacy of the proposed approach.

Combined Hybridizable Discontinuous Galerkin (HDG) and Runge-Kutta Discontinuous Galerkin (RK-DG) formulations for Green-Naghdi equations on unstructured meshes

In this paper, we introduce some new high-order discrete formulations on general unstructured meshes, especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations. Working with a recent family of optimized asymptotically equivalent equations, we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlinear shallow water equations supplemented with several algebraic source terms, which globally account for the non-hydrostatic effects through the introduction of auxiliary coupling variables. High-order discrete approximations of the main flow variables are obtained with a RK-DG method, while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second-order linear elliptic sub-problems with high-order HDG formulations. The combined use of hybrid unknowns and local post-processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches. The proposed formulation is then extended to a more complex family of three parameters enhanced Green-Naghdi equations. The resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies, showing good convergence properties and very good agreements with several sets of experimental data.

A Line Search Penalty-Free Method for Nonlinear Second-Order Cone Programming

In this paper, we propose a line search penalty-free method for solving nonlinear second-order cone programming (NSOCP) problem. Compared with the traditional SQP-type method for NSOCP, our method does not need the assumption that the subproblem is feasible. Besides that, it does not use penalty-function or filter technique. We first use a robust linear second-order cone programming subproblem to get a detective step and then compute an optimal step from a quadratic optimization subproblem. The search direction is a convex combination of the detective step and optimal step. This two-phase strategy and the penalty-free technique are employed to promote global convergence which is analyzed under mild assumptions. We report some numerical experiments whose results show that the proposed algorithm is applicable and efficient.

Multi-parameter wavefield reconstruction inversion for wavespeed and attenuation with bound constraints and total variation regularization

Wavefield reconstruction inversion (WRI) extends the search space of Full Waveform Inversion (FWI) by allowing for wave equation errors during wavefield reconstruction to match the data from the first iteration. Then, the wavespeeds are updated from the wavefields by minimizing the source residuals. Performing these two tasks in alternating mode breaks down the nonlinear FWI as a sequence of two linear subproblems, relaying on the bilinearity of the wave equation. We solve this biconvex optimization with the alternating-direction method of multipliers (ADMM) to cancel out efficiently the data and source residuals in iterations and stabilize the parameter estimation with appropriate regularizations.Here, we extend WRI to viscoacoustic media for attenuation imaging. Attenuation reconstruction is challenging because of the small imprint of attenuation in the data and the crosstalks with velocities. To address these issues, we recast the multivariate viscoacoustic WRI as a triconvex optimization and update wavefields, squared slowness, and attenuation factor in alternating mode at each WRI iteration. This requires to linearize the attenuation-estimation subproblem via an approximated trilinear viscoacoustic wave equation. The iterative defect correction embedded in ADMM corrects the errors generated by this linearization, while the operator splitting allows us to tailor ℓ1 regularization to each parameter class. A toy numerical example shows that these strategies mitigate crosstalk artifacts and noise from the attenuation reconstruction. A more realistic example representative of the North sea further shows the ability of the method to jointly reconstruct the wavespeed and the attenuation starting from a very crude attenuation-free initial model, the moderate strength of the crosstalk artifacts and the sensitivity of the multi-parameter reconstruction to noise.

A quick center-point algorithm for unit commitment with carbon emission trading

A novel global optimization algorithm, named center-point (CP) algorithm, is proposed to solve the unit commitment problem with carbon emission trading (UC-CET) in this paper. By solving a sequence of linear continuous-relaxed subproblems and performing line search procedures, the CP algorithm constructs a sequence of perspective-cuts to generate a tight linear approximation of UC-CET. Then the algorithm iteratively finds integer ellipsoid center of the current linear approximation as the trial solutions, and makes the trial solutions close to the optimal solution by searching the neighborhood of these solutions and adding new linear constraints. We build two types of UC-CET models, one of which considers power balance constraints and the other considers DC power flow constraints. Then, based on these models, we compared the CP algorithm with state-of-the-art solver CPLEX. The simulation results show that the proposed algorithm can find high-quality solutions significantly faster than CPLEX and it is suitable to solve large-scale UC-CET problem.

Output-Space Branch-and-Bound Reduction Algorithm for a Class of Linear Multiplicative Programs

In this paper, a new relaxation bounding method is proposed for a class of linear multiplicative programs. Although the 2 p − 1 variable is introduced in the construction of equivalence problem, the branch process of the algorithm is only carried out in p − dimensional space. In addition, a super-rectangular reduction technique is also given to greatly improve the convergence rate. Furthermore, we construct an output-space branch-and-bound reduction algorithm based on solving a series of linear programming sub-problems, and prove the convergence and computational complexity of the algorithm. Finally, to verify the feasibility and effectiveness of the algorithm, we carried out a series of numerical experiments and analyzed the advantages and disadvantages of the algorithm by numerical results.

The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming

This paper presents a new two-phase method for solving convex mixed-integer nonlinear programming (MINLP) problems, called Decomposition-based Outer Approximation Algorithm (DECOA). In the first phase, a sequence of linear integer relaxed sub-problems (LP phase) is solved in order to rapidly generate a good linear relaxation of the original MINLP problem. In the second phase, the algorithm solves a sequence of mixed integer linear programming sub-problems (MIP phase). In both phases the outer approximation is improved iteratively by adding new supporting hyperplanes by solving many easier sub-problems in parallel. DECOA is implemented as a part of Decogo (Decomposition-based Global Optimizer), a parallel decomposition-based MINLP solver implemented in Python and Pyomo. Preliminary numerical results based on 70 convex MINLP instances up to 2700 variables show that due to the generated cuts in the LP phase, on average only 2–3 MIP problems have to be solved in the MIP phase.

Noise-Immune Model Identification and State-of-Charge Estimation for Lithium-Ion Battery Using Bilinear Parameterization

Accurate estimation of state of charge (SOC) is critical to the safe and efficient utilization of a battery system. Model-based SOC observers have been widely used due to their high accuracy and robustness, but they rely on a well-parameterized battery model. This article scrutinizes the effect of measurement noises on model parameter identification and SOC estimation. A novel parameterization method combining instrumental variable (IV) estimation and bilinear principle is proposed to compensate for the noise-induced biases of model identification. Specifically, the IV estimator is used to reformulate an overdetermined system so as to allow coestimating the model parameters and noise variances. The coestimation problem is then decoupled into two linear subproblems which are solved efficiently by a two-stage least squares algorithm in a recursive manner. The parameterization method is further combined with a Luenberger observer to estimate the SOC in real time. Simulations and experiments are performed to validate the proposed method. Results reveal that the proposed method is superior to existing method in terms of the immunity to noise corruption.

Efficient time-splitting Hermite-Galerkin spectral method for the coupled nonlinear Schrödinger equations

The paper focuses on efficient time-splitting Hermite-Galerkin spectral approximation of the coupled nonlinear Schrödinger equations on the whole line. The original problem is decomposed into one nonlinear subproblem and one linear subproblem by time-splitting method. At each time step, the nonlinear subproblem is solved exactly. While the linear subproblem is efficiently solved by choosing suitable Hermite basis functions with matrix decomposition technique. Numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed method.

Sensor Selection with Nonsmooth Design Criteria Based on Semi-Infinite Programming

A problem of optimal node activation in large-scale sensor networks is considered. The resulting measurements are supposed to be used to estimate unknown parameters of a spatiotemporal process described by a partial differential equation. In this setting, the sensor subset selection problem may quickly become computationally intractable when an excessively complex sensor location algorithm is employed. The is even more pronounced when the design criterion is nondifferentiable. A vital example of this criterion is the sum of an arbitrary number of smallest eigenvalues of the Fisher information matrix, being a generalization of the well-known E-optimality criterion. A simple branch-and-bound algorithm is exposed here to maximize this criterion. Its key component to produce upper bounds to the maximum of the objective function implements a relaxation procedure for solving semi-infinite programming problems. It alternates between solving a linear programming subproblem and evaluation of the eigenvalues and eigenvectors of the current information matrix, which makes it extremely easy to implement. The paper is complemented with a numerical example of computing actual sensor locations.

The operator splitting method with semi-implicit spectral deferred correction for molecular beam epitaxial growth models

This paper presents a high order time discretization method by combining the time splitting method with semi-implicit spectral deferred correction method to simulate the molecular beam epitaxial growth models with and without slope selection. The original problem is split into linear and nonlinear subproblems. The Fourier spectral method is adopted for the linear part, and a second-order SSP-RK method together with the Fourier spectral method is used for the nonlinear part. The scheme takes advantage of avoiding nonlinear iteration. However, the temporal error is dominated by the operator splitting error, which is second order for Strang splitting. In order to achieve higher order numerical algorithm in time, we consider a semi-implicit spectral deferred correction (SDC) method to reduce the splitting error. Specifically, the temporal order is increased by one with each correction loop in the SDC framework. Numerical results are given to illustrate that the high order temporal algorithm is a practical, accurate and efficient simulation tool for molecular beam epitaxial growth models.

An Efficient Tensor Completion Method Via New Latent Nuclear Norm

In tensor completion, the latent nuclear norm is commonly used to induce low-rank structure, while substantially failing to capture the global information due to the utilization of unbalanced unfolding scheme. To overcome this drawback, a new latent nuclear norm equipped with a more balanced unfolding scheme is defined for low-rank regularizer. Moreover, the new latent nuclear norm together with the Frank-Wolfe (FW) algorithm is developed as an efficient completion method by utilizing the sparsity structure of observed tensor. Specifically, both FW linear subproblem and line search only need to access the observed entries, by which we can instead maintain the sparse tensors and a set of small basis matrices during iteration. Most operations are based on sparse tensors, and the closed-form solution of FW linear subproblem can be obtained from rank-one SVD. We theoretically analyze the space-complexity and time-complexity of the proposed method, and show that it is much more efficient over other norm-based completion methods for higher-order tensors. Extensive experimental results of visual-data inpainting demonstrate that the proposed method is able to achieve state-of-the-art performance at smaller costs of time and space, which is very meaningful for the memory-limited equipment in practical applications.

Privacy-Preserving Collaborative Operation of Networked Microgrids With the Local Utility Grid Based on Enhanced Benders Decomposition

In this paper, a privacy-preserving decision-making framework is developed for a collaborative operation of networked microgrids and the local utility grid, where the distribution system operator (DSO) interacts with microgrid master controllers (MCs) in a leader-followers fashion. First, the paper models the decision-making in the collaborative operation presented as an integrated mixed-integer second-order cone programming problem which satisfies various technical and operational requirements in a look-ahead manner. Then, the integrated optimization problem is decomposed into a master problem and a set of subproblems in order to decentralize the decision-making process of DSO and MCs. The master problem is solved by DSO for optimizing the local utility grid operation, and the subproblems are solved independently by MCs for optimizing the operation of individual microgrids. We propose an enhanced Benders decomposition algorithm to guide the iterations between the master problem and subproblems. In particular, an enhanced approach is introduced in the Benders decomposition algorithm for generating valid cutting planes from mixed-integer linear programming subproblems. Mathematically, the proposed algorithm can find the globally optimal solution after a finite number of iterations. Accordingly, DSO and MCs manage to collaborate on making operational decisions that would maximize the social welfare without compromising independent decision-making and privacy provisions. Numerical experiments are conducted on a simplified two-bus system and a modified IEEE 123-bus system. The results validate the efficiency, the robustness, and the scalability of the proposed collaborative decision-making framework in power systems.

Hypersonic Heat Flux Reconstruction with Distributed Temperature Sensors

This paper presents a new method for the multidimensional reconstruction of the heat flux via distributed surface temperature measurements and its validation using hypersonic wind tunnel tests. The first innovation is the formulation of the reconstruction in terms of the Green’s function, approximated using a weak Galerkin technique together with the method of variation of parameters. When compared with previous research, the method does not require nullification of the first derivative of the polynomial basis at boundaries of the second kind, and it applies seamlessly to complex geometries. The second innovation is the use of homotopy to reduce the nonlinear conduction at high temperature to a series of linear subproblems. Our main contribution is the determination of the conditions for the quadrature formulas to yield a convergent homotopy sequence. The third innovation is the use of a multidimensional B-spline basis to reconstruct the heat flux on curved surfaces and in time. The inverse problem projected in the B-spline basis is well-posed and does not require regularization parameters. The fourth innovation is the introduction of a piecewise Green’s function on partitioned multimaterial domains by imposing continuity of heat flux and temperature. The partitioned approach improves the accuracy of the reconstruction in the linear and nonlinear regimes.

Level 2 Reformulation Linearization Technique–Based Parallel Algorithms for Solving Large Quadratic Assignment Problems on Graphics Processing Unit Clusters

This paper discusses efficient parallel algorithms for obtaining strong lower bounds and exact solutions for large instances of the quadratic assignment problem (QAP). Our parallel architecture is comprised of both multicore processors and compute unified device architecture–enabled NVIDIA graphics processing units (GPUs) on the Blue Waters Supercomputing Facility at the University of Illinois at Urbana–Champaign. We propose novel parallelization of the Lagrangian dual ascent algorithm on the GPUs, which is used for solving a QAP formulation based on the level-2 reformulation linearization technique. The linear assignment subproblems in this procedure are solved using our accelerated Hungarian algorithm [Date K, Rakesh N (2016) GPU-accelerated Hungarian algorithms for the linear assignment problem. Parallel Computing 57:52–72.]. We embed this accelerated dual-ascent algorithm in a parallel branch-and-bound scheme and conduct extensive computational experiments on single and multiple GPUs, using problem instances with up to 42 facilities from the quadratic assignment problem library (QAPLIB). The experiments suggest that our GPU-based approach is scalable, and it can be used to obtain tight lower bounds on large QAP instances. Our accelerated branch-and-bound scheme is able to comfortably solve Nugent and Taillard instances (up to 30 facilities) from the QAPLIB, using a modest number of GPUs.