Nonlinear Subproblems Research Articles

SS-DNN: A hybrid strang splitting deep neural network approach for solving the Allen–Cahn equation

The Allen–Cahn equation is a fundamental partial differential equation that describes phase separation and interface motion in materials science, physics, and various other scientific domains. The presence of interfacial width (ϵ) between two stable phases, associated with a nonlinear term, is a small positive parameter which makes the problem more challenging to solve as ϵ approaches zero. This paper proposes a novel hybrid deep splitting method to efficiently and accurately solve the Allen–Cahn equation in a convex polygonal domain in Rd(d=1,2,3). The method combines the benefits of deep learning and splitting strategies, leveraging the strengths of both approaches. Essentially, a second-order splitting method is employed to split the Allen–Cahn equation into two simpler linear and non-linear sub-problems. While the nonlinear sub-problem can be solved analytically, the deep neural network is utilized to approximate the linear sub-problem. By integrating deep learning into the splitting strategy, we achieve a more efficient and accurate solution for the Allen–Cahn equation, demonstrating promising results. We also derive an error estimate for the proposed hybrid method. Modified space adaptivity and transform learning techniques are employed to enhance the efficiency of the neural network.

A fast explicit time-splitting spectral scheme for the viscous Cahn–Hilliard equation with nonlocal diffusion operator

The viscous Cahn–Hilliard equation (VCH) is a parameter-dependent model encapsulated Cahn–Hilliard (CH) model and constrained Allen–Cahn (CAC) model. In this paper, we propose a fast explicit time-splitting spectral scheme for the VCH with nonlocal diffusion operator. The numerical scheme is constructed based on the second-order symmetric time-splitting method, which results in a splitting of the original problem into nonlocal linear part and nonlinear part, respectively. The nonlocal linear subproblem can be solved exactly by applying the Fourier spectral discretization in space and thus has no stability restriction on the time-step size. The nonlinear subproblem is solved via a second-order strong stability preserving Runge–Kutta method. The new scheme is time symmetric, fully explicit and conserves the discrete mass exactly. Theoretical analysis shows that the stability of the scheme depends on both the viscosity coefficient and the range of nonlocal interactions. In addition, a rigorously convergence analysis of the scheme is established, and the convergence rate is proved to be second-order in time and spectral-order in space, respectively. Four numerical experiments are performed to demonstrate the effectiveness of the proposed scheme.

On the number of subproblem iterations per coupling step in partitioned fluid‐structure interaction simulations

AbstractIn literature, the cost of a partitioned fluid‐structure interaction scheme is typically assessed by the number of coupling iterations required per time step, while ignoring the internal iterations within the nonlinear subproblems. In this work, we demonstrate that these internal iterations have a significant influence on the computational cost of the coupled simulation. Particular attention is paid to how limiting the number of iterations within each solver call can shorten the overall run time, as it avoids polishing the subproblem solution using unconverged coupling data. Based on systematic parameter studies, we investigate the optimal number of subproblem iterations per coupling step. In addition, this work proposes a new convergence criterion for partitioned algorithms that is based solely on the number of subproblem iterations required to reach the subproblem residual tolerances and therefore does not require any additional convergence tolerance for the coupling loop.

Coupling-strength criteria for sequential implicit formulations

The Fully Implicit Method (FIM) is used widely in the reservoir simulation community, especially for the so-called black-oil model. The primary reason for resorting to FIM is its unconditional stability, but this comes at the high computational cost of solving coupled nonlinear systems of all the conservation equations simultaneously. In addition, having a scalable FIM simulator relies on specialized multi-stage linear solvers. In the last few years, the Sequential Fully Implicit (SFI) formulation has received a great deal of attention. SFI decomposes the fully coupled nonlinear problem into two nonlinear sub-problems: flow and transport. The flow problem is solved implicitly for pressure, and the transport problem is solved implicitly for the saturation(s). The splitting usually assumes that the total-velocity is fixed during the transport computations. Because the pressure has a near-elliptic behavior and the saturation(s) have near-hyperbolic behavior, the flow-transport splitting of SFI allows for the use of dedicated scalable nonlinear and linear solvers for each sub-problem.At convergence, SFI and FIM lead to the same numerical solution within the specified tolerances on the residuals. SFI, however, can suffer from convergence difficulties of the overall nonlinear iterations between flow and transport. The convergence difficulties of SFI are intimately related to the splitting, and they depend on the type and strength of coupling between the nonlinear flow and transport problems. In this paper, we develop indicators that quantify the nature and strength of coupling across the systems of equations and solution variables. The indicators quantify the coupling in both space and time. Our strategy employs terms in the FIM Jacobian matrix to quantifying the sensitivity of the mass conservation equations and the global volume-balance to changes in pressure and saturation.These ‘coupling strength’ indicators are used in a domain-decomposition strategy to define local sub-domains for which a fully coupled correction is performed instead of the usual fixed-point iteration update. We demonstrate the accuracy and utility of these indicators for several challenging problems, including buoyancy driven lock-exchange - with or without gas solubility, as well as, gas and water injection problems in heterogeneous reservoir models.

A Fast Temporal Decomposition Procedure for Long-Horizon Nonlinear Dynamic Programming

We propose a fast temporal decomposition procedure for solving long-horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit step size is accepted locally for the approximate search direction and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme (Na et al. 2022). However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, whereas we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method. Funding: This work was supported by the National Science Foundation [Grant CNS-1545046] and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research [Grant DE-AC02-06CH11347].

A high order accurate numerical algorithm for the space-fractional Swift-Hohenberg equation

This paper presents a high order time discretization method by combining the time splitting scheme with semi-implicit spectral deferred correction method for the space-fractional Swift-Hohenberg equation (SFSH). Based on the operator splitting method, the original problem is split into linear and nonlinear subproblems, respectively. The Fourier spectral method is adopted for the linear part, and a first-order accurate finite difference scheme in time together with the Fourier spectral method in space is used for the nonlinear part. The stability and convergence of the obtained numerical scheme are analyzed theoretically. Moreover, the spectral deferred correction (SDC) method is then employed to improve the temporal accuracy. Various two and three dimensional numerical experiments are performed to validate the theoretical results and efficiency of the proposed method.

Branch-and-price algorithms for large-scale mission-oriented maintenance planning problems

This paper presents a novel column-generation-based approach for solving large-scale instances of the joint selective maintenance and repairperson assignment problem (JSM-RAP) for mission-oriented systems in industrial settings. Such systems perform consecutive missions separated by scheduled finite-duration breaks during which some of their components are imperfectly maintained by repairpersons, aiming to maximize system reliability in subsequent missions. The resulting mathematical model is computationally expensive, even for problems of moderate size. The proposed approach decomposes the JSM-RAP into a master problem and multiple subproblems that are solved to generate maintenance patterns, i.e., columns. Two methods are developed to handle the mixed-integer nonlinear subproblems: a piecewise-linear approximation and an exact reformulation into mixed-integer exponential conic programs. Branch-and-price algorithms are developed by embedding the column-generation method into a branch-and-bound tree to restore solution integrality and guarantee its optimality. Furthermore, we use a stabilization scheme to accelerate convergence. Numerical experiments validate the proposed approach and demonstrate its added value in terms of computation time and solution quality. Problem instances of very-large size, similar to real-life industrial production plants, are solved efficiently. Results also show that increasing the number of maintenance levels grants more flexibility to the optimizer to find combinations of components and maintenance actions that better use the limited resources.

A split-step Fourier pseudo-spectral method for solving the space fractional coupled nonlinear Schrödinger equations

In this paper, we propose a split-step Fourier pseudo-spectral method for solving the space fractional coupled nonlinear Schrödinger (CNLS) equations. The space fractional CNLS equations can be split into two subproblems such that one of them is linear. The solution of the nonlinear subproblem is computed exactly. The Riesz space fractional derivative is approximated by a Fourier pseudo-spectral method. The unconditional stability, convergence, discrete charge and multi-symplectic preserving properties for the proposed method are investigated. Then, the proposed method is extended for solving the two-dimensional problem. Finally, some numerical experiments are performed to confirm our theoretical analysis and illustrate the efficiency of the proposed scheme.

An effective operator splitting method based on spectral deferred correction for the fractional Gray–Scott model

This paper presents a method by combining the semi-implicit spectral deferred correction (SDC) method with the operator splitting scheme to simulate the fractional Gray-Scott (GS) model. We start with the second-order operator splitting scheme, which is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using the Fourier spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size. The nonlinear subproblem is solved via the Crank-Nicolson formula and Rubin-Graves linearization technique, which can be solved effectively. The stability and convergence of this method are analyzed in L2-norm. Moreover, the scheme also takes advantage of the semi-implicit SDC method to improve the temporal accuracy. Numerical results are given to illustrate that the proposed method is a practical, accurate and efficient simulation tool for solving fractional GS problems.

Outer Approximation for Mixed-Integer Nonlinear Robust Optimization

Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or do not guarantee robust protection. In this work, we develop an algorithm for convex mixed-integer nonlinear robust optimization problems where a key feature is that the method does not rely on a specific structure of the inner worst-case (adversarial) problem and allows the latter to be non-convex. A major challenge of such a general nonlinear setting is ensuring robust protection, as this calls for a global solution of the non-convex adversarial problem. Our method is able to achieve this up to a tolerance, by requiring worst-case evaluations only up to a certain precision. For example, the necessary assumptions can be met by approximating a non-convex adversarial via piecewise relaxations and solving the resulting problem up to any requested error as a mixed-integer linear problem.In our approach, we model a robust optimization problem as a nonsmooth mixed-integer nonlinear problem and tackle it by an outer approximation method that requires only inexact function values and subgradients. To deal with the arising nonlinear subproblems, we render an adaptive bundle method applicable to this setting and extend it to generate cutting planes, which are valid up to a known precision. Relying on its convergence to approximate critical points, we prove, as a consequence, finite convergence of the outer approximation algorithm.As an application, we study the gas transport problem under uncertainties in demand and physical parameters on realistic instances and provide computational results demonstrating the efficiency of our method.

An Investigation on Semismooth Newton based Augmented Lagrangian Method for Image Restoration

The augmented Lagrangian method (also called as method of multipliers) is an important and powerful optimization method for lots of smooth or nonsmooth variational problems in modern signal processing, imaging and optimal control. However, one usually needs to solve a coupled and nonlinear system of equations, which is very challenging. In this paper, we propose several semismooth Newton methods to solve arising nonlinear subproblems for image restoration in finite dimensional spaces, which leads to several highly efficient and competitive algorithms for imaging processing. With the analysis of the metric subregularities of the corresponding functions, we give both the global convergence and local linear convergence rate for the proposed augmented Lagrangian methods with semismooth Newton solvers.

A charge-preserving compact splitting method for solving the coupled stochastic nonlinear Schrödinger equations

In this paper, a splitting method for solving a system of coupled stochastic nonlinear Schrödinger (CSNLS) equations is proposed. Based on its Hamiltonian structure, the CSNLS equations split into two subproblems such that one of them is linear. The solution of the nonlinear subproblem is computed exactly, and the linear subproblem is discritized through a compact method in the spatial direction and the symplectic implicit midpoint rule with respect to the time variable. Stability, discrete stochastic multi-symplectic conservation law, discrete charge and energy conservation laws for the proposed method are investigated. Numerical results are reported for various amplitudes of noise. We present the propagation and collision of solitons which are different from the deterministic case. Finally, the mean-square order of convergence of the proposed algorithm in the temporal and spatial directions for the CSNLS equations is investigated in a numerical example.

High order compact split step finite difference method for two-dimensional coupled nonlinear Schrödinger system

In this paper, a decoupled and efficient finite difference method is developed for two-dimensional coupled nonlinear Schrödinger (CNLS) system. The proposed method uses split step technique for the temporal discretization and high order compact (HOC) difference approximation for the spatial discretization. The original problem is decomposed into a two-dimensional linear subproblem and a two-dimensional nonlinear subproblem. For the two-dimensional linear subproblem, the Lie-Trotter splitting formula is adopted in time to reduce computational cost. While for the nonlinear subproblem, it can be integrated directly and exactly. By the von Neumann approach, it is showed that the proposed method is unconditionally stable. Numerical examples are conducted to compare it with other scheme and numerical results verified the superiority of the proposed method in terms of accuracy and efficiency. The new method also exhibits good numerical performance in long-time simulation.

Emergency medical response in mass casualty incidents considering the traffic congestions in proximity on-site and hospital delays

The response efficiency of Emergency Medical Service (EMS) is related to the survival rate of patients in Mass Casualty Incidents (MCIs). This study aims to reduce delays caused by congestions in the transportation network and also that in hospitals. A patient transportation and assignment model is proposed considering routing of ambulances and operational conditions of hospitals for an efficient MCI response. The proposed model is composed of a Cell Transmission Model (CTM) and a nonlinear treatment impedance function. A Lagrangian heuristic is utilized to expedite model solution by decomposing the problem into two relatively tractable sub-problems: one linear and the other nonlinear. The linear sub-problem is solved by the simplex algorithm, and is related to ambulance routing, while the nonlinear sub-problem is solved by a gradient projection algorithm for the optimization of patients’ hospital assignment. A case study and several benchmark networks were tested, and the proposed methodology outperformed a typical Lagrangian Relaxation approach. This work has the potential to enable a more efficient patient assignment in MCIs.

An efficient augmented Lagrangian method with semismooth Newton solver for total generalized variation

<p style='text-indent:20px;'>Total generalization variation (TGV) is a very powerful and important regularization for various inverse problems and computer vision tasks. In this paper, we propose a semismooth Newton based augmented Lagrangian method for solving this problem. The augmented Lagrangian method (also called as method of multipliers) is widely used for lots of smooth or nonsmooth variational problems. However, its efficiency heavily depends on solving the corresponding coupled and nonlinear system together and simultaneously. With efficient primal-dual semismooth Newton methods for the challenging and highly coupled nonlinear subproblems involving total generalized variation, we develop a highly efficient and competitive augmented Lagrangian method compared with some fast first-order method. With the analysis of the metric subregularities of the corresponding functions, we give both the global convergence and local linear convergence rate for the proposed augmented Lagrangian methods.</p>

A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes and general space domains. The stability results are proved for all algorithms and the accuracy of obtained approximations is estimated by solving well-known test problems. A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem. Results of computational experiments are presented and analyzed.

Partially distributed outer approximation

This paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression.

Optimizing over the properly efficient set of convex multi-objective optimization problems

Optimizing over the efficient set of a multi-objective optimization problem is among the difficult problems in global optimization because of its nonconvexity, even in the linear case. In this paper, we consider only properly efficient solutions which are characterized through weighted sum scalarization. We propose a numerical method to tackle this problem when the objective functions and the feasible set of the multi-objective optimization problem are convex. This algorithm penalizes progressively iterates that are not properly efficient and uses a sequence of convex nonlinear subproblems that can be solved efficiently. The proposed algorithm is shown to perform well on a set of standard problems from the literature, as it allows to obtain optimal solutions in all cases.

Multi‐slack power flow for islanded microgrids with radial and meshed topologies

This study proposes a new power flow formulation for islanded microgrids. The proposed power flow is based on the effect of the superposition principle and the solution of a small non-linear subproblem to determine the frequency of the microgrid and the voltage magnitude in the angular reference node in each iteration. These variables are determined by considering the following equations in the non-linear subproblem: active and reactive power balance equations and specified phase at the angular reference node. The application of the superposition principle allowed to obtain two versions of the proposed technique: one for radial networks - based on the current summation method - and another for meshed networks - based on the Gauss-Zbus method. Therefore, the iterative framework prosed in this study expands in a simple and integrated way the two most commonly used power flow methods in conventional distribution networks for islanded microgrids. The tests in microgrids with 33, 310 and 1438 nodes showed that the proposed approach has the same accuracy as Newton-Raphson algorithm, but with significantly lower computational cost in large scale microgrids. In addition, the proposed method for island microgrid showed good accuracy and convergence for the most common load models applied in power flow studies of islanded microgrids.

Optimization with nonstationary, nonlinear monolithic fluid‐structure interaction

SummaryWithin this work, we consider optimization settings for nonlinear, nonstationary fluid‐structure interaction (FSI). The problem is formulated in a monolithic fashion using the arbitrary Lagrangian‐Eulerian framework to set‐up the fluid‐structure forward problem. In the optimization approach, either optimal control or optimal design problems are treated. In the latter, the stiffness of the solid is estimated from given reference values. In the numerical solution, the optimization problem is solved with a gradient‐based solution algorithm. The nonlinear subproblems of the FSI forward problem are solved with a Newton method including line search. Specifically, we will formally provide the backward‐in‐time running adjoint state used for gradient computations. Our algorithmic developments are demonstrated with some numerical examples as, for instance, extensions of the well‐known fluid‐structure benchmark settings and a flapping membrane test in a channel flow with elastic walls.