Inexact Newton Algorithms Research Articles

Accelerating aeroelastic UVLM simulations by inexact Newton algorithms

AbstractWe consider the aeroelastic simulation of flexible mechanical structures submerged in subsonic fluid flows at low Mach numbers. The nonlinear kinematics of flexible bodies are described in the total Lagrangian formulation and discretized by finite elements. The aerodynamic loads are computed using the unsteady vortex-lattice method wherein a free wake is tracked over time. Each implicit time step in the dynamic simulation then requires solving a nonlinear equation system in the structural variables with additional aerodynamic load terms. Our focus here is on the efficient numerical solution of this system by accelerating the Newton algorithm. The particular structure of the aeroelastic nonlinear system suggests the structural derivative as an approximation to the full derivative in the linear Newton system. We investigate and compare two promising algorithms based on this approximation, a quasi-Newton type algorithm and a novel inexact Newton algorithm. Numerical experiments are performed on a flexible plate and on a wind turbine. Our computational results show that the approximation can indeed accelerate the Newton algorithm substantially. Surprisingly, the theoretically preferable inexact Newton algorithm is much slower than the quasi-Newton algorithm, which motivates further research to speed up derivative evaluations.

Privacy-preserving energy trading management in networked microgrids via data-driven robust optimization assisted by machine learning

Trading mechanism design, uncertainty treatment and privacy protection are the main issues in the energy management of networked microgrids (MGs). To address these issues in a comprehensive manner, this paper proposes a data-driven two-level transactive energy management framework, where the upper-level determines the optimal strategies of internal scheduling within MGs and external trading between MGs while the lower-level formulates a Nash bargaining game model for the fair allocation of trading benefits. The uncertainties of renewable energy sources are fully captured by an adjustable data-driven robust optimization approach with an uncertainty set constructed using the robust kernel density estimation (RKDE) as a machine learning technique. The resulting uncertainty set can provide robust scheduling and trading schemes even when the power generation data of wind turbines and photovoltaic systems are contaminated with anomalous samples, whereas conventional sets are not reliable in the case of contaminated data. To preserve the operational independence and information privacy of MGs, the proposed model is solved in a distributed manner by the alternating direction method of multipliers (ADMM) and the augmented Lagrange-based alternating direction inexact Newton (ALADIN) algorithms. ADMM is commonly used in previous studies, but it has low computational efficiency for handling the consensus process among a large number of MGs. This paper applies ADMM and ALADIN to enhance the applicability of the proposed model when the size and complexity of the networks increase. Numerical tests show the effectiveness of the proposed framework and solution methodology in terms of system cost, solution robustness, and convergence speed.

Bound-preserving inexact Newton algorithms on parallel computers for wormhole propagation in porous media

Simulating wormhole propagation during reactive dissolution of carbonates is often significantly challenging, because of the high nonlinearity of the governing equations with complex fluid physics. High resolution grids are often required to represent the complex geological heterogeneity, which demands massively parallel computers with a large number of processors. Herein, we present a parallel and scalable simulator on parallel computers for the fully implicit solution of the wormhole propagation model. Our approach is based on a family of mixed finite element methods for the spatial discretization and the implicit Backward Euler scheme for the temporal integration, to handle the combination of complicated flow physics and high resolution grids in their full complexity. Moreover, the active-set reduced-space method, as a class of bound-preserving inexact Newton algorithms, is proposed for the resultant nonlinear system of equations, to guarantee the nonlinear consistency of the fully implicit discretization in a monolithic way and to ensure the boundedness requirement of the solution. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed simulator for a set of heterogeneous medium problems. Large-scale results are provided to show the scalability for reservoir simulation with hundreds of millions of unknowns by using several thousand processors.

Direct quantitative photoacoustic tomography for realistic acoustic media

Quantitative photo-acoustic tomography (QPAT) seeks to reconstruct a distribution of optical attenuation coefficients inside a sample from a set of time series of pressure data that is measured outside the sample. The associated inverse problems involve two steps, namely acoustic and optical, which can be solved separately or as a direct composite problem. We adopt the latter approach for realistic acoustic media that possess heterogeneous and often not accurately known distributions for sound speed and ambient density, as well as an attenuation following a frequency power law that is evident in tissue media. We use a diffusion approximation (DA) model for the optical portion of the problem. We solve the corresponding composite inverse problem using three total variation (TV) regularised optimisation approaches. Accordingly, we develop two Krylov-subspace inexact-Newton algorithms that utilise the Jacobian matrix in a matrix-free manner in order to handle the computational cost. Additionally, we use a gradient-based algorithm that computes a search direction using the L-BFGS method, and applies a TV regularisation based on the alternating direction method of multipliers (ADMM) as a benchmark, because this method is popular for QPAT and direct QPAT. The results indicate the superiority of the developed inexact Newton algorithms over gradient-based quasi-Newton approaches for a comparable computational complexity.

A Flexible Iterative Solver for Nonconvex, Equality-Constrained Quadratic Subproblems

We present an iterative primal-dual solver for nonconvex equality-constrained quadratic optimization subproblems. The solver constructs the primal and dual trial steps from the subspace generated by the generalized Arnoldi procedure used in flexible GMRES (FGMRES). This permits the use of a wide range of preconditioners for the primal-dual system. In contrast with FGMRES, the proposed method selects the subspace solution that minimizes a quadratic penalty function over a trust region. Analysis of the method indicates the potential for fast asymptotic convergence near the solution, which is corroborated by numerical experiments. The results also demonstrate the effectiveness and efficiency of the method in the presence of nonconvexity. Overall, the iterative solver is a promising matrix-free linear algebra kernel for inexact-Newton optimization algorithms and is well-suited to partial differential equation--constrained optimization.

Field-Split Preconditioned Inexact Newton Algorithms

The multiplicative Schwarz preconditioned inexact Newton (MSPIN) algorithm is presented as a complement to additive Schwarz preconditioned inexact Newton (ASPIN). At an algebraic level, ASPIN and MSPIN are variants of the same strategy to improve the convergence of systems with unbalanced nonlinearities; however, they have natural complementarity in practice. MSPIN is naturally based on partitioning of degrees of freedom in a nonlinear PDE system by field type rather than by subdomain, where a modest factor of concurrency can be sacrificed for physically motivated convergence robustness. ASPIN, originally introduced for decompositions into subdomains, is natural for high concurrency and reduction of global synchronization. We consider both types of inexact Newton algorithms in the field-split context, and we augment the classical convergence theory of ASPIN for the multiplicative case. Numerical experiments show that MSPIN can be significantly more robust than Newton methods based on global linearizations...

A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithms

We propose and test a new class of two-level nonlinear additive Schwarz preconditioned inexact Newton algorithms (ASPIN). The two-level ASPIN combines a local nonlinear additive Schwarz preconditioner and a global linear coarse preconditioner. This approach is more attractive than the two-level method introduced in [X.-C. Cai, D.E. Keyes, L. Marcinkowski, Nonlinear additive Schwarz preconditioners and applications in computational fluid dynamics, Int. J. Numer. Methods Fluids, 40 (2002), 1463–1470], which is nonlinear on both levels. Since the coarse part of the global function evaluation requires only the solution of a linear coarse system rather than a nonlinear coarse system derived from the discretization of original partial differential equations, the overall computational cost is reduced considerably. Our parallel numerical results based on an incompressible lid-driven flow problem show that the new two-level ASPIN is quite scalable with respect to the number of processors and the fine mesh size when the coarse mesh size is fine enough, and in addition the convergence is not sensitive to the Reynolds numbers.

Non‐linear additive Schwarz preconditioners and application in computational fluid dynamics

AbstractThe focus of this paper is on the numerical solution of large sparse non‐linear systems of algebraic equations on parallel computers. Such non‐linear systems often arise from the discretization of non‐linear partial differential equations, such as the Navier–Stokes equations for fluid flows, using finite element or finite difference methods. A traditional inexact Newton method, applied directly to the discretized system, does not work well when the non‐linearities in the algebraic system become unbalanced. In this paper, we study some preconditioned inexact Newton algorithms, including the single‐level and multilevel non‐linear additive Schwarz preconditioners. Some results for solving the high Reynolds number incompressible Navier–Stokes equations are reported. Copyright © 2002 John Wiley & Sons, Ltd.

Nonlinearly Preconditioned Inexact Newton Algorithms

Inexact Newton algorithms are commonly used for solving large sparse nonlinear system of equations $F(u^{\ast})=0$ arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of $\|F\|$, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution $u^{\ast}$, one may want to solve instead an equivalent nonlinearly preconditioned system ${\cal F}(u^{\ast})=0$ whose nonlinearities are more balanced. In this paper, we propose and study a nonlinear additive Schwarz-based parallel nonlinear preconditioner and show numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails.

Inexact-Newton methods for semismooth systems of equations with block-angular structure

Systems of equations with block-angular structure have applications in evolution problems coming from physics, engineering and economy. Many times, these systems are time-stage formulations of mathematical models that consist of mathematical programming problems, complementarity, or other equilibrium problems, giving rise to nonlinear and nonsmooth equations. The final versions of these dynamic models are nonsmooth systems with block-angular structure. If the number of state variables and equations is large, it is sensible to adopt an inexact-Newton strategy for solving this type of systems. In this paper we define two inexact-Newton algorithms for semismooth block-angular systems and we prove local and superlinear convergence.

Preconditioned Newton methods using incremental unknowns methods for the resolution of a steady-state Navier-Stokes-like problem

In a previous work, one of the authors has studied a numerical treatment (by fully implicit discretizations) of a two-dimensional Navier-Stokes-like problem and has proved existence and convergence results for the resulting discretized systems with homogeneous Dirichlet boundary conditions. In this work, we propose some new preconditioned multilevel versions of inexact-Newton algorithms to solve these equations. We also develop another multilevel preconditioner for a nonlinear GMRES algorithm. All of the preconditioners are based on incremental unknowns formulations.

Fully coupled finite volume solutions of the incompressible Navier–Stokes and energy equations using an inexact Newton method

AbstractAn inexact Newton method is used to solve the steady, incompressible Navier–Stokes and energy equation. Finite volume differencing is employed on a staggered grid using the power law scheme of Patankar. Natural convection in an enclosed cavity is studied as the model problem. Two conjugate‐gradient ‐like algorithms based upon the Lanczos biorthogonalization procedure are used to solve the linear systems arising on each Newton iteration. The first conjugate‐gradient‐like algorithm is the transpose‐free quasi‐minimal residual algorithm (TFQMR) and the second is the conjugate gradients squared algorithm (CGS). Incomplete lower‐upper (ILU) factorization of the Jacobian matrix is used as a right preconditioner. The performance of the Newton‐ TFQMR algorithm is studied with regard to different choices for the TFQMR convergence criteria and the amount of fill‐in allowed in the ILU factorization. Performance data are compared with results using the Newton‐CGS algorithm and previous results using LINPACK banded Gaussian elimination (direct‐Newton). The inexact Newton algorithms were found to be CPU competetive with the direct‐Newton algorithm for the model problem considered. Among the inexact Newton algorithms, Newton‐CGS outperformed Newton‐ TFQMR with regard to CPU time but was less robust because of the sometimes erratic CGS convergence behaviour.

Protein electrostatics: rapid multigrid-based Newton algorithm for solution of the full nonlinear Poisson-Boltzmann equation.

A new method for solving the full nonlinear Poisson-Boltzmann equation is outlined. This method is robust and efficient, and uses a combination of the multigrid and inexact Newton algorithms. The novelty of this approach lies in the appropriate combination of the two methods, neither of which by themselves are capable of solving the nonlinear problem accurately. Features of the Poisson-Boltzmann equation are fully exploited by each component of the hybrid algorithm to provide robustness and speed. The advantages inherent in this method increase with the size of the problem. The efficacy of the method is illustrated by calculations of the electrostatic potential around the enzyme Superoxide Dismutase. The CPU time required to solve the full nonlinear equation is less than half that needed for a conjugate gradient solution of the corresponding linearized Poisson-Boltzmann equation. The solutions reveal that the field around the active sites is significantly reduced as compared to that obtained by solving the corresponding linearized Poisson-Boltzmann equation. This new method for the nonlinear Poisson-Boltzmann equation will enable fast and accurate solutions of large protein electrostatics problems.