Large Systems Of Nonlinear Equations Research Articles

Trefftz method for a polynomial-based boundary identification in two-dimensional Laplacian problems

The paper addresses a two-dimensional boundary identification (reconstruction) problem in steady-state heat conduction. Having found the solution to the Laplace equation by superpositioning T-complete functions, the unknown boundary of a plane region is approximated by polynomials of an increasing degree. The provided examples indicate that sufficient accuracy can be obtained with a use of polynomials of a relatively low degree, which allows avoidance of large systems of nonlinear equations. Numerical simulations for assessing the performance of the proposed algorithm show better than 1% accuracy after a few iterations and very low sensitivity to small data errors.

Amplified and directional spontaneous emission from arbitrary composite bodies: A self-consistent treatment of Purcell effect below threshold

We study amplified spontaneous emission (ASE) from wavelength-scale composite bodies--complicated arrangements of active and passive media--demonstrating highly directional and tunable radiation patterns, depending strongly on pump conditions, materials, and object shapes. For instance, we show that under large enough gain, $\mathcal{PT}$ symmetric dielectric spheres radiate mostly along either active or passive regions, depending on the gain distribution. Our predictions are based on a recently proposed fluctuating volume--current (FVC) formulation of electromagnetic radiation that can handle inhomogeneities in the dielectric and fluctuation statistics of active media, e.g. arising from the presence of non-uniform pump or material properties, which we exploit to demonstrate an approach to modelling ASE in regimes where Purcell effect (PE) has a significant impact on the gain, leading to spatial dispersion and/or changes in power requirements. The nonlinear feedback of PE on the active medium, captured by the Maxwell--Bloch equations but often ignored in linear formulations of ASE, is introduced into our linear framework by a self-consistent renormalization of the (dressed) gain parameters, requiring the solution of a large system of nonlinear equations involving many linear scattering calculations.

Optimization Integrator for Large Time Steps.

Practical time steps in today's state-of-the-art simulators typically rely on Newton's method to solve large systems of nonlinear equations. In practice, this works well for small time steps but is unreliable at large time steps at or near the frame rate, particularly for difficult or stiff simulations. We show that recasting backward Euler as a minimization problem allows Newton's method to be stabilized by standard optimization techniques with some novel improvements of our own. The resulting solver is capable of solving even the toughest simulations at the [Formula: see text] frame rate and beyond. We show how simple collisions can be incorporated directly into the solver through constrained minimization without sacrificing efficiency. We also present novel penalty collision formulations for self collisions and collisions against scripted bodies designed for the unique demands of this solver. Finally, we show that these techniques improve the behavior of Material Point Method (MPM) simulations by recasting it as an optimization problem.

Utilizing algorithmic differentiation to efficiently compute chromatograms and parameter sensitivities

The efficiency of numerical methods for solving mass transfer equations, such as in chromatography modeling, crucially depends on the availability of specific derivatives. In particular, vector products with the system Jacobian are frequently required, i.e. derivatives of the right hand side of the spatially discretized partial differential equations with respect to the state variables. More specifically, large systems of non-linear equations are solved in each time step by Newton iterations, typically contributing more than 80% of the total compute time. We apply algorithmic differentiation (AD) combined with block and band compression for efficiently computing the required system Jacobian. Results are compared with a symbolically derived Jacobian based on non-overlapping domain decomposition. The symbolic approach turns out to be roughly two times faster, at the price of rather tedious and error-prone manual derivation of the mathematical equations. AD helps to simplify implementation of the computational code and is much more flexible when model variants are considered. In addition, AD is combined with a forward sensitivity approach for computing parameter sensitivities that are frequently applied in parameter estimation and process optimization. Results are compared with traditional finite differences. Forward sensitivity analysis is more straightforward to apply, as no perturbation factor needs to be chosen, and even computes faster than finite differences for a benchmark example with three chemical components and load, wash and elution phases.

Orthogonal polynomials of equilibrium measures supported on Cantor sets

The equilibrium measure of a compact set is a fundamental object in logarithmic potential theory. We compute numerically this measure and its orthogonal polynomials, when the compact set is a Cantor set, defined by an Iterated Function System.We first construct sequences of discrete measures, via the solution of large systems of non-linear equations, that converge weakly to the equilibrium measure. Successively, we compute their Jacobi matrices via standard procedures, enhanced for the scope. Numerical estimates of the convergence rate to the limit Jacobi matrix are provided, which show stability and efficiency of the whole procedure. As a companion result, we also compute Jacobi matrices in two other cases: equilibrium measures on finite sets of intervals, and balanced measures of Iterated Function Systems.These algorithms can reach large polynomial orders: therefore, we study the asymptotic behavior of the orthogonal polynomials and, by a natural extension of the concept of regular root asymptotics, we derive an efficient scheme for the computation of complex Green’s functions and of related conformal mappings.

A component framework for the parallel solution of the incompressible Navier–Stokes equations with Radau‐IIA methods

SummaryAn efficient solution strategy for the simulation of incompressible fluids needs adequate and accurate space and time discretization schemes. In this paper, for the space discretization, we use an inf–sup stable finite element method and for the time discretization, Radau‐IIA methods of higher order, which have the advantage that the pressure component has convergence order s in time, where s is the number of internal stages. The disadvantage of this approach is that we have a high computational amount of work, because large nonlinear systems of equations have to solved. In this paper, we use a transformation of the coefficient matrix and the simplified Newton method. This approach has the effect that our large nonlinear systems split into smaller ones, which can now also be solved in parallel. For the parallelization of the code we use the software component technology and the Component Template Library. Numerical examples show that high order in the pressure component can be achieved and that the proposed solution technique is very effective. Copyright © 2015 John Wiley & Sons, Ltd.

REDUCED-ORDER MODELS FOR THE IMPLIED VARIANCE UNDER LOCAL VOLATILITY

Implied volatility is a key value in financial mathematics. We discuss some of the pros and cons of the standard ways to compute this quantity, i.e. numerical inversion of the well-known Black–Scholes formula or asymptotic expansion approximations, and propose a new way to directly calculate the implied variance in a local volatility framework based on the solution of a quasilinear degenerate parabolic partial differential equation. Since the numerical solution of this equation may lead to large nonlinear systems of equations and thus high computation times compared to the classical approaches, we apply model order reduction techniques to achieve computational efficiency. Our method of choice for the derivation of a reduced-order model (ROM) will be proper orthogonal decomposition (POD). This strategy is additionally combined with the discrete empirical interpolation method (DEIM) to deal with the nonlinear terms. Numerical results prove the quality of our approach compared to other methods.

On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric jacobian matrices

Preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) method is an unconditionally convergent iterative method for solving large sparse complex symmetric systems of linear equations. By making use of the PMHSS iteration as the inner solver to approximately solve the Newton equations, we establish a modified Newton-PMHSS method for solving large systems of nonlinear equations. Motivated by the idea in Chen et al. (2014), we analyze the local convergence properties under the Holder continuous condition, which is weaker than the assumptions used in modified Newton-HSS method proposed by Wu and Chen (2013). Numerical results are given to confirm the effectiveness of our method.

On unconditionally positive implicit time integration for the DG scheme applied to shallow water flows

SUMMARYWe present a new unconditionally positivity‐preserving (PP) implicit time integration method for the DG scheme applied to shallow water flows. This novel time discretization enhances the currently used PP DG schemes, because in the majority of previous work, explicit time stepping is implemented to deal with wetting and drying. However, for explicit time integration, linear stability requires very small time steps. Especially for locally refined grids, the stiff system resulting from space discretization makes implicit or partially implicit time stepping absolutely necessary. As implicit schemes require a lot of computational time solving large systems of nonlinear equations, a much larger time step is necessary to beat explicit time stepping in terms of CPU time. Unfortunately, the current PP implicit schemes are subject to time step restrictions due to a so‐called strong stability preserving constraint. In this work, we hence give a novel approach to positivity preservation including its theoretical background. The new technique is based on the so‐called Patankar trick and guarantees non‐negativity of the water height for any time step size while still preserving conservativity. In the DG context, we prove consistency of the discretization as well as a truncation error of the third order away from the wet–dry transition. Because of the proposed modification, the implicit scheme can take full advantage of larger time steps and is able to beat explicit time stepping in terms of CPU time. The performance and accuracy of this new method are studied for several classical test cases. Copyright © 2014 John Wiley & Sons, Ltd.

As-rigid-as-possible spherical parametrization

In this paper, we present an efficient approach for parameterizing a genus-zero triangular mesh onto the sphere with an optimal radius in an as-rigid-as-possible (ARAP) manner, which is an extension of planar ARAP parametrization approach to spherical domain. We analyze the smooth and discrete ARAP energy and formulate our spherical parametrization energy from the discrete ARAP energy. The solution is non-trivial as the energy involves a large system of non-linear equations with additional spherical constraints. To this end, we propose a two-step iterative algorithm. In the first step, we adopt a local/global iterative scheme to calculate the parametrization coordinates. In the second step, we optimize a best approximate sphere on which parametrization triangles can be embedded in a rigidity-preserving manner. Our algorithm is simple, robust, and efficient. Experimental results show that our approach provides almost isometric spherical parametrizations with lowest rigidity distortion over state-of-the-art approaches.

Divide and Conquer: A New Approach to Dynamic Discrete Choice with Serial Correlation

In this paper, we develop a method to efficiently estimate dynamic discrete choice models with AR(n) type serial correlation of the errors. First, to approximate the expected value function of the underlying dynamic problem, we use Gaussian quadrature, interpolation over an adaptively refined grid, and solve a potentially large non-linear system of equations. Second, to evaluate the likelihood function, we decompose the integral over the unobserved state variables in the likelihood function into a series of lower dimensional integrals, and successively approximate them using Gaussian quadrature rules. Finally, we solve the maximum likelihood problem using a nested fixed point algorithm. We then apply this method to obtain point estimates of the parameters of the bus engine replacement model of Rust [Econometrica, 55 (5): 999–1033, (1987)]: First, we verify the algorithm's ability to recover the parameters of an artificial data set, and second, we estimate the model using the original data, finding significant serial correlation for some sub-samples.

On unconditionally positivity preserving DG schemes for shallow water flows with shock capturing by adaptive filtering procedures

AbstractIn this work, we present an unconditionally positivity preserving implicit time integration scheme for the DG method applied to shallow water flows. For locally refined grids with very small elements, the ODE resulting from space discretization is stiff and requires implicit or partially implicit time stepping. However, for simulations including wetting and drying or regions with small water height, severe time step restrictions have to be imposed due to positivity preservation. Nevertheless, as implicit time stepping demands a significant amount of computational time in order to solve a large system of nonlinear equations for each time step we need large time steps to obtain an efficient scheme. In this context, we propose a novel approach to the strategy of positivity preservation. This new technique is based on the so‐called Patankar trick and guarantees non‐negativity of the water height for any time step size while still preserving conservativity. Due to this modification, the implicit scheme can take full advantage of larger time steps and is therefore able to beat explicit time stepping in terms of CPU time. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical Simulation of Dynamic Processes of Elastoplastic Interaction between Three-Dimensional Heterogeneous Bodies on the basis of Semi-Analytical Finite Element Method. Part 1. Computational Relationships of the Semi-Analytical Finite Element Method and Algorithms for the Study of Transient Processes of Dynamic Deformation of Heterogeneous Prismatic Bodies and Bodies of Revolution

On the basis of a semi-analytical finite element method, an effective approach has been developed for studying transient processes of dynamic deformation of three-dimensional heterogeneous bodies of revolution and prismatic bodies of complex shape and structure by the action of time-and space-varying pulsed stressing with allowance for the plastic properties of the material and time-varying contact interaction conditions. New types of finite elements have been created, on the basis of which computational relationships of the semi-analytical finite element method (SAFEM) for problems of dynamics have been constructed. Modified relationships of the Newmark method have been obtained, which have been formulated for the amplitude subsystems of SAFEM. Effective block iteration algorithms for the solution of large systems of nonlinear equations of SAFEM have been developed and realized.

Solving equations describing the steady-state model of MSF desalination process using Solver Optimization Tool of MATLAB software

This work presents an algorithm for solving the large system of non-linear equations describing the steady state model of the two main layouts for the MSF process (the multistage flashing with brine recirculation and the once-through multistage flash desalination (MSF-OT)). The model accounts for the geometry of the stage, the mechanism of heat transfer, and the variation of the thermophysical properties of various fluids with temperature and salinity. In addition, the overall heat transfer coefficient was evaluated at each stage using convective heat transfer coefficient for internal and external flows, tube thermal conductivity and fouling resistance. Furthermore, the model takes into consideration the dependence of thermodynamic losses on stream flow rate, temperature, and the brine salinity. The system of equations was solved through an iterative procedure by using Solver Optimization Tool of MATLAB software. For each stage, the developed model was then used for evaluating the temperatures of the brine, distillate and cooling brine, the flow rates of brine outlet, distillate production, and steam heating. Finally, the resolution method was validated against the simulation results reported in the literature and the actual plant data at MSF-OT installation in Doha, Kuwait. The agreement was found to be good.

A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number

Numerical simulation of three-dimensional incompressible flows at high Reynolds number using the unsteady Navier---Stokes equations is challenging. In order to obtain accurate simulations, very fine meshes are necessary, and such simulations are increasingly important for modern engineering practices, such as understanding the flow behavior around high speed trains, which is the target application of this research. To avoid the time step size constraint imposed by the CFL number and the fine spacial mesh size, we investigate some fully implicit methods, and focus on how to solve the large nonlinear system of equations at each time step on large scale parallel computers. In most of the existing implicit Navier---Stokes solvers, segregated velocity and pressure treatment is employed. In this paper, we focus on the Newton---Krylov---Schwarz method for solving the monolithic nonlinear system arising from the fully coupled finite element discretization of the Navier---Stokes equations on unstructured meshes. In the subdomain, LU or point-block ILU is used as the local solver. We test the algorithm for some three-dimensional complex unsteady flows, including flows passing a high speed train, on a supercomputer with thousands of processors. Numerical experiments show that the algorithm has superlinear scalability with over three thousand processors for problems with tens of millions of unknowns.

Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations

Hermitian and skew-Hermitian splitting(HSS) method has been proved quite successfully in solving large sparse non-Hermitian positive definite systems of linear equations. Recently, by making use of HSS method as inner iteration, Newton-HSS method for solving the systems of nonlinear equations with non-Hermitian positive definite Jacobian matrices has been proposed by Bai and Guo. It has shown that the Newton-HSS method outperforms the Newton-USOR and the Newton-GMRES iteration methods. In this paper, a class of modified Newton-HSS methods for solving large systems of nonlinear equations is discussed. In our method, the modified Newton method with R-order of convergence three at least is used to solve the nonlinear equations, and the HSS method is applied to approximately solve the Newton equations. For this class of inexact Newton methods, local and semilocal convergence theorems are proved under suitable conditions. Moreover, a globally convergent modified Newton-HSS method is introduced and a basic global convergence theorem is proved. Numerical results are given to confirm the effectiveness of our method.

Parallel fully implicit two‐grid methods for distributed control of unsteady incompressible flows

SUMMARYIn this paper, we investigate some fully coupled parallel two‐grid Lagrange–Newton–Krylov–Schwarz algorithms for the suboptimal distributed control of unsteady incompressible flows governed by the Navier–Stokes equations. The algorithms include two major parts: a two‐grid Newton method for the nonlinear part of the problem and a two‐level Schwarz preconditioner for the linear part of the problem. Most of the existing approaches for distributed control problems are based on the so‐called reduced space method, which is easier to implement, but may have convergence issues in some situations. In the full space approach, we couple the state variables, the control variables, and the adjoint variables in a single large system of nonlinear equations. The coupled system is considerably more ill‐conditioned than its sub‐systems; however, with the powerful two‐grid approach, we are able to solve these difficult systems efficiently on large scale parallel computers. We show numerically that such an approach is scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly independent of the grid size, the number of processors, and the Reynolds numbers. We present numerical experiments for some suboptimal control problems obtained on supercomputers with more than two thousand processors. Copyright © 2012 John Wiley & Sons, Ltd.

Iterative strategies for solving linearized discrete mean field games systems

Mean fields games (MFG) describe the asymptotic behavior of stochastic differential games in which thenumber of players tends to $+\infty$. Under suitable assumptions,they lead to a new kind of system of two partial differential equations: a forward Bellman equation coupled with a backward Fokker-Planck equation.In earlier articles, finite difference schemes preserving the structure of the system have been proposed and studied.They lead to large systems of nonlinear equations in finite dimension. A possible way of numerically solving the latter is to use inexact Newton methods: a Newton step consists of solving a linearized discrete MFG system.The forward-backward character of the MFG system makes it impossible to use time marching methods. In the present work, we propose three families of iterative strategies for solving the linearized discrete MFG systems,most of which involve suitable multigrid solvers or preconditioners.

Hierarchical elements for the iterative solving of turbulent flow problems on anisotropic meshes

Accurate solution of industrial turbulent flow problems requires very fine meshes resulting in large systems of non-linear equations and huge computational costs. Efficient iterative methods are therefore necessary. Mesh adaptation, and in particular anisotropic mesh adaptation, allows to reduce considerably meshes size while preserving the accuracy of the solution. Unfortunately, iterative methods and anisotropic meshes do not come along easily and convergence problems may occur. In this work, we show how quadratic elements, expressed in a hierarchical basis, can be used to develop efficient iterative methods for the numerical simulation of turbulent flows on strongly anisotropic meshes.

Hierarchical elements for the iterative solving of turbulent flow problems on anisotropic meshes

Accurate solution of industrial turbulent flow problems requires very fine meshes resulting in large systems of non-linear equations and huge computational costs. Efficient iterative methods are therefore necessary. Mesh adaptation, and in particular anisotropic mesh adaptation, allows to reduce considerably meshes size while preserving the accuracy of the solution. Unfortunately, iterative methods and anisotropic meshes do not come along easily and convergence problems may occur. In this work, we show how quadratic elements, expressed in a hierarchical basis, can be used to develop efficient iterative methods for the numerical simulation of turbulent flows on strongly anisotropic meshes.