Fixed-point Iteration Scheme Research Articles

A Fixed-point Scheme for the Numerical Construction of Magnetohydrostatic Atmospheres in Three Dimensions.

Magnetohydrostatic models of the solar atmosphere are often based on idealized analytic solutions because the underlying equations are too difficult to solve in full generality. Numerical approaches, too, are often limited in scope and have tended to focus on the two-dimensional problem. In this article we develop a numerical method for solving the nonlinear magnetohydrostatic equations in three dimensions. Our method is a fixed-point iteration scheme that extends the method of Grad and Rubin (Proc. 2nd Int. Conf. on Peaceful Uses of Atomic Energy31, 190, 1958) to include a finite gravity force. We apply the method to a test case to demonstrate the method in general and our implementation in code in particular.

Convective Fins Problem with Variable Thermal Conductivity: An Approach Based on Embedding Green’s Functions into Fixed Point Iterative Schemes

Abstract This article introduces a new numerical approach to solve the equation that models a rectangular purely convecting fin with temperature-dependent thermal conductivity. The algorithm embeds an integral operator, defined in terms of Green’s function, into Krasnoselskii–Mann’s fixed point iteration scheme. The validity of the method is demonstrated by a number of examples that consist of a range of values of the parameters that appear in the model. In addition, the evaluation of the fin efficiency is presented. The residual error computations show that the current method provides highly accurate approximations.

Minimal polynomial and reduced rank extrapolation methods are related

Minimal Polynomial Extrapolation (MPE) and Reduced Rank Extrapolation (RRE) are two polynomial methods used for accelerating the convergence of sequences of vectors {xm}. They are applied successfully in conjunction with fixed-point iterative schemes in the solution of large and sparse systems of linear and nonlinear equations in different disciplines of science and engineering. Both methods produce approximations sk to the limit or antilimit of {xm} that are of the form sk=źi=0kźixi$\boldsymbol {s}_{k}={\sum }^{k}_{i=0}\gamma _{i}\boldsymbol {x}_{i}$ with źi=0kźi=1${\sum }^{k}_{i=0}\gamma _{i}=1$, for some scalars źi. The way the two methods are derived suggests that they might, somehow, be related to each other; this has not been explored so far, however. In this work, we tackle this issue and show that the vectors skMPE$\boldsymbol {s}_{k}^{\textit {{\tiny {MPE}}}}$ and skRRE$\boldsymbol {s}_{k}^{\textit {{\tiny {RRE}}}}$ produced by the two methods are related in more than one way, and independently of the way the xm are generated. One of our results states that RRE stagnates, in the sense that skRRE=skź1RRE$\boldsymbol {s}_{k}^{\textit {{\tiny {RRE}}}}=\boldsymbol {s}_{k-1}^{\textit {{\tiny {RRE}}}}$, if and only if skMPE$\boldsymbol {s}_{k}^{\textit {{\tiny {MPE}}}}$ does not exist. Another result states that, when skMPE$\boldsymbol {s}_{k}^{\textit {{\tiny {MPE}}}}$ exists, there holds μkskRRE=μkź1skź1RRE+źkskMPEwithμk=μkź1+źk,$$\mu_{k}\boldsymbol{s}_{k}^{\textit{{\tiny{RRE}}}}=\mu_{k-1}\boldsymbol{s}_{k-1}^{\textit{{\tiny{RRE}}}}+ \nu_{k}\boldsymbol{s}_{k}^{\textit{{\tiny{MPE}}}}\quad \text{with}\quad \mu_{k}=\mu_{k-1}+\nu_{k}, $$for some positive scalars μk, μkź1, and źk that depend only on skRRE$\boldsymbol {s}_{k}^{\textit {{\tiny {RRE}}}}$, skź1RRE$\boldsymbol {s}_{k-1}^{\textit {{\tiny {RRE}}}}$, and skMPE$\boldsymbol {s}_{k}^{\textit {{\tiny {MPE}}}}$, respectively. Our results are valid when MPE and RRE are defined in any weighted inner product and the norm induced by it. They also contain as special cases the known results pertaining to the connection between the method of Arnoldi and the method of generalized minimal residuals, two important Krylov subspace methods for solving nonsingular linear systems.

Sensitivity of the ECMWF Model to Semi-Lagrangian Departure Point Iterations

Accurate estimation of the position of the departure points (d.p.) is crucial for the accuracy of a semi-Lagrangian NWP model. This calculation is often performed applying an implicit discretization to a kinematic equation solved by a fixed-point iteration scheme. A small number of iterations is typically used, assuming that this is sufficient for convergence. This assumption, derived from a past theoretical analysis, is revisited here. Analyzing the convergence of a generic d.p. iteration scheme and testing the ECMWF Integrated Forecast System (IFS) model, it is demonstrated that 2–3 iterations may not be sufficient for convergence to satisfactory accuracy in a modern high-resolution global model. Large forecast improvements can be seen by increasing the number of iterations. The extratropical geopotential error decreases and the simulated vertical structure of tropical cyclones improves. These findings prompted the implementation of an algorithm in which stopping criteria based on estimated convergence rates are used to “dynamically” stop d.p. iterations when an error tolerance criterion is satisfied. This is applied consistently to the nonlinear forecast, tangent linear, and adjoint models used by the ECMWF data assimilation system (4DVAR). Although the additional benefit of dynamic iteration is small, its testing reinforces the conclusion that a larger number of iterations is needed in regions of strong winds and shear. Furthermore, experiments suggest that dynamic iteration may prevent occasional 4DVAR failures in cases of strong stratospheric cross-polar flow in which the tangent linear model becomes unstable.

Two-Step Fixed-Point Proximity Algorithms for Multi-block Separable Convex Problems

Multi-block separable convex problems recently received considerable attention. This class of optimization problems minimizes a separable convex objective function with linear constraints. The algorithmic challenges come from the fact that the classic alternating direction method of multipliers (ADMM) for the problem is not necessarily convergent. However, it is observed that ADMM outperforms numerically many of its variants with guaranteed theoretical convergence. The goal of this paper is to develop convergent and computationally efficient algorithms for solving multi-block separable convex problems. We first characterize the solutions of the optimization problems by proximity operators of the convex functions involved in their objective function. We then design a two-step fixed-point iterative scheme for solving these problems based on the characterization. We further prove convergence of the iterative scheme and show that it has O(1/k) convergence rate in the ergodic sense and the sense of the partial primal-dual gap, where k denotes the iteration number. Moreover, we derive specific two-step fixed-point proximity algorithms (2SFPPA) from the proposed iterative scheme and establish their global convergence. Numerical experiments for solving the sparse MRI problem demonstrate the numerical efficiency of the proposed 2SFPPA.

Optimisation of Ducted Propellers for Hybrid Air Vehicles Using High-Fidelity CFD

ABSTRACTThis paper presents performance analysis and design of ducted propellers for lighter-than-air vehicles. High-fidelity computational fluid dynamics simulations were first performed on a detailed model of the propulsor, and the results were in very good agreement with available experimental data. Additional simulations were performed using a simplified geometry, to quantify the effect of the duct and of the blade twist on the propeller performance. It was shown that the duct is particularly effective at low flight speed and that the blades with relatively high twist have better performance over the flight envelope. Design of the optimal twist distribution and of the duct shape was also attempted by coupling the flow solver with a quasi-Newton optimisation method. Flow gradients were computed by solving the discrete adjoint of the Reynolds-averaged Navier-Stokes equations using a fixed-point iteration scheme or a nested Krylov method with deflated restarting. The results show that the ducted propeller propulsive efficiency can be increased by 2%.

Nonlinear dynamics of hidden modes in a system with internal symmetry

We consider a discrete dynamical system with internal degrees of freedom (DOF). Due to the symmetry between the internal DOFs, certain internal modes cannot be excited by external forcing (in a case of linear interactions) and thus are considered “hidden”. If such a system is weakly asymmetric, the internal modes remain approximately “hidden” from the external excitation, given that small damping is taken into account. However, already in the case of weak cubic nonlinearity, these hidden modes can be excited, even as the exact symmetry is preserved. This excitation occurs through parametric resonance. Floquet analysis reveals instability patterns for the explored modes. To perform this analysis with the required accuracy, we suggest a special method for obtaining the Fourier series of the unperturbed solution for the nonlinear normal mode. This method does not require explicit integration of the arising quadratures. Instead, it employs expansion of the solution at the stage of the implicit quadrature in terms of Chebyshev polynomials. The emerging implicit equations are solved by using a fixed-point iteration scheme. Poincaré sections help to clarify the correspondence between the loss of stability of the modes and the global structure of the dynamical flow. In particular, the conditions for intensive energy exchange in the system are characterized.

Sparse decomposition by iterating Lipschitzian-type mappings

This paper provides the analysis of a fast iterative method for finding sparse solutions to underdetermined linear systems. It is based on a fixed-point iteration scheme which combines nonconvex Lipschitzian-type mappings with canonical orthogonal projectors. The former are aimed at uniformly enhancing the sparsity level by shrinkage effects, the latter are used to project back onto the space of feasible solutions. The iterative process is driven by an increasing sequence of a scalar parameter that mainly contributes to approach the sparsest solutions. It is shown that the minima are locally asymptotically stable for a specific smooth ℓ0-norm. Furthermore, it is shown that the points yielded by this iterative strategy are related to the optimal solutions measured in terms of a suitable smooth ℓ1-norm. Numerical simulations on phase transition show that the performances of the proposed technique overcome those yielded by well known methods for sparse recovery.

Confined Pressuremeter Tests for the Assessment of the Theoretically Back-Calculated Cross-Anisotropic Elastic Moduli

Abstract This study considered the determination of the elastic cross-anisotropic moduli of two base course materials. The cross-anisotropic moduli were back-calculated from a series of vertical and horizontal laboratory pressuremeter tests in a confined mold. First, a theoretical model of an expanding cylindrical cavity in an elastic cross-anisotropic confined medium was developed. Subsequently, an estimate of four (Ghh, μ′hh, μ′vh, μ′hv) of the five independent cross-anisotropic moduli were determined based on a fixed point iteration scheme, which processed the analytically back-calculated test results. Finally, in order to assess the estimated moduli, results of the finite element simulations of the pressuremeter lab tests were compared with the actual measurements.

Combining stochastic programming and optimal control to decompose multistage stochastic optimization problems

The paper suggests a possible cooperation between stochastic programming and optimal control for the solution of multistage stochastic optimization problems. We propose a decomposition approach for a class of multistage stochastic programming problems in arborescent form (i.e. formulated with implicit non-anticipativity constraints on a scenario tree). The objective function of the problem can be either linear or nonlinear, while we require that the constraints are linear and involve only variables from two adjacent periods (current and lag 1). The approach is built on the following steps. First, reformulate the stochastic programming problem into an optimal control one. Second, apply a discrete version of Pontryagin maximum principle to obtain optimality conditions. Third, discuss and rearrange these conditions to obtain a decomposition that acts both at a time stage level and at a nodal level. To obtain the solution of the original problem we aggregate the solutions of subproblems through an enhanced mean valued fixed point iterative scheme.

Fast proximity-gradient algorithms for structured convex optimization problems

We introduce in this paper fixed-point proximity-gradient algorithms for solving a class of structured convex optimization problems arising from image restoration. The objective function of such optimization problems is the sum of three convex functions. We study in this paper the scenario where one of the convex functions involved is differentiable with a Lipschitz continuous gradient and another convex function is composed by an affine transformation. We first characterize the solutions of the optimization problem as fixed-points of a mapping defined in terms of the gradient of the differentiable function and the proximity operators of the other two functions. Then, a fixed-point proximity-gradient iterative scheme is developed based on the fixed-point equation which characterizes the solutions. We establish the convergence of the proposed iterative scheme by the notion of averaged nonexpansive operators. Moreover, we obtain that in general the proposed iterative scheme has O(1k) convergence rate in the ergodic sense and the sense of partial primal–dual gap. Under stronger assumptions on the convex functions involved the proposed iterative scheme will converge linearly. We in particular derive fixed-point proximity-gradient algorithms from the proposed iterative scheme. The quasi-Newton and the overrelaxation strategies are designed to accelerate the algorithms. Numerical experiments for the computerized tomography reconstruction problem demonstrate that the proposed algorithms perform favorably and the quasi-Newton as well as the overrelaxation strategies significantly accelerate the convergence of the algorithms.

Bratu’s problem: A novel approach using fixed-point iterations and Green’s functions

In this article, the one-dimensional non-linear Bratu’s boundary value problem is solved via a novel approach that combines Green’s function and fixed point iterative schemes, such as Picard’s and Krasnoselskii–Mann’s. The convergence of the introduced iterative algorithm is proved using the contraction principle. The method is supported by considering a number of numerical examples that correspond to different cases of eigenvalues. The procedure underlying the strategy reduces calculations and provides highly accurate results in comparison with the exact solution and/or numerical solutions provided in the literature. The current method overcomes the difficulty of treating the problem for eigenvalues near and at the critical value, such as λ=3 and λ=3.51, and handles them reliably and very efficiently.

Proof-of-principle of high-fidelity coupled CRUD deposition and cycle depletion simulation

A multiphysics framework for the high-fidelity simulation of CRUD deposition is developed to better understand the coupled physics and their respective feedback mechanisms. This framework includes the primary physics of lattice depletion, computational fluid dynamics, and CRUD chemistry. The three physics are coupled together via the operator-splitting technique, where predictor–corrector and fixed-point iteration schemes are utilized to converge the nonlinear solution. High-fidelity simulations may provide a means to predict and assess potential operating issues, including CRUD induced power shift and CRUD induced localized corrosion, known as CIPS and CILC, respectively. As a proof-of-principle, a coupled 500-day cycle depletion simulation of a pressurized water reactor fuel pin cell was performed using the coupled code suite; a burnup of 31MWd/kgHM was reached. The simulation recreated the classic striped CRUD pattern often seen on pulled fuel rods containing CRUD. It is concluded that the striping is caused by the flow swirl induced by spacer grid mixing vanes. Two anti-correlated effects contribute to the striping: (1) the flow swirl yields significant azimuthal temperature variations, which impact the locations where CRUD deposits, and (2) the flow swirl is correlated to increased shear stress along the cladding surface and subsequent erosion of the CRUD layer. The CIPS condition of the core is concluded to be primarily controlled by lithium tetraborate precipitation, referred to as boron hideout, which occurs in regions experiencing subcooled nucleate boiling as soluble boron and lithium species reach their solubility limit within the CRUD layer. Subsequently, a localized reduction in power occurs due to the high neutron absorption cross section of boron-10.

Spectral recycling strategies for the solution of nonlinear eigenproblems in thermoacoustics

SummaryIn this work, we consider the numerical solution of large nonlinear eigenvalue problems that arise in thermoacoustic simulations involved in the stability analysis of large combustion devices. We briefly introduce the physical modeling that leads to a nonlinear eigenvalue problem that is solved using a nonlinear fixed point iteration scheme. Each step of this nonlinear method requires the solution of a complex non‐Hermitian linear eigenvalue problem. We review a set of state‐of‐the‐art eigensolvers and discuss strategies to recycle spectral information from one nonlinear step to the next. More precisely, we consider the Jacobi–Davidson algorithm, the Implicitly Restarted Arnoldi method, the Krylov–Schur solver and its block‐variant, and the subspace iteration method with Chebyshev acceleration. On a small test example, we study the relevance of the different approaches and illustrate on a large industrial test case the performance of the parallel solvers best suited to recycle spectral information for large‐scale thermoacoustic stability analysis. Copyright © 2015 John Wiley & Sons, Ltd.

Multigrid-based reconstruction algorithm for quantitative photoacoustic tomography.

This paper proposes a multigrid inversion framework for quantitative photoacoustic tomography reconstruction. The forward model of optical fluence distribution and the inverse problem are solved at multiple resolutions. A fixed-point iteration scheme is formulated for each resolution and used as a cost function. The simulated and experimental results for quantitative photoacoustic tomography reconstruction show that the proposed multigrid inversion can dramatically reduce the required number of iterations for the optimization process without loss of reliability in the results.

Fixed point sets of set-valued mappings

AbstractWe present new results regarding fixed point sets of various set-valued mappings using the concept of fixed point iteration schemes and the newly defined concept of fixed point resolutions. In particular, we prove that the fixed point sets of certain nonexpansive set-valued mappings are contractible.

A Class of Boundary Value Problems Arising in Mathematical Physics: A Green’s Function Fixed-point Iteration Method

Abstract This paper presents a method based on embedding Green’s function into a well-known fixed-point iteration scheme for the numerical solution of a class of boundary value problems arising in mathematical physics and geometry, in particular the Yamabe equation on a sphere. Convergence of the numerical method is exhibited and is proved via application of the contraction principle. A selected number of cases for the parameters that appear in the equation are discussed to demonstrate and confirm the applicability, efficiency, and high accuracy of the proposed strategy. The numerical outcomes show the superiority of our scheme when compared with existing numerical solutions.

Numerical analysis of bump solutions for neural field equations with periodic microstructure

We study numerically single bump solutions of a homogenized Amari equation with periodic microvariation. Two attempts are made to detect single bumps that depend on the microvariable. The first attempt which is based on a pinning function technique is applicable in the Heaviside limit of the firing rate function. In the second attempt, we develop a numerical scheme which combines the two-scale convergence theory and an iteration procedure for the corresponding heterogeneous Amari equation. The numerical simulations in both attempts indicate the nonexistence of single bump solutions that depend on the microvariable. Motivated by this result, we finally develop a fixed point iteration scheme for the construction of single bump solutions that are independent of the microvariable when the firing rate function is given by a sigmoidal firing rate function.

A novel fixed point scheme: Proper setting of variational iteration method for BVPs

The aim of this article is to introduce a novel approach based on embedding Green’s function into a tailored linear integral operator then apply a well-known fixed point iterative scheme, such as Picard’s and Mann’s, for the numerical solution of initial and boundary value problems (IVPs and BVPs). The strategy provides insight into the proper setting of the variational iteration method (VIM) for BVPs. A convergence analysis of the method is included. A number of examples consisting of second and third order BVPs are presented to demonstrate the efficiency and applicability of the scheme.

Nonlinear MPC with fixed-point iterations: numerical and experimental validation

The paper demonstrates the applicability of a recently presented MPC scheme for nonlinear systems with control constraints. The algorithm is based on fixed-point iterations that in principal consist of combined forward/backward integrations over the MPC horizon. Numerical and runtime results for a nonlinear chemical reactor model show the efficiency of the fixed-point iterative MPC scheme. Experimental results for a magnetic levitation system confirm its practical applicability.