Corrector Step Research Articles

A New Primal–Dual Predictor–Corrector Interior-Point Method for Linear Programming Based on a Wide Neighbourhood

In this paper, we propose a new predictor---corrector interior-point algorithm for linear programming based on a wide neighbourhood. In each iteration, the algorithm computes the Ai-Zhang's predictor direction (SIAM J. Optim. 16(2):400---417, 2005) and a new corrector direction, in an attempt to improve its performance. We drive that the duality gap reduces in both predictor and corrector steps. Moreover, we also prove that the complexity of the algorithm coincides with the best iteration bound for small neighbourhood algorithms. Finally, some numerical experiments are provided which reveal capability and effectiveness of the proposed method.

A Constraint-Reduced Algorithm for Semidefinite Optimization Problems with Superlinear Convergence

Constraint reduction is an essential method because the computational cost of the interior point methods can be effectively saved. Park and O'Leary proposed a constraint-reduced predictor---corrector algorithm for semidefinite programming with polynomial global convergence, but they did not show its superlinear convergence. We first develop a constraint-reduced algorithm for semidefinite programming having both polynomial global and superlinear local convergences. The new algorithm repeats a corrector step to have an iterate tangentially approach a central path, by which superlinear convergence can be achieved. This study proves its convergence rate and shows its effective cost saving in numerical experiments.

Algorithm 956

Pseudo-arclength continuation is a well-established method for generating a numerical curve approximating the solution of an underdetermined system of nonlinear equations. It is an inherently sequential predictor-corrector method in which new approximate solutions are extrapolated from previously converged results and then iteratively refined. Convergence of the iterative corrections is guaranteed only for sufficiently small prediction steps. In high-dimensional systems, corrector steps are extremely costly to compute and the prediction step length must be adapted carefully to avoid failed steps or unnecessarily slow progress. We describe a parallel method for adapting the step length employing several predictor-corrector sequences of different step lengths computed concurrently. In addition, the algorithm permits intermediate results of correction sequences that have not converged to seed new predictions. This strategy results in an aggressive optimization of the step length at the cost of redundancy in the concurrent computation. We present two examples of convoluted solution curves of high-dimensional systems showing that speed-up by a factor of two can be attained on a multicore CPU while a factor of three is attainable on a small cluster.

A subluminal relativistic magnetohydrodynamics scheme with ADER-WENO predictor and multidimensional Riemann solver-based corrector

The relativistic magnetohydrodynamics (RMHD) set of equations has recently seen increased use in astrophysical computations. Even so, RMHD codes remain fragile. The reconstruction can sometimes yield superluminal velocities in certain parts of the mesh. In this paper we present a reconstruction strategy that overcomes this problem by making a single conservative to primitive transformation per cell followed by higher order WENO reconstruction on a carefully chosen set of primitives that guarantee subluminal reconstruction of the flow variables. For temporal evolution via a predictor step we also present second, third and fourth order accurate ADER methods that keep the velocity subluminal during the predictor step. The RMHD system also requires the magnetic field to be evolved in a divergence-free fashion. In the treatment of classical numerical MHD the analogous issue has seen much recent progress with the advent of multidimensional Riemann solvers. By developing multidimensional Riemann solvers for RMHD, we show that similar advances extend to RMHD. As a result, the face-centered magnetic fields can be evolved much more accurately using the edge-centered electric fields in the corrector step. Those edge-centered electric fields come from a multidimensional Riemann solver for RMHD which we present in this paper. In this paper we also develop several new test problems for RMHD. We show that RMHD vortices can be designed that propagate on the computational mesh as self-preserving structures. These RMHD vortex test problems provide a means to do truly multidimensional accuracy testing for RMHD codes. Several other stringent test problems are presented. We show the importance of resolution in certain test problems. Our tests include a demonstration that RMHD vortices are stable when they interact with shocks.

A New Wide Neighborhood Primal-Dual Predictor-Corrector Interior-Point Method for Linear Programming

ABSTRACTIn this article, we present a new predictor-corrector interior-point method, based on a wide neighborhood of the central path, for linear programming problems. In the corrector step ofthe proposed method, we derive the step size and corrector directions that guarantee that each iteration lies in a wide neighborhood of the central path. We also prove that the algorithm has iteration complexity, which coincides with the best bound derived for linear programming problems.

Adaptive burnup stepsize selection using control theory for 2-D lattice depletion simulations

A control theory approach is adopted to determine the temporal discretization during two-dimensional lattice physics depletion simulations. Two primary applications of automated and adaptive stepsize control are identified: (i) the presence of strong absorbers such as gadolinium, where the accurate burnout of the isotopes requires a depletion stepsize smaller than typically required, and (ii) high fidelity multiphysics simulations, e.g. loosely coupled physics, where the coupled physics are nonlinear in time and stepsize changes may be necessary to obtain an accurate coupled solution. A conventional predictor–corrector method is used to address the nonlinearity of the nuclide transmutation and neutron flux. An adaptive stepsize method is developed based on monitoring the one-group scalar neutron flux at both the predictor and corrector steps to approximate the convergence residual of the nonlinear solution. A user-specified tolerance on the L2 relative error norm of the scalar neutron flux is utilized by the stepsize controller. Controllers that include integral, proportional, and/or derivative components are investigated and parameterized using Latin hypercube sampling of the controller input parameters. Three distinct fuel loadings of pressurized water reactor 17 × 17 fuel pin assemblies are considered, including no burnable absorbers, Integral Fuel Burnable Absorber, and gadolinium fuel pins. The required depletion stepsizes, as predicted throughout the cycle by the controller, are compared with a very small stepsize (0.01 MW d/kgHM) reference solution and a solution obtained by a typical rule of thumb depletion stepsize sequence.

A fractional-step lattice Boltzmann flux solver for axisymmetric thermal flows

ABSTRACTA fractional-step lattice Boltzmann flux solver is proposed in this work for effective simulation of axisymmetric thermal flows with rotating walls. The predictor and corrector steps are introduced in the solver. In the predictor step, excluding axisymmetric effects, the intermediate flow variables are predicted by the lattice Boltzmann flux solver (LBFS), which applies the finite-volume method to discretize the conservative equations recovered by the standard lattice Boltzmann method (LBM). The fluxes of the LBFS at the cell interfaces are reconstructed by local application of the lattice Boltzmann (LB) model with three distribution functions. These three distribution functions are used respectively for calculating axial and radial velocities, azimuthal velocity, and internal energy. In the corrector step, the intermediate flow variables are corrected by considering the axisymmetric effects. The present method not only retains the simplicity of the LBM but also eliminates the complicated derivation process in the axisymmetric LB model. The reliability of the proposed solver is examined by its application to simulate natural convection in an annulus, the Rayleigh-Benard convection, mixed convection in a vertical tall annulus, and Wheeler’s benchmark problem in crystal growth. The numerical results obtained agree well with the published data.

A New Total Variation Diminishing Predictor Corrector Approach for Two-Dimensional Shallow Water Flow

This paper presents a new efficient and robust hydrodynamic model for simulation of unsteady shallow water flow. The governing equations of shallow water flows in two dimensional forms are solved using a new total variation diminishing (TVD) MacCormack predictor corrector scheme. In this numerical technique an additional TVD term is added after the traditional predictor corrector steps. The advantage of the present TVD term is that it is very simple and gives accurate results at the same time removing the numerical oscillations. Further, application of semi implicit treatment of the friction slope term helps in flow simulation even with very low water depth. Finally the model is used to analyze a wide variety of hydraulic problems including quiescent water above irregular bed, steady flow over irregular bed, steady flow over irregular bed with a shock, dam break flow over dry bed and dam break flow over wet bed. For each of the cases numerical results are compared with available analytical solution and known experimental data. The agreements between the results are satisfactory.

A Predictor–Corrector Algorithm for Monotone Linear Complementarity Problems in a Wide Neighborhood

We propose a new primal-dual interior-point predictor–corrector algorithm in Ai and Zhang’s wide neighborhood for solving monotone linear complementarity problems (LCP). Based on the understanding of this neighborhood, we use two new directions in the predictor step and in the corrector step, respectively. Especially, the use of new corrector direction also reduces the duality gap in the corrector step, which has good effects on the algorithm’s convergence. We prove that the new algorithm has a polynomial complexity of [Formula: see text], which is the best complexity result so far. In the paper, we also prove a key result for searching for the best step size along some direction. Considering local convergence, we revise the algorithm to be a variant, which enjoys both complexity of [Formula: see text] and Q-quadratical convergence. Finally, numerical result shows the effectiveness and superiority of the two new algorithms for monotone LCPs.

A Mizuno-Todd-Ye predictor-corrector infeasible-interior-point method for linear programming over symmetric cones

We propose a Mizuno---Todd---Ye predictor-corrector infeasible-interior-point method for linear programming over symmetric cones by using a wide neighborhood. In the corrector step, we adopt a special strategy, which can ensure the existence of a step size to keep every iteration in the given small neighborhood. By using an elegant analysis, we obtain the iteration bounds for a commutative class of directions. In particular, the iteration bound is źź(rlogźź1)$\mathcal {O}(r\log \varepsilon ^{-1})$ for the Nesterov-Todd search direction, and źź(r3/2logźź1)$\mathcal {O}(r^{3/2}\log \varepsilon ^{-1})$ for the xs and sx search direction. To our knowledge, the obtained iteration bounds match the currently best known iteration bounds for infeasible-interior-point method. Some preliminary numerical results are provided as well.

Error estimation approach for controlling the comunication step size for semi‐implicit co‐simulation methods

AbstractIn this paper, an approach for controlling the macro‐step size in connection with co‐simulation methods [1, 2, 4] is suggested. The investigated step‐size controller is tailored for semi‐implicit co‐simulation techniques. Concretely, we consider predictor/corrector co‐simulation approaches [3]. By comparing variables from the predictor and the corrector step, an error estimator for the local error can be constructed. Making use of the estimated local error, a step‐size controller for the macro step‐size can be implemented. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modified liquid entrainment fraction correlation for varying pipe orientation and system pressure

2015 Elsevier Ltd. All rights reserved. Review Cioncolini and Thome (2012) proposed a correlation to predict the liquid entrainment fraction in vertical upward gas–liquid annular two phase flow. Their correlation is based on a two-step method of predictor and corrector technique. This technique calculates an approximate value of liquid entrainment fraction in the first (predictor) step and then recalculates its corrected value in the second (corrector) step. In the predictor step, as shown in Eq. (1), the predicted value of gas core Weber number (Wec,p) is calculated using gas density (qG) [kg/m], superficial gas velocity (USG) [m/s], pipe diameter (D) [m] and the gas–liquid interface surface tension (r) [N/m]. Based on this predicted Weber number, the first estimate of liquid entrainment fraction (Ep) is calculated from Eq. (2) where the multiplying factor (n) of 279.6 and exponents of 0.8395 and 2.209 are determined from the experimental data for vertical upward flow by solving for a nonlinear regression problem and are based on least absolute residuals. Next, the gas core density (qc) [kg/m] based on two phase flow quality (x) and the gas and liquid phase density (qG and qL) is calculated using Ep in Eq. (3) and the corrected values of gas core Weber number (Wec) and liquid entrainment fraction (E) are calculated in Eqs. (4) and (5), respectively. Wec;p 1⁄4 DqGU 2 SG r ð1Þ Ep 1⁄4 ð1þ n We 0:8395 c;p Þ 2:209 where n 1⁄4 279:6 ð2Þ qc 1⁄4 xþ Epð1 xÞ ðx=qGÞ þ Epð1 xÞ=qL ð3Þ Wec 1⁄4 DqcU 2 SG r ð4Þ E 1⁄4 ð1þ n We 0:8395 c Þ 2:209 where n 1⁄4 279:6 ð5Þ The correlation of Cioncolini and Thome (2012) is based on the experimental data set consisting of 2293 data points for circular pipes and 71 data points for non-circular pipes in vertical upward flow. Their correlation also used 96 data points in horizontal two phase flow for preliminary comparison and to validate their correlation (based on data in vertical upward two phase flow). For horizontal flow, Cioncolini and Thome (2012) found a significant scatter between the measured data and the predicted values however; they concluded that the general trend of the liquid entrainment in horizontal flow was correctly captured by their correlation.

A mass-conservative finite volume predictor–corrector solution of the 1D Richards’ equation

Summary Numerical solution of the Richards’ equation (RE) in variably saturated soils continues to be a challenge due to its highly non-linear behavior. This is particularly true as soils approach saturation and the behavior of the fundamental partial differential equation changes from elliptic to parabolic. In this paper, a finite volume predictor–corrector method with adaptive time-stepping was developed to solve the 1D vertical RE. The numerical method was mass-conservative and non-iterative. In the predictor step, the pressure head-based form of the RE was solved using the cell-centered finite volume method and the pressure head was updated. In the corrector step, the soil water content was calculated by solving the mixed form RE. Five different schemes to evaluate the inter-cell hydraulic conductivity were investigated. The robustness and accuracy of the numerical model were demonstrated through simulation of experimental tests, including free drainage, field infiltration into wet and dry soils, and laboratory infiltration with falling water table. Numerical results were compared against laboratory measurements, simulation results from the Hydrus-1D program, or analytical solution when available. Results showed that the developed scheme is robust and accurate in simulating variably saturated flows with various boundary conditions. The arithmetic mean and Szymkiewicz’s mean of inter-cell hydraulic conductivity performed better than other methods especially in the case of infiltration into very dry soil.

An immersed boundary-lattice Boltzmann flux solver and its applications to fluid–structure interaction problems

An immersed boundary-lattice Boltzmann flux solver (IB–LBFS) for the simulation of two-dimensional fluid–structure interaction (FSI) problems is presented in this paper. The IB–LBFS applies the fractional-step method to split the overall solution process into the predictor step and the corrector step. In the predictor step, the intermediate flow field is predicted by applying the LBFS (lattice Boltzmann flux solver) without considering the presence of immersed object. The LBFS applies the finite volume method to solve N–S (Navier–Stokes) equations for the flow variables at cell centers. At each cell interface, the LBFS evaluates its viscous and inviscid fluxes simultaneously through local reconstruction of the LBE (lattice Boltzmann equation) solutions. In the corrector step, the intermediate flow field is corrected by the implicit boundary condition-enforced immersed boundary method (IBM) so that the no-slip boundary conditions can be accurately satisfied. The IB–LBFS effectively combines the advantages of the LBFS in solving the flow field and the flexibility of the IBM in dealing with boundary conditions. Consequently, the IB–LBFS presents a much simpler and more effective approach for simulating complex FSI problems on non-uniform grids. Several test cases, including flows past one and two cylinders with prescribed motions, are firstly simulated to examine the accuracy of present solver. After that, two strongly coupled fluid–structure interaction problems, i.e., particle sedimentations and vortex-induced vibrations of a circular cylinder are investigated. Good agreements between the present results and those in literature verify the capability and flexibility of IB–LBFS for simulating FSI problems.

A numerical method to simulate turbulent cavitating flows

The objective of this paper is to develop a numerical method for simulating multiphase cavitating flows on unstructured grids. The multiphase medium is represented using a homogeneous mixture model that assumes thermal equilibrium between the liquid and vapor phases. We develop a predictor–corrector approach to solve the governing Navier–Stokes equations for the liquid/vapor mixture, together with the transport equation for the vapor mass fraction. While a non-dissipative and symmetric scheme is used in the predictor step, a novel characteristic-based filtering scheme with a second order TVD filter is developed for the corrector step to handle shocks and material discontinuities in non-ideal gases and mixtures. Additionally, a sensor based on vapor volume fraction is proposed to localize dissipation to the vicinity of discontinuities. The scheme is first validated for simple one dimensional canonical problems to verify its accuracy in predicting jump conditions across material discontinuities and shocks. It is then applied to two turbulent cavitating flow problems – over a hydrofoil using RANS and over a wedge using LES. Our results show that the simulations are in good agreement with experimental data for the above tested cases, and that the scheme can be successfully applied to both RANS and LES methodologies.

On operator split technique for the time integration within finite strain viscoplasticity in explicit FEM

AbstractIn this contribution, the operator split technique is applied to the time integration within viscoplasticity for explicit FEM. As an example, the finite strain viscoplastic material model of Shutov and Kreißig is analyzed. In the new solution scheme, some evolution equations are solved using an explicit update formula for implicit time stepping. The solution procedure is split into three steps: an elastic predictor and two viscoplastic corrector steps. Aspects of accuracy and stability of the algorithm are discussed. As shown, the proposed method is superior compared to a fully explicit integration of evolution equations. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Predictor-Corrector Algorithm for P ∗(κ)-Linear Complementarity Problems Based on a Specific Self-Regular Proximity Function

First-order predictor-corrector methods working in a large neighborhood of the central path are among the most efficient interior point methods. In Peng et al. (SIAM J. Optim. 15(4):1105–1127, 2005), based on a specific proximity function, a wide neighborhood of the central path is defined which matches the standard large neighborhood defined by the infinity norm. In this paper, we extend the predictor-corrector algorithm proposed for linear optimization in Peng et al. (SIAM J. Optim. 15(4):1105–1127, 2005) to P ∗(κ)-linear complementarity problems. Our algorithm performs two kinds of steps. In corrector steps, we use the specific self-regular proximity function to compute the search directions. The predictor step is the same as the predictor step of standard predictor-corrector method in the interior point method literature. We prove that our predictor-corrector algorithm has an \({\mathcal {O}}\left ((1+2\kappa )\sqrt {n}\log n\log \frac {(x^{0})^{T} s^{0}}{\epsilon }\right )\) iteration bound, which is the best known iteration complexity for problems of this type.

A framework to study QV-constraint exchange points in the maximum loadability analysis

This paper presents a framework to analyse critical levels of the power system loadability from the point of view of reactive power generation. Three main aspects are addressed: (1) the preliminary identification of the critical points of reactive power generation in terms of the power system loadability through a predictor scheme, (2) the solution of an optimisation problem to precisely determine these points in the corrector step, and (3) the classification of these critical points in terms of bifurcation type by using by-products of the optimisation process. The conventional power flow equations are expressed in rectangular coordinates, since this type of formulation allows both an accurate estimation of the power network variables in the predictor step and the computation of fast and robust solutions in the correction step. Numerical results obtained with two power networks are shown to illustrate the performance of the proposed methodology.

Semi-implicit co-simulation approach for solver coupling

Co-simulation methods are used to couple two or more subsystem solvers in time domain. Considering mechanical systems, the subsystems can be coupled in two different ways, namely either by applied forces/torques (i.e., by physical force/torque laws) or by constraint forces/torques (i.e., by algebraic constraint equations). Here, we present and analyze a semi-implicit co-simulation method for the case that the subsystems are coupled by applied forces/torques. The semi-implicit approach (predictor/corrector approach) analyzed here is more stable than explicit coupling methods and numerically more efficient than full-implicit coupling schemes. Furthermore, the approach takes only into account partial derivatives of the state variables with respect to the coupling variables so that the Jacobian matrix, which is required for the corrector step, has very small dimensions. For that reason, this semi-implicit approach can be implemented very easily and very time-efficiently, especially for the case that commercial codes without full solver access have to be coupled. Within this paper, we focus on mechanical systems. However, the presented method may also be applied to couple arbitrary non-mechanical systems and arbitrary solvers.

The Koiter–Newton approach using von Kármán kinematics for buckling analyses of imperfection sensitive structures

The Koiter–Newton approach is a reduced-basis method for nonlinear structural analyses. The method combines Koiter’s approach as a predictor step and the Newton arc-length technique as a corrector step to trace the entire load–displacement equilibrium path of a structure in a step by step manner. This computationally highly efficient and accurate solution method has recently demonstrated a superior performance in nonlinear analyses compared to standard techniques. In this paper we propose an extension to buckling imperfection sensitivity analyses exploiting the method’s stability and reliability. We present two different modeling techniques for imperfection loads that both profit from the Koiter–Newton approach reducing the initial computation steps to a small fraction thereof. We introduce von Kármán kinematics which neglect some nonlinear terms of the Green strain–displacement relations to further increase the computational efficiency of the analysis by an essential reduction of the computation effort required to obtain higher order derivatives of the strain energy. Using various numerical examples and benchmark tests we demonstrate the overall high quality and performance of the proposed method.