Fixed-point Iteration Algorithm Research Articles

Convergence results for fixed point iterative algorithms in metric spaces

In this paper we give some conditions on fn and f which imply the convergence of algorithm (2). In this way we improve some results given in [Rus, I. A., An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations, Fixed Point Theory, 13 (2012), No. 1, 179–192]. In our results, in general we do not suppose that, Ff 6= ∅. Some research directions are formulated.

A Comparison of BDD-Based Parity Game Solvers

Parity games are two player games with omega-winning conditions, played on finite graphs. Such games play an important role in verification, satisfiability and synthesis. It is therefore important to identify algorithms that can efficiently deal with large games that arise from such applications. In this paper, we describe our experiments with BDD-based implementations of four parity game solving algorithms, viz. Zielonka's recursive algorithm, the more recent Priority Promotion algorithm, the Fixpoint-Iteration algorithm and the automata based APT algorithm. We compare their performance on several types of random games and on a number of cases taken from the Keiren benchmark set.

Gaussian Process Regressions for Inverse Problems and Parameter Searches in Models of Ventricular Mechanics

Patient specific models of ventricular mechanics require the optimization of their many parameters under the uncertainties associated with imaging of cardiac function. We present a strategy to reduce the complexity of parametric searches for 3-D FE models of left ventricular contraction. The study employs automatic image segmentation and analysis of an image database to gain geometric features for several classes of patients. Statistical distributions of geometric parameters are then used to design parametric studies investigating the effects of: (1) passive material properties during ventricular filling, and (2) infarct geometry on ventricular contraction in patients after a heart attack. Gaussian Process regression is used in both cases to build statistical models trained on the results of biophysical FEM simulations. The first statistical model estimates unloaded configurations based on either the intraventricular pressure or the end-diastolic fiber strain. The technique provides an alternative to the standard fixed-point iteration algorithm, which is more computationally expensive when used to unload more than 10 ventricles. The second statistical model captures the effects of varying infarct geometries on cardiac output. For training, we designed high resolution models of non-transmural infarcts including refinements of the border zone around the lesion. This study is a first effort in developing a platform combining HPC models and machine learning to investigate cardiac function in heart failure patients with the goal of assisting clinical diagnostics.

Iterative inversion of deformation vector fields with feedback control.

Often, the inverse deformation vector field (DVF) is needed together with the corresponding forward DVF in four-dimesional (4D) reconstruction and dose calculation, adaptive radiation therapy, and simultaneous deformable registration. This study aims at improving both accuracy and efficiency of iterative algorithms for DVF inversion, and advancing our understanding of divergence and latency conditions. We introduce a framework of fixed-point iteration algorithms with active feedback control for DVF inversion. Based on rigorous convergence analysis, we design control mechanisms for modulating the inverse consistency (IC) residual of the current iterate, to be used as feedback into the next iterate. The control is designed adaptively to the input DVF with the objective to enlarge the convergence area and expedite convergence. Three particular settings of feedback control are introduced: constant value over the domain throughout the iteration; alternating values between iteration steps; and spatially variant values. We also introduce three spectral measures of the displacement Jacobian for characterizing a DVF. These measures reveal the critical role of what we term the nontranslational displacement component (NTDC) of the DVF. We carry out inversion experiments with an analytical DVF pair, and with DVFs associated with thoracic CT images of six patients at end of expiration and end of inspiration. The NTDC-adaptive iterations are shown to attain a larger convergence region at a faster pace compared to previous nonadaptive DVF inversion iteration algorithms. By our numerical experiments, alternating control yields smaller IC residuals and inversion errors than constant control. Spatially variant control renders smaller residuals and errors by at least an order of magnitude, compared to other schemes, in no more than 10 steps. Inversion results also show remarkable quantitative agreement with analysis-based predictions. Our analysis captures properties of DVF data associated with clinical CT images, and provides new understanding of iterative DVF inversion algorithms with a simple residual feedback control. Adaptive control is necessary and highly effective in the presence of nonsmall NTDCs. The adaptive iterations or the spectral measures, or both, may potentially be incorporated into deformable image registration methods.

A non-intrusive global/local approach applied to phase-field modeling of brittle fracture

This paper aims at investigating the adoption of non-intrusive global/local approaches while modeling fracture by means of the phase-field framework. A successful extension of the non-intrusive global/local approach to this setting would pave the way for a wide adoption of phase-field modeling of fracture, already well established in the research community, within legacy codes for industrial applications. Due to the extreme difference in stiffness between the global counterpart of the zone to be analized locally and its actual response when undergoing extensive cracking, the main foreseen issues are robustness, accuracy and efficiency of the fixed point iterative algorithm which is at the core of the method. These issues are tackled in this paper. We investigate the convergence performance when using the native global/local algorithm and show that the obtained results are identical to the reference phase-field solution. We also equip the global/local solution update procedure with relaxation/acceleration techniques such as Aitken’s Delta ^2-method, the Symmetric Rank One and Broyden’s methods and show that the iterative convergence can be improved significantly. Results indicate that Aitken’s Delta ^2-method is probably the most convenient choice for the implementation of the approach within legacy codes, as this method needs only tools already available for the so-called sub-modeling approach, a strategy routinely used in industrial contexts.

Numerical approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network.

In this work, we consider the numerical approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network. The mathematical model couples the 3D elastodynamics and bidomain equations for the electrophysiology in the myocardium with the 1D monodomain equation in the Purkinje network. For the numerical solution of the coupled problem, we consider a fixed-point iterative algorithm that enables a partitioned solution of the myocardium and Purkinje network problems. Different levels of myocardium-Purkinje network splitting are considered and analyzed. The results are compared with those obtained using standard strategies proposed in the literature to trigger the electrical activation. Finally, we present a numerical study that, although performed in an idealized computational domain, features all the physiological issues that characterize a heartbeat simulation, including the initiation of the signal in the Purkinje network and the systolic and diastolic phases.

An efficient method to compute solitary wave solutions of fractional Korteweg–de Vries equations

ABSTRACTConsidered here is an efficient technique to compute approximate profiles of solitary wave solutions of fractional Korteweg–de Vries equations. The numerical method is based on a fixed-point iterative algorithm along with extrapolation techniques of acceleration. This combination improves the performance in both the velocity of convergence and the computation of profiles for limiting values of the fractional parameter. The algorithm is described and numerical experiments of validation are presented. The accuracy attained by the procedure can be used to investigate additional properties of the waves. This approach is illustrated here by analysing the speed–amplitude relation.

Distribution Optimal Power Flow With Real-Time Price Elasticity

This letter investigates a new class of optimal power flow problem for distribution systems, where elastic loads respond to real-time nodal prices by adjusting their demands. A fixed-point iteration algorithm is suggested to identify an equilibrium. A concise criterion is devised to judge convergence.

Solving the large-scale knapsack feasibility problem using a distributed computation approach to integer programming

The knapsack feasibility problems have been intensively studied both because of their immediate applications in industry and financial management, but more pronounced for theoretical reasons, as knapsack problems frequently occur by relaxation of various integer programming problems. In this work, the large-scale knapsack feasibility problem is divided into two subproblems. The first subproblem is transforming of the knapsack feasibility problem into a polytope judgement problem which is based on lattice basis reduction. In the next subproblem, a distributed implementation of Dang and Ye’s fixed-point iterative algorithm is introduced to solve the polytope judgement problem generated in the former subproblem. Compared with the branch and bound method, numerical results show that this distributed fixed-point method is effective.

Maximum likelihood estimation for incomplete multinomial data via the weaver algorithm

In a multinomial model, the sample space is partitioned into a disjoint union of cells. The partition is usually immutable during sampling of the cell counts. In this paper, we extend the multinomial model to the incomplete multinomial model by relaxing the constant partition assumption to allow the cells to be variable and the counts collected from non-disjoint cells to be modeled in an integrated manner for inference on the common underlying probability. The incomplete multinomial likelihood is parameterized by the complete-cell probabilities from the most refined partition. Its sufficient statistics include the variable-cell formation observed as an indicator matrix and all cell counts. With externally imposed structures on the cell formation process, it reduces to special models including the Bradley–Terry model, the Plackett–Luce model, etc. Since the conventional method, which solves for the zeros of the score functions, is unfruitful, we develop a new approach to establishing a simpler set of estimating equations to obtain the maximum likelihood estimate (MLE), which seeks the simultaneous maximization of all multiplicative components of the likelihood by fitting each component into an inequality. As a consequence, our estimation amounts to solving a system of the equality attainment conditions to the inequalities. The resultant MLE equations are simple and immediately invite a fixed-point iteration algorithm for solution, which is referred to as the weaver algorithm. The weaver algorithm is short and amenable to parallel implementation. We also derive the asymptotic covariance of the MLE, verify main results with simulations, and compare the weaver algorithm with an MM/EM algorithm based on fitting a Plackett–Luce model to a benchmark data set.

Boxed-constraint least mean square algorithm and its performance analysis

In this paper, a novel adaptive filter algorithm, called boxed-constraint least mean square (BXCLMS) algorithm, is proposed for identifying the boxed-constrained system where the parameter to estimate is limited in a range from lower bound to upper bound. The proposed algorithm is derived by using the Karush-Kuhn-Tucker (KKT) conditions and fixed-point iteration algorithm. In addition, the stochastic behavior analysis of proposed algorithm is performed in terms of mean and mean square performance. Finally, simulations are carried out to demonstrate the performance of BXCLMS algorithm and verify the correctness of the analytical results.

An Adaptive Fixed-Point Iteration Algorithm for Finite-Element Analysis With Magnetic Hysteresis Materials

In this paper, an adaptive fixed-point iteration algorithm for 2-D/3-D finite-element analysis with hysteresis is proposed. The iteration starts with the $B$ -correction scheme. If the solution is not converged to a given accuracy after a certain number of iterations, the iteration will be continued by switching to the $H$ -correction scheme. Based on the combined use of the two correction schemes during the whole iteration process, the solution with the minimum error together with the scheme type is recorded and will be used as the final solution at the current time step. At the same time, the recorded scheme type will be used as the initial scheme type for the next time step. The numerical validation shows that the proposed algorithm not only has very fast convergence rate, but is also very stable.

Numerical approximation of a non-smooth phase-field model for multicomponent incompressible flow

We present a phase-field model for multiphase flow for an arbitrary number of immiscible incompressible fluids with variable densities and viscosities. The model consists of a system of the Navier−Stokes equations coupled to multicomponent Cahn−Hilliard variational inequalities. The proposed formulation admits a natural energy law, preserves physically meaningful constraints and allows for a straightforward modelling of surface tension effects. We propose a practical fully discrete finite element approximation of the model which preserves the energy law and the associated physical constraints. In the case of matched densities we prove convergence of the numerical scheme towards a weak solution of the continuous model. The convergence of the numerical approximations also implies the existence of weak solutions. Furthermore, we propose a convergent iterative fixed-point algorithm for the solution of the discrete nonlinear system of equations and present several computational studies of the proposed model.

New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation

ABSTRACTThe discrete coupled algebraic Riccati equation (DCARE) has wide applications in control theory and linear system. In general, for the DCARE, one discusses every term of the coupled term, respectively. In this paper, we consider the coupled term as a whole, which is different from the recent results. When applying eigenvalue inequalities to discuss the coupled term, our method has less error. In terms of the properties of special matrices and eigenvalue inequalities, we propose several upper and lower matrix bounds for the solution of DCARE. Further, we discuss the iterative algorithms for the solution of the DCARE. In the fixed point iterative algorithms, the scope of Lipschitz factor is wider than the recent results. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived results.

Structural Analysis Methods for Differential Algebraic Equations via Fixed-Point Iteration

Motivated by Pryce’s structural analysis method for differential algebraic equations (DAEs), we show the complexity of the fixed-point iteration algorithm (FPIA) and propose a fixed-point iteration method with parameters. It leads to a block fixed-point iteration method (BFPIM) which can be applied to immediately calculate the crucial canonical offsets for large-scale (coupled) DAE systems with block-triangular structure, and its complexity analysis is also given in this paper. Moreover, preliminary numerical experiments show that the time complexity of BFPIM can be reduced by at least O(l) compared to the FPIA.

Third-harmonic signal generation and enhancement in nonlinear photonic crystals with an efficient continuation finite-element method

A computational study of the third-harmonic (TH) generation (THG) and enhancement through an χ(3) process in one-dimensional nonlinear photonic crystals is presented in this paper. We have introduced a canonical variational problem with a small increment factor to solve the THG problem. The problem with a large input pump intensity or strong nonlinear susceptibilities can be handled by a versatile and accurate method, which combines finite-element methods and a continuation fixed-point iteration algorithm. The TH signal generation and enhancement through the direct χ(3) process are displayed by two periodic structures. Numerical experiments show that the TH signal is significantly enhanced when the frequencies of the fundamental wave and the TH wave are turned to the corresponding photonic band edges to meet the phase-matching condition. Results of numerical experiments also indicate that our proposed method is computationally efficient and precise to handle strong nonlinearities, i.e., large values of input pump power.

Phase-updated regularized SENSE for navigator-free multishot diffusion imaging.

Either SENSE+CG or POCS-ICE methods can be used to correct for motion-induced phase errors in navigator-free multishot diffusion imaging. SENSE+CG has the advantage of a fast convergence, however, occasionally the convergence can be unstable, thus degrading the image quality. POCS-ICE has a stable convergence and can be used with a high number of shots, but its convergence is slow, which limits its practical usage. The study here proposes an improved method based on both SENSE+CG and POCS-ICE, called Phase-updated Regularized SENSE (PR-SENSE), for navigator-free multishot diffusion imaging. In PR-SENSE, a total variation regularization method is used to solve the SENSE inverse problem instead of the conjugate gradient method used in SENSE+CG. This method is implemented by using a lagged diffusivity fixed point iteration algorithm. Additionally, the phase is updated during the iteration process to improve the image accuracy. Simulations and in vivo experiments demonstrated that PR-SENSE can successfully correct for the motion-induced phase errors in multi-shot DWI. It integrates the advantages of SENSE+CG and POCS-ICE, resulting in a fast and stable convergence with improved image quality. Given its advantages, PR-SENSE is a significant improvement over other methods for navigator-free high-resolution DWI. Magn Reson Med 78:172-181, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

The solution bounds and fixed point iterative algorithm for the discrete coupled algebraic Riccati equation applied to automatic control

The solution bounds and fixed point iterative algorithm for the discrete coupled algebraic Riccati equation applied to automatic control

SU‐F‐J‐79: Extension of Fixed‐Point Iteration Algorithm for Inverse Deformation

Purpose:(I) To present a modification of the fixed point iteration algorithm to assure the convergence for a wide range of cases such as rotation and scaling deformations and (II) to provide implementation details with results from simulations and clinical images and emphasize more on practical considerations, such as how to choose good initial values to improve the speed of convergence. Inverse deformations are often applied to images, contours, and dose.Methods:The 2D or 3D inverse deformation field is iteratively found by taking a partial step size parameter (< 1) in a given direction which depends on the forward deformation field as opposed to a full step size parameter (= 1) as in the original fixed‐point inverse deformation paper. The step size parameter controls the size of the change in the inverse deformation field during each iteration. The step size parameter is also dynamically changed to achieve convergence faster.Results:The new method was implemented in Matlab and tested for both 2D simulated cases and 3D clinical cases. We started the main iteration loop with step size parameter = 1 and then dynamically changed it to achieve convergence of maximum voxel error of 0.01. The same method was implemented for deformation fields from deformable registration of three patients to another set of patient data. Convergence of the inverse deformation fields was also achieved for the patient cases.Conclusion:If the step size parameter is small enough, then the improved iterative algorithm achieves convergence for a wider array of cases such as rotations and scaling of the image. The method is demonstrated to be successful for 2D simulated data and 3D patient data sets and allows for the finding of inverse deformation fields for a wider range of cases.Supported by Cancer Prevention & Research Institute of Texas (CPRIT) ‐ ID RP150485

Proximal point algorithms for nonsmooth convex optimization with fixed point constraints

The problem of minimizing the sum of nonsmooth, convex objective functions defined on a real Hilbert space over the intersection of fixed point sets of nonexpansive mappings, onto which the projections cannot be efficiently computed, is considered. The use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed for solving the problem. With the focus on fixed point approximation techniques, two algorithms are devised for solving the problem. One blends an incremental subgradient method, which is a useful algorithm for nonsmooth convex optimization, with a Halpern-type fixed point iteration algorithm. The other is based on an incremental subgradient method and the Krasnosel’skiĭ–Mann fixed point algorithm. It is shown that any weak sequential cluster point of the sequence generated by the Halpern-type algorithm belongs to the solution set of the problem and that there exists a weak sequential cluster point of the sequence generated by the Krasnosel’skiĭ–Mann-type algorithm, which also belongs to the solution set. Numerical comparisons of the two proposed algorithms with existing subgradient methods for concrete nonsmooth convex optimization show that the proposed algorithms achieve faster convergence.