Iterative Projection Method Research Articles

A distributed memory parallel randomized Kaczmarz for sparse system of equations

AbstractKaczmarz algorithm is an iterative projection method for solving system of linear equations that arise in science and engineering problems in various application domains. In addition to classical Kaczmarz, there are randomized and parallel variants. The main challenge of the parallel implementation is the dependency of each Kaczmarz iteration on its predecessor. Because of this dependency, frequent communication is required which results in a substantial overhead. In this study, a new distributed parallel method that reduces the communication overhead is proposed. The proposed method partitions the problem so that the Kaczmarz iterations on different blocks are less dependent. A frequency parameter is introduced to see the effect of communication frequency on the performance. The communication overhead is also decreased by allowing communication between processes only if they have shared non‐zero columns. The experiments are performed using problems from various domains to compare the effects of different partitioning methods on the communication overhead and performance. Finally, parallel speedups of the proposed method on larger problems are presented.

A sharp immersed method for electrohydrodynamic flows accompanied by charge evaporation

AbstractThis article presents a sharp immersed method for simulating electrohydrodynamic (EHD) flows that involve charge evaporation. This well‐known multi‐scale, multi‐physics problem is widely used in various fields, including industry and medicine. The method adopts a fully sharp model, where surface tension and Maxwell stress are treated as surface forces and free charges are concentrated on the zero thickness liquid‐vacuum interface. Incorporating charge evaporation imposes strict restrictions on the time‐step, as the rate of evaporation sharply increases with surface evolution. To overcome this challenge, an iterative algorithm that couples the electric field and surface charge density is proposed to obtain accurate results, even with significantly large time‐steps. To mitigate the numerical residuals near the interface, which may introduce parasitic flows and cause numerical instability, an immersed interface method‐based iterative projection method for the Navier–Stokes equations is proposed, in which a traction boundary condition involving multiple surface forces is imposed on the sharp interface. Numerical experiments were carried out, and the results show that the method is splitting‐error‐free and stable. The sharp immersed method is applied to simulate the electric‐induced deformation of an ionic liquid drop with charge evaporation. The results indicate that charge evaporation can suppress the sharp development of Taylor cones at the ends of the drops. These findings have significant implications for the design and optimization of EHD systems in various applications.

Subspace recursive Fermi-operator expansion strategies for large-scale DFT eigenvalue problems on HPC architectures.

Quantum mechanical calculations for material modeling using Kohn-Sham density functional theory (DFT) involve the solution of a nonlinear eigenvalue problem for N smallest eigenvector-eigenvalue pairs, with N proportional to the number of electrons in the material system. These calculations are computationally demanding and have asymptotic cubic scaling complexity with the number of electrons. Large-scale matrix eigenvalue problems arising from the discretization of the Kohn-Sham DFT equations employing a systematically convergent basis traditionally rely on iterative orthogonal projection methods, which are shown to be computationally efficient and scalable on massively parallel computing architectures. However, as the size of the material system increases, these methods are known to incur dominant computational costs through the Rayleigh-Ritz projection step of the discretized Kohn-Sham Hamiltonian matrix and the subsequent subspace diagonalization of the projected matrix. This work explores the potential of polynomial expansion approaches based on recursive Fermi-operator expansion as an alternative to the subspace diagonalization of the projected Hamiltonian matrix to reduce the computational cost. Subsequently, we perform a detailed comparison of various recursive polynomial expansion approaches to the traditional approach of explicit diagonalization on both multi-node central processing unit and graphics processing unit architectures and assess their relative performance in terms of accuracy, computational efficiency, scaling behavior, and energy efficiency.

Enhancing Block Cimmino for Sparse Linear Systems with Dense Columns via Schur Complement

The block Cimmino is a parallel hybrid row-block projection iterative method successfully used for solving general sparse linear systems. However, the convergence of the method degrades when angles between subspaces spanned by the row-blocks are far from being orthogonal. The density of columns as well as the numerical values of their nonzeros are more likely to contribute to the nonorthogonality between row-blocks. We propose a novel scheme to handle such “dense” columns. The proposed scheme forms a reduced system by separating these columns and the respective rows from the original coefficient matrix and handling them via the Schur complement. Then the angles between subspaces spanned by the row-blocks of the reduced system are expected to be closer to orthogonal, and the reduced system is solved efficiently by the block conjugate gradient (CG) accelerated block Cimmino in fewer iterations. We also propose a novel metric for selecting “dense” columns considering the numerical values. The proposed metric establishes an upper bound on the sum of inner products between row-blocks. Then we propose an efficient algorithm for computing the proposed metric for the columns. Extensive numerical experiments for a wide range of linear systems confirm the effectiveness of the proposed scheme by achieving fewer iterations and faster parallel solution time compared to the classical CG accelerated block Cimmino algorithm.

An Approximation Solution of Linear Differential Equation using Kantorovich Methods

In our work, we constructed a numerical approximations method to deal with approximations of a linear differential equation. We explained the general framework of the projection method which helps to clarify the basic ideas of the Kantorovich methods. We applied the iterative projection methods and presented a theorem to show the convergence of the constructed solutions to the exact solution. Also, most of the expressions encountered earlier can be used to define functions. Here are some illustrations. A great deal of information can be learned about a functioning relationship by studying its graph. A fundamental objective of section 4, is to acquaint with the graphs of some important functions and develop basic graphing procedures.

Iterative projection algorithm for retrieval of angle-resolved single-molecule dipoles from high-harmonic spectra

High-order harmonic spectroscopy (HHS), as an emerging technique, provides a way to detect molecular structures with \aa{}ngstr\"om spatial resolution and ultrafast electron dynamics in molecules on an attosecond timescale. However, the implementation of HHS still encounters great difficulties due to the coherent average of single-molecule emissions from different alignment angles. Here, we provide a machine learning algorithm based on the iterative projection method to fully retrieve the complex single-molecule dipole of a linear molecule from the harmonic spectra measured at different delays and polarization angles between the alignment and driving pulses. We demonstrated our algorithm both theoretically and experimentally using the ${\mathrm{N}}_{2}$ molecule. The results show that our algorithm can accurately retrieve the complex single-molecule dipole from the harmonic spectra with good noise stability and robustness. From the retrieved single-molecule dipole, the contributions of multiple orbitals in high harmonic generation are identified even with a low degree of alignment.

Parallel Sparse Computation Toolkit

This paper presents a new software framework for solving large and sparse linear systems on current hybrid architectures, from small servers to high-end supercomputers, embedding multi-core CPUs and Nvidia GPUs at the node level. The framework has a modular structure and is composed of three main components, which separate basic functionalities for managing distributed sparse matrices and executing some sparse matrix computations involved in iterative Krylov projection methods, eventually exploiting multi-threading and CUDA-based programming models, from the functionalities for setup and application of different types of one-level and multi-level algebraic preconditioners.

Fast IPPM for Distributed Cooperative Localization With Hybrid Information

This letter addresses the problem of distributed cooperative localization using the joint range and angle measurements. With the appealing properties of fully distributed implementation and low complexity at each agent, a convex formulation named fast iterative parallel projection method (FastIPPM) is exploited to solve the cooperative localization problem. In pursuit of faster convergence, gradient descent with momentum is utilized. Simulation results validate the superior performance of the proposed algorithm compared with existing ones. After adopting the acceleration techniques, the FastIPPM shows a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\times$ </tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3\times$ </tex-math></inline-formula> speed-up. In addition, positioning accuracy and convergence stability in sparse networks have also been greatly enhanced.

Iterative methods for vector equilibrium and fixed point problems in Hilbert spaces

In this paper, algorithms for finding common solutions of strong vector equilibrium and fixed point problems of multivalued mappings are considered. First, a Minty vector equilibrium problem is introduced and the relationship between the Minty vector equilibrium problem and the strong equilibrium problem is discussed. Then, by applying the Minty vector equilibrium problem, projection iterative methods are proposed and some convergence results are established in Hilbert spaces. The main results obtained in this paper develop and improve some recent works in this field.

Multi-Scale Single-Bit RP-EMS Synthesis for Advanced Propagation Manipulation Through System-by-Design

A new method for synthesizing Single-Bit Reconfigurable Passive Electromagnetic Skins (1RP-EMSs) featuring advanced beam shaping capabilities is proposed. By using single-bit unit cells, the multi-scale problem of controlling 1RP-EMSs is formulated as a two-phase process. First, the macro-scale synthesis of the discrete surface current that radiates the electromagnetic (EM) field fitting user-designed requirements is performed by means of an innovative quantized version of the iterative projection method (QIPM). Successively, the meta-atom states of the 1RP-EMS are optimized with a customized implementation of the System-by-Design paradigm to yield a 1RP-EMS that supports such a feasible reference current. A representative set of numerical results is reported to assess the effectiveness of the proposed approach in designing and controlling single-bit meta-atom RP-EMSs that enable complex wave manipulations.

Solving a directed-percolation inverse problem.

We present a directed-percolation inverse problem for diode networks: Given information about which pairs of nodes allow current to percolate from one to the other, can one find a configuration of diodes consistent with the observed currents? We implement a divide-and-concur iterative projection method for solving the problem and demonstrate the supremacy of our method over an exhaustive approach for nontrivial instances of the problem. We find that the problem is most difficult when some but not all of the percolation data are hidden, and that the most difficult networks to reconstruct generally are those for which the currents are most sensitive to the addition or removal of a single diode.

An effective iterative projection method for variational inequalities in Hilbert spaces

An effective iterative projection method for variational inequalities in Hilbert spaces

Recognition and Detection of Wide Field Bionic Compound Eye Target Based on Cloud Service Network.

In this paper, a multidisciplinary cross-fusion of bionics, robotics, computer vision, and cloud service networks was used as a research platform to study wide-field bionic compound eye target recognition and detection from multiple perspectives. The current research status of wide-field bionic compound-eye target recognition and detection was analyzed, and improvement directions were proposed. The surface microlens array arrangement was designed, and the spaced surface bionic compound eye design principle cloud service network model was established for the adopted spaced-type circumferential hierarchical microlens array arrangement. In order to realize the target localization of the compound eye system, the content of each step of the localization scheme was discussed in detail. The distribution of virtual spherical targets was designed by using the subdivision of the positive icosahedron to ensure the uniformity of the targets. The spot image was pre-processed to achieve spot segmentation. The energy symmetry-based spot center localization algorithm was explored and its localization effect was verified. A suitable spatial interpolation method was selected to establish the mapping relationship between target angle and spot coordinates. An experimental platform of wide-field bionic compound eye target recognition and detection system was acquired. A super-resolution reconstruction algorithm combining pixel rearrangement and an improved iterative inverse projection method was used for image processing. The model was trained and evaluated in terms of detection accuracy, leakage rate, time overhead, and other evaluation indexes, and the test results showed that the cloud service network-based wide-field bionic compound eye target recognition and detection performs well in terms of detection accuracy and leakage rate. Compared with the traditional algorithm, the correct rate of the algorithm was increased by 21.72%. Through the research of this paper, the wide-field bionic compound eye target recognition and detection and cloud service network were organically provide more technical support for the design of wide-field bionic compound eye target recognition and detection system.

Avoiding traps in nonconvex problems

Iterative projection methods may become trapped at non-solutions when the constraint sets are nonconvex. Two kinds of parameters are available to help avoid this behavior and this study gives examples of both. The first kind of parameter, called a hyperparameter, includes any kind of parameter that appears in the definition of the iteration rule itself. The second kind comprises metric parameters in the definition of the constraint sets, a feature that arises when the problem to be solved has two or more kinds of variables. Through examples we show the importance of properly tuning both kinds of parameters and offer heuristic interpretations of the observed behavior.

Nonlinear iterative projection methods with multigrid in photon frequency for thermal radiative transfer

This paper presents nonlinear iterative methods for the fundamental thermal radiative transfer (TRT) model defined by the time-dependent multifrequency radiative transfer (RT) equation and the material energy balance (MEB) equation. The iterative methods are based on the nonlinear projection approach and use multiple grids in photon frequency. They are formulated by the high-order RT equation on a given grid in photon frequency and low-order moment equations on a hierarchy of frequency grids. The material temperature is evaluated in the subspace of lowest dimensionality from the MEB equation coupled to the effective grey low-order equations. The algorithms apply various multigrid cycles to visit frequency grids. Numerical results are presented to demonstrate convergence of the multigrid iterative algorithms in TRT problems with large number of photon frequency groups.

Hybrid projection methods for large-scale inverse problems with mixed Gaussian priors

When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix (e.g., in data assimilation) or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub–Kahan process, which is an extension of the generalized Golub–Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process. Furthermore, for problems where training data are available, various data-driven covariance matrices (including those based on learned covariance kernels) can be easily incorporated. Numerical examples from tomographic reconstruction demonstrate the potential for these methods.

A parallel, non-spatial iterative, and rotational pressure projection method for the nonlinear fluid-fluid interaction

In this paper, a parallel, non-spatial iterative, and rotational pressure projection method for the coupled Navier-Stokes equations is proposed and developed. As for each Navier-Stokes equation, one elliptic equation and one Possion equation at each time step determine the values of the velocity and pressure, respectively. Moreover, we only need to solve four simple linear equations, and the computational time of the whole system is greatly reduced. The rotational pressure projection method achieves the same accuracy as the traditional decoupled fractional time-stepping method. Furthermore, we apply the presented method, the rotational pressure projection method and the linear decoupled fractional time-stepping method to the submarine mountain problem. Compared to the streamline diagrams of these three methods from two aspects of accuracy and CPU times, we obtain that the presented method has good parallelism and is the most efficient method, the rotational pressure projection method is more efficient than the traditional decoupled fractional time-stepping method. Finally, numerical results verify that our proposed method has the high efficiency.

Macromolecular phasing using diffraction from multiple crystal forms.

A phasing algorithm for macromolecular crystallography is proposed that utilizes diffraction data from multiple crystal forms - crystals of the same molecule with different unit-cell packings (different unit-cell parameters or space-group symmetries). The approach is based on the method of iterated projections, starting with no initial phase information. The practicality of the method is demonstrated by simulation using known structures that exist in multiple crystal forms, assuming some information on the molecular envelope and positional relationships between the molecules in the different unit cells. With incorporation of new or existing methods for determination of these parameters, the approach has potential as a method for ab initio phasing.

Hybrid Projection Methods with Recycling for Inverse Problems

Iterative hybrid projection methods have proven to be very effective for solving large linear inverse problems due to their inherent regularizing properties as well as the added flexibility to sele...

Моделювання крайових задач для лінійних диференціально-різницевих рівнянь нейтрального типу

In space navigation problems, optimal control of systems with aftereffect, ecology and immunology problems, boundary value problems for differential-difference and integro-differential equations with delays arise, which are an important part of modern theory of differential-functional equations. Finding precise solutions of boundary value problems for differential-difference equations is possible only for the simplest classes of such problems. At present, projection-iterative methods, numerical-analytical method and others are suggested for boundary value problems with delay and of neutral type. The spline-collocation method for solving boundary value problems for differential-difference equations is one of the most efficient algorithms that allows building simple computational schemes. In this paper, we investigate the scheme of modeling boundary value problems for linear differential-difference equations of neutral type with many variable deviations of the argument. A functional space is defined to which the solutions of the considered boundary value problems belong, the properties of the solution smoothness are investigated depending on the structure of the argument deviations. Simple and verifiable sufficient conditions for the boundary value problem solution existence are given. For finding the solution of the boundary value problem, an iterative computational scheme based on the spline approximation method is described. In order to take into account possible discontinuities of the boundary value problem solution derivatives, cubic splines of defect two are used for neutral-type equations. Coefficient conditions for the initial equation which ensure the convergence of the iterative process are obtained. An estimate of the approximate solution error is conducted. A model example of a boundary value problem for a neutral type differential-difference equation is presented on which the iterative scheme is demonstrated. Numerical experiments confirm the obtained theoretical results.