Spectral Projection Method Research Articles

Superconvergent spectral projection and multi-projection methods for nonlinear Fredholm integral equations

In this article, we apply Galerkin and multi-Galerkin methods based on Kumar Sloan technique using Legendre polynomial basis functions for approximating the nonlinear Fredholm Hammerstein integral equations with the smooth kernels as well as the weakly singular algebraic and logarithmic kernels. We show that we are getting the improved superconvergence rates for Galerkin and multi-Galerkin methods based on Kumar Sloan technique without the need for the iterated Galerkin and the iterated multi-Galerkin methods. Infact without going to the iterated versions, we obtain the superconvergence rates equal to the convergence rates of iterated Galerkin and iterated multi-Galerkin methods. Theoretical results have been illustrated with numerical experiments.

Hybridized Brazilian–Bowein type spectral gradient projection method for constrained nonlinear equations

This paper proposes a hybridized Brazilian and Bowein derivative-free spectral gradient projection method for solving systems of convex-constrained nonlinear equations. The method avoids solving any subproblems in each iteration. Global convergence is established under appropriate assumptions on the functions involved. Additionally, numerical experiments are conducted to evaluate the algorithm’s performance, providing evidence of its efficiency compared to similar algorithms from the existing literature. The results demonstrate that the method outperforms some existing approaches in terms of the number of iterations, function evaluations, and time required to obtain a solution based on the examples considered.

Superconvergence results for hypersingular integral equation of first kind by Chebyshev spectral projection methods

In this article, we propose Chebyshev spectral projection methods to solve the hypersingular integral equation of first kind. The presence of strong singularity in Hadamard sense in the first part of the integral equation makes it challenging to get superconvergence results. To overcome this, we transform the first kind hypersingular integral equation into a second kind integral equation. This is achieved by defining a bounded inverse of the hypersingular integral operator in some suitable Hilbert space. Using iterated Chebyshev spectral Galerkin method on the equivalent second kind integral equation, we obtain improved convergence of O(N−2r), where N is the highest degree of Chebyshev polynomials employed in the approximation space and r is the smoothness of the solution. Further, using commutativity of projection operator and inverse of the hypersingular integral operator, we are able to obtain superconvergence of O(N−3r) and O(N−4r), by Chebyshev spectral multi-Galerkin method (CSMGM) and iterated CSMGM, respectively. Finally, numerical examples are presented to verify our theoretical results.

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

Spectral Projection Methods for Derivative Dependent Hammerstein Equations with Green’s Kernels

Spectral Projection Methods for Derivative Dependent Hammerstein Equations with Green’s Kernels

An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations

An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations

Normalized Newton method to solve generalized tensor eigenvalue problems

AbstractThe problem of generalized tensor eigenvalue is the focus of this paper. To solve the problem, we suggest using the normalized Newton generalized eigenproblem approach (NNGEM). Since the rate of convergence of the spectral gradient projection method (SGP), the generalized eigenproblem adaptive power (GEAP), and other approaches is only linear, they are significantly improved by our proposed method, which is demonstrated to be locally and cubically convergent. Additionally, the modified normalized Newton method (MNNM), which converges to symmetric tensors Z‐eigenpairs under the same ‐Newton stability requirement, is extended by the NNGEM technique. Using a Gröbner basis, a polynomial system solver (NSolve) generates all of the real eigenvalues for us. To illustrate the efficacy of our methodology, we present a few numerical findings.

A modified technique of spectral gradient projection method for solving nonlinear equations systems

This study suggests a new line search algorithm modified of Spectral gradient projection method (SGPM) called (D.A), combining the modified spectral gradient method and projection method for solving nonlinear equations systems. This approach can be more efficient because it requires less CPU time, less function evaluation, and fewer iterations. Furthermore, if this equation is Lipschitz continuous (LC) and monotone, we proved that this technique converges globally. The numerical results show how effective and beneficial this method is for solving nonlinear equations systems.

Combination of Karhunen-Loève and intrusive polynomial chaos for uncertainty quantification of thermomagnetic convection problem with stochastic boundary condition

Uncertainty propagation analysis plays a crucial role in understanding the impact of variations in initial condition, boundary condition, and fluid physical properties on simulation results for thermomagnetic convection heat transfer problems. To effectively analyze the uncertainty problem of thermomagnetic convection caused by random temperature fluctuations, a novel and comprehensive computational framework based on intrusive polynomial chaos approach was proposed in this paper. Our proposed approach aims to represent random temperature boundaries using Karhunen-Loève expansion (KLE) and stochastic output response using polynomial chaos expansion (PCE). In addition, we use spectral projection method to transform the stochastic control equation into a group of deterministic control equations. We obtained the uncertainty characteristics of the stochastic output response by solving each polynomial chaos basis. Compared with Monte Carlo simulation (MCS), our approach accurately and efficiently predicts the uncertainty characteristics of thermomagnetic convection problems under stochastic temperature boundary conditions while significantly reducing computational resources.

Superconvergence of Legendre spectral projection methods for [formula omitted]th order integro-differential equations with weakly singular kernels

In this article, we apply Legendre spectral Galerkin, Legendre spectral multi-projection methods and their iterated versions to find the approximate solution of mth order Fredholm integro-differential equations with weakly singular kernel. Motivated by Mandal et al. (2023), we use Cauchy repeated integral theorem to transform the integro-differential equation to an single integral equation and obtain superconvergence results by iterated Legendre spectral Galerkin method, in spite of the singularity in the kernel function and unbounded differential operator. We have further improved the convergence rate of the approximate solution by using iterated Legendre spectral multi-projection method. Global Legendre polynomials are used as a basis for the approximating space, to reduce the computational cost of our proposed methods. In this article, we have derived theoretical error bounds and obtained the global convergence rates for all the discussed methods in both L2 and infinity norms. Numerical methods are implemented on examples to justify the theoretical results.

A modified multivariate spectral gradient projection method for nonlinear complementarity problems

A modified multivariate spectral gradient projection method for nonlinear complementarity problems

Jacobi spectral projection methods for Fredholm integral equations of the first kind

Jacobi spectral projection methods for Fredholm integral equations of the first kind

Uncertainty quantification for mineral precipitation and dissolution in fractured porous media

In this work we present an uncertainty quantification analysis to determine the influence and importance of some physical parameters in a reactive transport model in fractured porous media. An accurate description of flow and transport in the fractures is key to obtain reliable simulations, however, fractures geometry and physical characteristics pose several challenges from both the modeling and implementation side. We adopt a mixed-dimensional approximation, where fractures and their intersections are represented as objects of lower dimension. To simplify the presentation, we consider only two chemical species: one solute, transported by water, and one precipitate attached to the solid skeleton. A global sensitivity analysis to uncertain input data is performed exploiting the Polynomial Chaos expansion along with spectral projection methods on sparse grids.

An Inertial Spectral CG Projection Method Based on the Memoryless BFGS Update

An Inertial Spectral CG Projection Method Based on the Memoryless BFGS Update

Legendre spectral projection methods for Fredholm integral equations of first kind

Abstract In this paper, we discuss the Legendre spectral projection method for solving Fredholm integral equations of the first kind using Tikhonov regularization. First, we discuss the convergence analysis under an a priori parameter strategy for the Tikhonov regularization using Legendre polynomial basis functions, and we obtain the optimal convergence rates in the uniform norm. Next, we discuss Arcangeli’s discrepancy principle to find a suitable regularization parameter and obtain the optimal order of convergence in uniform norm. We present numerical examples to illustrate the theoretical results.

New inertial-based spectral projection method for solving system of nonlinear equations with convex constraints

In this paper, a new spectral projection method for solving nonlinear system of equations with convex constraints is proposed based on inertial effect. The inertial technique is integrated into the new proposed search direction with the aim of enhancing the numerical performance. Interestingly, the convergence result of the new method is established based on the assumption that the underlying function is pseudomonotone. This assumption is weaker than monotonicity which is used in many existing methods to prove the convergence. The new method is suitable for large scale problems as well as nonsmooth problems. Numerical experiments presented validate the efficiency of the new method which also outperforms some existing methods in the literature

Chebyshev Spectral Projection Methods for Two-Dimensional Fredholm Integral Equations of Second Kind

In this paper, we approximate the two-dimensional linear Fredholm integral equations (fies) with smooth kernels using Chebyshev spectral Galerkin and collocation methods. The existence and convergence analysis for the problem have been discussed. The errors between approximated solutions with exact solution have been evaluated in $$L^2_\omega $$ norm applying both these methods. We also solve the associated eigenvalue problems (evps) by using above methods and obtain the errors between approximated eigenelements with the exact eigenelements in $$L^2_\omega $$ and $$L^\infty _\omega $$ norms. Numerical examples are presented to illustrate the theoretical results.

Coarse-grained spectral projection: A deep learning assisted approach to quantum unitary dynamics

We propose the coarse-grained spectral projection method (CGSP), a deep learning assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show that CGSP can extract spectral components of many-body quantum states systematically with a sophisticated neural network quantum ansatz. CGSP fully exploits the linear unitary nature of the quantum dynamics and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on one-dimensional XXZ models with periodic boundary conditions are carried out to demonstrate the practicality of CGSP.

UNCERTAINTY QUANTIFICATION IN STEADY STATE SIMULATIONS OF A MOLTEN SALT SYSTEM USING POLYNOMIAL CHAOS EXPANSION ANALYSIS

Uncertainty Quantification (UQ) of numerical simulations is highly relevant in the study and design of complex systems. Among the various approaches available, Polynomial Chaos Expansion (PCE) analysis has recently attracted great interest. It belongs to nonintrusive spectral projection methods and consists of constructing system responses as polynomial functions of the stochastic inputs. The limited number of required model evaluations and the possibility to apply it to codes without any modification make this technique extremely attractive. In this work, we propose the use of PCE to perform UQ of complex, multi-physics models for liquid fueled reactors, addressing key design aspects of neutronics and thermal fluid dynamics. Our PCE approach uses Smolyak sparse grids designed to estimate the PCE coefficients. To test its potential, the PCE method was applied to a 2D problem representative of the Molten Salt Fast Reactor physics. An in-house multi-physics tool constitutes the reference model. The studied responses are the maximum temperature and the effective multiplication factor. Results, validated by comparison with the reference model on 103 Monte-Carlo sampled points, prove the effectiveness of our PCE approach in assessing uncertainties of complex coupled models.

A Family of Modified Spectral Projection Methods for Nonlinear Monotone Equations with Convex Constraint

In this paper, we propose a family of modified spectral projection methods for nonlinear monotone equations with convex constraints, where the spectral parameter is mainly determined by a convex combination of the modified long Barzilai–Borwein stepsize and the modified short Barzilai–Borwein stepsize. We obtain a trial point by the spectral method and then get the iteration point by the projection technique. The algorithm can generate a bounded iterative sequence automatically, and we obtain the global convergence of the proposed method in the sense that every limit point is a solution of the nonlinear equation. The proposed method can be used to resolve the large-scale nonlinear monotone equations with convex constraints including smooth and nonsmooth equations. Numerical results for nonlinear equation problems and the ℓ 1 -norm regularization problem in compressive sensing demonstrate the efficiency and efficacy of our method.