A novel method based on ant colony optimization （ACO）, algorithm for solving the ill-conditioned linear systems of equations is proposed. ACO is a parallelized bionic optimization algorithm which is inspired from th...A novel method based on ant colony optimization （ACO）, algorithm for solving the ill-conditioned linear systems of equations is proposed. ACO is a parallelized bionic optimization algorithm which is inspired from the behavior of real ants. ACO algorithm is first introduced, a kind of positive feedback mechanism is adopted in ACO. Then, the solu- tion problem of linear systems of equations was reformulated as an unconstrained optimization problem for solution by an ACID algorithm. Finally, the ACID with other traditional methods is applied to solve a kind of multi-dimensional Hilbert ill-conditioned linear equations. The numerical results demonstrate that ACO is effective, robust and recommendable in solving ill-conditioned linear systems of equations.展开更多
Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics...Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.展开更多
In geodesy and geophysics,many large-scale over-determined linear equations need to be solved which are often ill-conditioned.When the conjugate gradient method is used,their ill-conditioning effects to the solutions ...In geodesy and geophysics,many large-scale over-determined linear equations need to be solved which are often ill-conditioned.When the conjugate gradient method is used,their ill-conditioning effects to the solutions must be overcome,which is studied in this paper.Through the regularization ideas,the conjugate gradient method is improved,and the regularization iterative solution based on controlling condition number is put forward.Firstly by constructing the interference source vector,a new equation is derived with ill-condition diminished greatly,which has the same solution to the original normal equation.Then the new equation is solved by conjugate gradient method.Finally the effectiveness of the new method is verified by some numerical experiments of airborne gravity downward to the earth surface.In the numerical experiments the new method is compared with LS,CG and Tikhonov methods,and its accuracy is the highest.展开更多
FeP/FeO was prepared on carbon cloth(CC)via hydrothermal method,heat treatment in air,and phosphorization in argon.FeP/FeO/CC presents a porous and loose morphology which is conducive to the exposure of active sites a...FeP/FeO was prepared on carbon cloth(CC)via hydrothermal method,heat treatment in air,and phosphorization in argon.FeP/FeO/CC presents a porous and loose morphology which is conducive to the exposure of active sites and the transfer of reactants.FeP/FeO/CC requires the low overpotentials of 257 and 117 mV(vs.reversible hydrogen electrode(RHE))to achieve the current density of 10 mA_·cm^(-2)for oxygen evolution reaction(OER)and hydrogen evolution reaction(HER)in alkaline KOH solution,respectively.The small Tafel slope values of 36.1 mV_·dec^(-1)(for OER)and 96.2 mV_·dec^(-1)(for HER)indicate that FeP/FeO/CC exhibits the fast electrocatalytic reactive kinetics for OER and HER.In particular,the reaction kinetics of FeP/FeO/CC accelerated with the progress of HER.The charge-transfer resistance(R_(ct))of FeP/FeO/CC is only 11Ω.Excellent bifunctional electrocatalytic performances of FeP/FeO/CC should be attributed to the porous morphology and the lower charge-transfer resistance.展开更多
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
文摘A novel method based on ant colony optimization （ACO）, algorithm for solving the ill-conditioned linear systems of equations is proposed. ACO is a parallelized bionic optimization algorithm which is inspired from the behavior of real ants. ACO algorithm is first introduced, a kind of positive feedback mechanism is adopted in ACO. Then, the solu- tion problem of linear systems of equations was reformulated as an unconstrained optimization problem for solution by an ACID algorithm. Finally, the ACID with other traditional methods is applied to solve a kind of multi-dimensional Hilbert ill-conditioned linear equations. The numerical results demonstrate that ACO is effective, robust and recommendable in solving ill-conditioned linear systems of equations.
基金supported by Open Fund of Engineering Laboratory of Spatial Information Technology of Highway Geological Disaster Early Warning in Hunan Province(Changsha University of Science&Technology,kfj150602)Hunan Province Science and Technology Program Funded Projects,China(2015NK3035)+1 种基金the Land and Resources Department Scientific Research Project of Hunan Province,China(2013-27)the Education Department Scientific Research Project of Hunan Province,China(13C1011)
文摘Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.
基金The National Natural Science Foundation of China(41174005,41474009).
文摘In geodesy and geophysics,many large-scale over-determined linear equations need to be solved which are often ill-conditioned.When the conjugate gradient method is used,their ill-conditioning effects to the solutions must be overcome,which is studied in this paper.Through the regularization ideas,the conjugate gradient method is improved,and the regularization iterative solution based on controlling condition number is put forward.Firstly by constructing the interference source vector,a new equation is derived with ill-condition diminished greatly,which has the same solution to the original normal equation.Then the new equation is solved by conjugate gradient method.Finally the effectiveness of the new method is verified by some numerical experiments of airborne gravity downward to the earth surface.In the numerical experiments the new method is compared with LS,CG and Tikhonov methods,and its accuracy is the highest.
基金Funded by the Natural Science Foundation of Anhui Higher Education Institution of China(No.2023AH040160)the Natural Science Foundation of Anhui Province(No.1808085QE126)+1 种基金the Hefei Normal University High level Talent Research Startup Fund Project(No.2022rcjj55)the Collaborative Innovation Project of Colleges and Universities in Anhui Province(No.GXXT-2021-091)。
文摘FeP/FeO was prepared on carbon cloth(CC)via hydrothermal method,heat treatment in air,and phosphorization in argon.FeP/FeO/CC presents a porous and loose morphology which is conducive to the exposure of active sites and the transfer of reactants.FeP/FeO/CC requires the low overpotentials of 257 and 117 mV(vs.reversible hydrogen electrode(RHE))to achieve the current density of 10 mA_·cm^(-2)for oxygen evolution reaction(OER)and hydrogen evolution reaction(HER)in alkaline KOH solution,respectively.The small Tafel slope values of 36.1 mV_·dec^(-1)(for OER)and 96.2 mV_·dec^(-1)(for HER)indicate that FeP/FeO/CC exhibits the fast electrocatalytic reactive kinetics for OER and HER.In particular,the reaction kinetics of FeP/FeO/CC accelerated with the progress of HER.The charge-transfer resistance(R_(ct))of FeP/FeO/CC is only 11Ω.Excellent bifunctional electrocatalytic performances of FeP/FeO/CC should be attributed to the porous morphology and the lower charge-transfer resistance.
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
