Dear Editor,This letter focuses on the remaining useful life(RUL)prediction task under limited labeled samples.Existing machine-learning-based RUL prediction methods for this task usually pay attention to mining degra...Dear Editor,This letter focuses on the remaining useful life(RUL)prediction task under limited labeled samples.Existing machine-learning-based RUL prediction methods for this task usually pay attention to mining degradation information to improve the prediction accuracy of degradation value or health indicator for the next epoch.However,they ignore the cumulative prediction error caused by iterations before reaching the failure point.展开更多
In this paper, supose Γ be a boundary of a Jordan domain D and Γ satisfied Альпер condition, the order that rational type interpolating operators at Fejer's points of f(z)∈C(Γ) converge to f(z) in the se...In this paper, supose Γ be a boundary of a Jordan domain D and Γ satisfied Альпер condition, the order that rational type interpolating operators at Fejer's points of f(z)∈C(Γ) converge to f(z) in the sense of uniformly convergence is obtained.展开更多
Results regarding best approximation and best Simultaneous approximation on convex metric spaces are Obtained.Existence of fixed points for an ultimately nonexpansive semigroup of mappings is also shown.
We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We ...We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We prove mean square convergence and convergence almost sure (a.s.) of iterative approximations and establish both asymptotic and nonasymptotic estimates of the convergence rate in degenerate and non-degenerate cases. Previously the stochastic approximation algorithms were studied mainly for optimization problems.展开更多
The main purpose of this paper is to prove some common fixed point theorems for pointwise R-subweakly commuting maps on non-starshaped domains in p-normed spaces and locally convex topological vector spaces. As applic...The main purpose of this paper is to prove some common fixed point theorems for pointwise R-subweakly commuting maps on non-starshaped domains in p-normed spaces and locally convex topological vector spaces. As applications, invariant approximation results are established. This work provides extension as well as substantial improvement of several results in the existing literature.展开更多
Some new characterizations and immediate explicit expressions of best L(1≤p≤∞) approximation and their deviations by an n-dimensional subspace on a set of n+1 points are given.
For a subset K of a metric space(X,d)and x∈X,Px(x)={y∈K:d(x,y)=d(x,K)≡inf{d(x,k):k∈K}}is called the set of best K-approximant to x.An element go E K is said to be a best simulta-neous approximation of the pair y1,...For a subset K of a metric space(X,d)and x∈X,Px(x)={y∈K:d(x,y)=d(x,K)≡inf{d(x,k):k∈K}}is called the set of best K-approximant to x.An element go E K is said to be a best simulta-neous approximation of the pair y1,y2 E∈if max{d(y1,go),d(y2,go)}=inf g∈K max{d(y1,g),d(y2,g)}.In this paper,some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved.For self mappings T and S on K,results are proved on both T-and S-invariant points for a set of best simultaneous approximation.Some results on best K-approximant are also deduced.The results proved generalize and extend some results of I.Beg and M.Abbas^[1],S.Chandok and T.D.Narang^[2],T.D.Narang and S.Chandok^[11],S.A.Sahab,M.S.Khan and S.Sessa^[14],P.Vijayaraju^[20]and P.Vijayaraju and M.Marudai^[21].展开更多
Using a recent result regarding the fixed points of multivalued mappings, the existence of invariant best simultaneous approximation in chainable metric space is proved.
This paper proposes two kinds of approximate proximal point algorithms （APPA） for monotone variational inequalities, both of which can be viewed as two extended versions of Solodov and Svaiter＇s APPA in the paper ...This paper proposes two kinds of approximate proximal point algorithms （APPA） for monotone variational inequalities, both of which can be viewed as two extended versions of Solodov and Svaiter＇s APPA in the paper ＂Error bounds for proximal point subproblems and associated inexact proximal point algorithms＂ published in 2000. They are both prediction- correction methods which use the same inexactness restriction; the only difference is that they use different search directions in the correction steps. This paper also chooses an optimal step size in the two versions of the APPA to improve the profit at each iteration. Analysis also shows that the two APPAs are globally convergent under appropriate assumptions, and we can expect algorithm 2 to get more progress in every iteration than algorithm 1. Numerical experiments indicate that algorithm 2 is more efficient than algorithm 1 with the same correction step size,展开更多
In this paper an introduction of the moving least squares approach is presented in the context of data approximation and interpolation problems in Geodesy.An application of this method is presented for geoid height ap...In this paper an introduction of the moving least squares approach is presented in the context of data approximation and interpolation problems in Geodesy.An application of this method is presented for geoid height approximation and interpolation using different polynomial basis functions for the approximant and interpolant,respectively,in a regular grid of geoid height data in the region 16.0417°≤φ≤47.9583°and 36.0417°≤λ≤69.9582°,with increment 0.0833°in both latitudal and longitudal directions.The results of approximation and interpolation are then compared with the geoid height data from GPS-Levelling approach.Using the standard deviation of the difference of the results,it is shown that the planar interpolant,with reciprocal of distance as weight function,is the best choice in this local approximation and interpolation problem.展开更多
The structure of any a.s. self-similar set K(w) generated by a class of random elements {gn,wσ} taking values in the space of contractive operators is given and the approximation of K(w) by the fixed points {Pn,wσ} ...The structure of any a.s. self-similar set K(w) generated by a class of random elements {gn,wσ} taking values in the space of contractive operators is given and the approximation of K(w) by the fixed points {Pn,wσ} of {gn,ow} is obtained. It is useful to generate the fractal in computer.展开更多
Abstract We extend the concept of R-subeommuting maps due to Shahzad to the case of non-starshaped domain and obtain a common fixed point result for this class of maps on non-starshaped domain in the setup of p-norra...Abstract We extend the concept of R-subeommuting maps due to Shahzad to the case of non-starshaped domain and obtain a common fixed point result for this class of maps on non-starshaped domain in the setup of p-norraed spaces. As applications, we establish noncommutative versions of various best approximation results for generalized I-nonexpansive maps on non-starshaped domain. Our results unify and extend that of Al- Thagafi, Dotson, IIabiniak, Jungck and Senna, Latif, Sahab, Khan and Sessa and Shahzad.展开更多
Let K be a closed convex subset of a real reflexive Banach space E, T：K→K be a nonexpansive mapping, and f：K→K be a fixed weakly contractive （may not be contractive） mapping. Then for any t∈（0, 1）, let x1∈K ...Let K be a closed convex subset of a real reflexive Banach space E, T：K→K be a nonexpansive mapping, and f：K→K be a fixed weakly contractive （may not be contractive） mapping. Then for any t∈（0, 1）, let x1∈K be the unique fixed point of the weak contraction x1→tf（x）＋（1-t）Tx. If T has a fixed point and E admits a weakly sequentially continuous duality mapping from E to E^＊, then it is shown that {xt} converges to a fixed point of T as t→0. The results presented here improve and generalize the corresponding results in （Xu, 2004）.展开更多
Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approx...Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions ofPPA （APPA） are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm Ⅰ; in the same way, Algorithm Ⅱ is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm Ⅱ usually outperforms Algorithm Ⅰ. For computation practice, we consider a class of MVI with a special structure, and choose the extending Algorithm Ⅱ to implement, which is inspired by the idea of Gauss-Seidel iteration method making full use of information about the latest iteration. And in particular, self-adaptive techniques are adopted to adjust relevant parameters for faster convergence. Finally, some numerical experiments are reported on the separated MVI. Numerical results showed that the extending Algorithm II is feasible and easy to implement with relatively low computation load.展开更多
Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been anal...Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.展开更多
In this paper, a new concept of weakly ,convex graph for set-valued mappings is introduced and studied. By using the concept , some new coincidence, the bestapproximation and fixed point-theorems are obta...In this paper, a new concept of weakly ,convex graph for set-valued mappings is introduced and studied. By using the concept , some new coincidence, the bestapproximation and fixed point-theorems are obtained.展开更多
We propose a class of iteration methods searching the best approximately generalized polynomial, which has parallel computational function and converges to the exact solution quadratically. We first transform it into ...We propose a class of iteration methods searching the best approximately generalized polynomial, which has parallel computational function and converges to the exact solution quadratically. We first transform it into a special system of nonlinear equations with constraint, then by using to certain iteration method, we combine the two basic processes of the Remes method into a whole such that the iterative process of the system of nonlinear equations and the computation of the solution to the system of linear equations proceed alternately. A lot of numerical examples show that this method not only has good convergence property but also always converges to the exact solution of the problem accurately and rapidly for almost all initial approximations .展开更多
In order to find roots of maximal monotone operators, this paper introduces and studies the modified approximate proximal point algorithm with an error sequence {e k} such that || ek || \leqslant hk || xk - [(x)\tilde...In order to find roots of maximal monotone operators, this paper introduces and studies the modified approximate proximal point algorithm with an error sequence {e k} such that || ek || \leqslant hk || xk - [(x)\tilde]k ||\left\| { e^k } \right\| \leqslant \eta _k \left\| { x^k - \tilde x^k } \right\| with ?k = 0￥ ( hk - 1 ) < + ￥\sum\limits_{k = 0}^\infty {\left( {\eta _k - 1} \right)} and infk \geqslant 0 hk = m\geqslant 1\mathop {\inf }\limits_{k \geqslant 0} \eta _k = \mu \geqslant 1 . Here, the restrictions on {η k} are very different from the ones on {η k}, given by He et al (Science in China Ser. A, 2002, 32 (11): 1026–1032.) that supk \geqslant 0 hk = v < 1\mathop {\sup }\limits_{k \geqslant 0} \eta _k = v . Moreover, the characteristic conditions of the convergence of the modified approximate proximal point algorithm are presented by virtue of the new technique very different from the ones given by He et al.展开更多
We pressent new Ky Fan type best approximation theorems for a discontinuous multivalued map on metrizable topological vector spaces and hyperconvex spaces. In addition, fixed point results are derived for the map stud...We pressent new Ky Fan type best approximation theorems for a discontinuous multivalued map on metrizable topological vector spaces and hyperconvex spaces. In addition, fixed point results are derived for the map studied. Our work generalizes severl results in approximation theory.展开更多
We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F<sub>1</sub> and F<sub>2</sub> are cont...We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F<sub>1</sub> and F<sub>2</sub> are continuous functions on a closed interval [a,b], S is an n-dimensional Chebyshev subspace of C<span style="white-space:normal;"> [<em style="white-space:normal;">a<span style="white-space:normal;">,<em style="white-space:normal;">b<span style="white-space:normal;">] and s<sub>1</sub>* & <span style="white-space:normal;">s<sub>2</sub>* are the best uniform approximations to F<sub>1</sub> and F<sub>2</sub> from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F<sub>1</sub><span style="white-space:nowrap;">−s<sub>1</sub>* and <em style="white-space:normal;">F<sub style="white-space:normal;">2</sub>−<em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">*, <em style="white-space:normal;">s<sub style="white-space:normal;">1</sub><span style="white-space:normal;">* or <em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">* is also a best A(1) simultaneous approximation to F<sub>1</sub> and F<sub>2</sub> from S with F<sub>1</sub><span style="white-space:nowrap;">≥F<sub>2</sub> and n=2.展开更多
基金supported in part by the National Natural Science Foundation of China(U2034209)the Postdoctoral Science Foundation of Chongqing(cstc2021jcyj-bsh X0047)+1 种基金the Fundamental Research Funds for the Central Universities(2022CDJJMRH-008)the National Natural Science Foundation of China(62203075)
文摘Dear Editor,This letter focuses on the remaining useful life(RUL)prediction task under limited labeled samples.Existing machine-learning-based RUL prediction methods for this task usually pay attention to mining degradation information to improve the prediction accuracy of degradation value or health indicator for the next epoch.However,they ignore the cumulative prediction error caused by iterations before reaching the failure point.
文摘In this paper, supose Γ be a boundary of a Jordan domain D and Γ satisfied Альпер condition, the order that rational type interpolating operators at Fejer's points of f(z)∈C(Γ) converge to f(z) in the sense of uniformly convergence is obtained.
文摘Results regarding best approximation and best Simultaneous approximation on convex metric spaces are Obtained.Existence of fixed points for an ultimately nonexpansive semigroup of mappings is also shown.
文摘We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We prove mean square convergence and convergence almost sure (a.s.) of iterative approximations and establish both asymptotic and nonasymptotic estimates of the convergence rate in degenerate and non-degenerate cases. Previously the stochastic approximation algorithms were studied mainly for optimization problems.
文摘The main purpose of this paper is to prove some common fixed point theorems for pointwise R-subweakly commuting maps on non-starshaped domains in p-normed spaces and locally convex topological vector spaces. As applications, invariant approximation results are established. This work provides extension as well as substantial improvement of several results in the existing literature.
文摘Some new characterizations and immediate explicit expressions of best L(1≤p≤∞) approximation and their deviations by an n-dimensional subspace on a set of n+1 points are given.
文摘For a subset K of a metric space(X,d)and x∈X,Px(x)={y∈K:d(x,y)=d(x,K)≡inf{d(x,k):k∈K}}is called the set of best K-approximant to x.An element go E K is said to be a best simulta-neous approximation of the pair y1,y2 E∈if max{d(y1,go),d(y2,go)}=inf g∈K max{d(y1,g),d(y2,g)}.In this paper,some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved.For self mappings T and S on K,results are proved on both T-and S-invariant points for a set of best simultaneous approximation.Some results on best K-approximant are also deduced.The results proved generalize and extend some results of I.Beg and M.Abbas^[1],S.Chandok and T.D.Narang^[2],T.D.Narang and S.Chandok^[11],S.A.Sahab,M.S.Khan and S.Sessa^[14],P.Vijayaraju^[20]and P.Vijayaraju and M.Marudai^[21].
文摘Using a recent result regarding the fixed points of multivalued mappings, the existence of invariant best simultaneous approximation in chainable metric space is proved.
文摘This paper proposes two kinds of approximate proximal point algorithms （APPA） for monotone variational inequalities, both of which can be viewed as two extended versions of Solodov and Svaiter＇s APPA in the paper ＂Error bounds for proximal point subproblems and associated inexact proximal point algorithms＂ published in 2000. They are both prediction- correction methods which use the same inexactness restriction; the only difference is that they use different search directions in the correction steps. This paper also chooses an optimal step size in the two versions of the APPA to improve the profit at each iteration. Analysis also shows that the two APPAs are globally convergent under appropriate assumptions, and we can expect algorithm 2 to get more progress in every iteration than algorithm 1. Numerical experiments indicate that algorithm 2 is more efficient than algorithm 1 with the same correction step size,
文摘In this paper an introduction of the moving least squares approach is presented in the context of data approximation and interpolation problems in Geodesy.An application of this method is presented for geoid height approximation and interpolation using different polynomial basis functions for the approximant and interpolant,respectively,in a regular grid of geoid height data in the region 16.0417°≤φ≤47.9583°and 36.0417°≤λ≤69.9582°,with increment 0.0833°in both latitudal and longitudal directions.The results of approximation and interpolation are then compared with the geoid height data from GPS-Levelling approach.Using the standard deviation of the difference of the results,it is shown that the planar interpolant,with reciprocal of distance as weight function,is the best choice in this local approximation and interpolation problem.
基金Supported by NNSF of China and the Foundation of Wuhan University
文摘The structure of any a.s. self-similar set K(w) generated by a class of random elements {gn,wσ} taking values in the space of contractive operators is given and the approximation of K(w) by the fixed points {Pn,wσ} of {gn,ow} is obtained. It is useful to generate the fractal in computer.
文摘Abstract We extend the concept of R-subeommuting maps due to Shahzad to the case of non-starshaped domain and obtain a common fixed point result for this class of maps on non-starshaped domain in the setup of p-norraed spaces. As applications, we establish noncommutative versions of various best approximation results for generalized I-nonexpansive maps on non-starshaped domain. Our results unify and extend that of Al- Thagafi, Dotson, IIabiniak, Jungck and Senna, Latif, Sahab, Khan and Sessa and Shahzad.
文摘Let K be a closed convex subset of a real reflexive Banach space E, T：K→K be a nonexpansive mapping, and f：K→K be a fixed weakly contractive （may not be contractive） mapping. Then for any t∈（0, 1）, let x1∈K be the unique fixed point of the weak contraction x1→tf（x）＋（1-t）Tx. If T has a fixed point and E admits a weakly sequentially continuous duality mapping from E to E^＊, then it is shown that {xt} converges to a fixed point of T as t→0. The results presented here improve and generalize the corresponding results in （Xu, 2004）.
基金Project (No. 1027054) supported by the National Natural Science Foundation of China
文摘Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions ofPPA （APPA） are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm Ⅰ; in the same way, Algorithm Ⅱ is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm Ⅱ usually outperforms Algorithm Ⅰ. For computation practice, we consider a class of MVI with a special structure, and choose the extending Algorithm Ⅱ to implement, which is inspired by the idea of Gauss-Seidel iteration method making full use of information about the latest iteration. And in particular, self-adaptive techniques are adopted to adjust relevant parameters for faster convergence. Finally, some numerical experiments are reported on the separated MVI. Numerical results showed that the extending Algorithm II is feasible and easy to implement with relatively low computation load.
文摘Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.
文摘In this paper, a new concept of weakly ,convex graph for set-valued mappings is introduced and studied. By using the concept , some new coincidence, the bestapproximation and fixed point-theorems are obtained.
文摘We propose a class of iteration methods searching the best approximately generalized polynomial, which has parallel computational function and converges to the exact solution quadratically. We first transform it into a special system of nonlinear equations with constraint, then by using to certain iteration method, we combine the two basic processes of the Remes method into a whole such that the iterative process of the system of nonlinear equations and the computation of the solution to the system of linear equations proceed alternately. A lot of numerical examples show that this method not only has good convergence property but also always converges to the exact solution of the problem accurately and rapidly for almost all initial approximations .
基金Supported both by the Teaching and Research Award Fund for Outstanding Young Teachers inHigher Educational Institutions of MOEChinaand by the Dawn Program Fund in Shanghai
文摘In order to find roots of maximal monotone operators, this paper introduces and studies the modified approximate proximal point algorithm with an error sequence {e k} such that || ek || \leqslant hk || xk - [(x)\tilde]k ||\left\| { e^k } \right\| \leqslant \eta _k \left\| { x^k - \tilde x^k } \right\| with ?k = 0￥ ( hk - 1 ) < + ￥\sum\limits_{k = 0}^\infty {\left( {\eta _k - 1} \right)} and infk \geqslant 0 hk = m\geqslant 1\mathop {\inf }\limits_{k \geqslant 0} \eta _k = \mu \geqslant 1 . Here, the restrictions on {η k} are very different from the ones on {η k}, given by He et al (Science in China Ser. A, 2002, 32 (11): 1026–1032.) that supk \geqslant 0 hk = v < 1\mathop {\sup }\limits_{k \geqslant 0} \eta _k = v . Moreover, the characteristic conditions of the convergence of the modified approximate proximal point algorithm are presented by virtue of the new technique very different from the ones given by He et al.
文摘We pressent new Ky Fan type best approximation theorems for a discontinuous multivalued map on metrizable topological vector spaces and hyperconvex spaces. In addition, fixed point results are derived for the map studied. Our work generalizes severl results in approximation theory.
文摘We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F<sub>1</sub> and F<sub>2</sub> are continuous functions on a closed interval [a,b], S is an n-dimensional Chebyshev subspace of C<span style="white-space:normal;"> [<em style="white-space:normal;">a<span style="white-space:normal;">,<em style="white-space:normal;">b<span style="white-space:normal;">] and s<sub>1</sub>* & <span style="white-space:normal;">s<sub>2</sub>* are the best uniform approximations to F<sub>1</sub> and F<sub>2</sub> from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F<sub>1</sub><span style="white-space:nowrap;">−s<sub>1</sub>* and <em style="white-space:normal;">F<sub style="white-space:normal;">2</sub>−<em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">*, <em style="white-space:normal;">s<sub style="white-space:normal;">1</sub><span style="white-space:normal;">* or <em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">* is also a best A(1) simultaneous approximation to F<sub>1</sub> and F<sub>2</sub> from S with F<sub>1</sub><span style="white-space:nowrap;">≥F<sub>2</sub> and n=2.
