Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, submitted.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....have been formulated in an infinite dimensional setting. Moreover, the algorithms are restricted to functionals of the form (1. 2) Interesting numerical results with this infinite dimensional approach have been obtained by Chen, Zhou, and Ni [7] We refer to this paper and to the related papers [9, 14, 15] for additional information. We propose the elastic string algorithm for the computation of mountain passes in finitedimensional problems. This algorithm is derived from the mountain pass characterization (1.1) by approximating # with the set of piecewise linear paths. We analyze the convergence ....

....but this algorithm is based on an intuitive notion of how systems transition between stable stages. In particular, there is no clear relationship between the nudged elastic band algorithm and the characterization (1. 1) The elastic string algorithm also di#ers from the algorithms (for example, [9, 7, 14, 15]) based on the infinite dimensional approach of Choi and McKenna [8] for functionals of the form (1.2) These algorithms use only steepest descent searches and thus are unlikely to be e#cient on general problems. On the other hand, the finite dimensional approach based on the elastic string ....

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Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDEs, SIAM J. Sci. Comput., 23 (2001), pp. 840--865.

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Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, submitted.

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X. Yao and J. Zhou, A Minimax Method for Finding Multiple Critical Points in Banach Spaces and Its Application to Quasilinear Elliptic PDE, submitted.

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Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, SIAM J. Sci. Comp., 23(2001), 840-865.

....maximum points of J on v . A single valued mapping p: S L # H is a peak selection of J w.r.t. L if p(v) For v we say that J has a local peak selection w.r.t. L at v, if there is a neighborhood (v) of v and a mapping p: H s.t. p(u) P (u) #u # N S L #. Lemma 2. 1 [18] Let v # S L #. If J has a local peak selection p w.r.t. L at v # s.t. p is continuous at v # and dis(p(v # ) L) # 0 for some # 0, then either J # (p(v # ) 0 or for any # 0 with # (p(v # ) # #, there exists s 0 0, s.t. J(p(v(s) J(p(v # ) ###v(s) v #, #0 s s 0 ....

....# s.t. p is continuous at v # and dis(p(v # ) L) # 0 for some # 0, then either J # (p(v # ) 0 or for any # 0 with # (p(v # ) # #, there exists s 0 0, s.t. J(p(v(s) J(p(v # ) ###v(s) v #, #0 s s 0 with v(s) v # sd # sd# , d = # (p(v # ) Theorem 2. 1 [18] If J has a local peak selection p w.r.t. L at v 0 s.t. a) p is continuous at v 0 , b) dis(p(v 0 ) L) 0 and (c) v 0 is a local minimum point of J(p(v) on S L #, then p(v 0 ) is a critical point of J . Define a solution set p(v) v S L # . When L = 0 , M corresponds to Nehari ....

Y. Li and J. Zhou, "A minimax method for finding multiple critical points and its applications to semilinear PDE", SIAM J. Sci. Comp., 23(2001), 840-865.

....work is to develop some tools for local instability analysis of saddle points which can be computationally carried out. To fulfill the objective, first, one has to find a method to numerically approximate an unstable solution in a stable way. This is a vary challenging task. Our previous works [20], 21] have laid a solid foundation. Then one needs to find a way to measure local instabilities of saddle points. Usually this is done by defining certain local instability index. Here we want to define a local instability index for a saddle point which is general enough to be applied to usual ....

....direction space of J at u . # The definition of a local linking lacks of characterization and it is still too di#cult to compute numerically. So far no constructive method to compute such an index is available in the literature. In this paper, we use a local minimax method developed in [20], 21] to define a new local instability index which is known beforehand and can help in finding a saddle point numerically. Throughout this paper, when the Morse index is involved, we always assume that ) is a self adjoint Fredholm, linear operator from H # H where u is a critical point ....

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Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, SIAM J. Sci. Comp., 23 (2001), 840-865.

.... and Stable Method for Computing Multiple Saddle Points with Symmetries ZHI QIANG WANG and JIANXIN ZHOU i Abstract In this paper, an efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method established in [12, 13]. First an invariant space is defined in a more general sense and a Principle of Invariant Criticality is proved for the generalization. Then the orthogonal projection operator to the invariant space is used both to preserve the invariance and to reduce computational error across iterations. ....

....points have been made in [7] for MI=i and in [10] for MI:2 which were motivated by theoretical (global minimax) characterizations of saddle points in [1] and [23] respectively. Inspired by [7, 10] and an idea in [9] a local minimax method (we shall refer it as LMM in the paper) was developed in [12, 13] and many multiple solutions were numerically computed for a class of semilinear elliptic equations. Its convergence results are obtained in [13] Several results in instability analysis of saddle points are established in [14,25] We briefly recall LMM here. The basic idea of LMM is to define a ....

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Y. Li and J. Zhou, A minimax method for finding multiple critical points and its appli- cations to semilinear PDE, SIAM Sci. Comp. 23(2001), 840-865

....adaptations are necessary. One of the main components of such adaptations is the localization of the global procedure in the Mountain Pass Lemma, resulting in several possible versions of MMA. Only recently, convergence analysis for a local MMA has been successfully carried out; see [13, 14, 15]. In this paper, our main interest is in SIA, the scaling iterative algorithm. The version of SIA as presented in [4, pp. 1573 1574] see more details in (2.3) 2.4) in Section 2) was found to be very e#ective in computing numerical solutions of a certain class of unstable solutions of ....

....term in numerical computation and there is a stepsize before the steepest descent direction in the algorithm, this stepsize can absorb a positve factor anyway, we can simply use the negative gradient to replace the steepest descent direction in the algorithm. Following similar arguments as in [13, 14], we can prove Theorem 3.1. Theorem 4.3 in [13] Assume that Conditions (h1) h5 ) in [13] are satisfied and that there exist a 5 0 and a 6 0 s.t. for s as specified in (h2) f # (x, #) a 5 a 6 # s 1 . 3.4) Then the peak selection p of J is C . Now we assume that ....

[Article contains additional citation context not shown here]

Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, SIAM J. Sci. Comp. 23 (2001), 840--865.

....in critical point theory. However, most minimax theorems in the literature (See [1] 16] 17] 19] 24] such as the mountain pass, various linking and saddle point theorems, require one to solve a two level global optimization problem and therefore not for algorithm implementation. In [11], motivated by the numerical works of Choi McKenna [7] and Ding Costa Chen 9, the Morse theory and the idea to define a solution submanifold, new local minimax theorems which characterize critical points as solutions to a two level local optimization problem are established. Based on the local ....

....local optimization problem are established. Based on the local characterization, a new numerical minimax method for finding multiple critical points is devised. The numerical method is implemented successfully to solve a class of semilinear elliptic PDE on various domains for multiple solutions [11]. Although Morse index has been printed for each numerical solution obtained in [11] their mathematical verifications have not been established. In [2] by using a global minimax principle, A. Bahri and P.L. Lions established some lower bound estimates for the Morse indices of solutions to a ....

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Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, SIAM J. on Scientic Computing, to appear.

.... Saddle Points JIANXIN ZHOU Department of Mathematics, Texas A M University College Station, TX 77843 Abstract This paper is concerned with some theoretical and practical issues related to a peak selection, an important notion defined in the development of the local minimax method in [10,11], which laid a mathematical foundation for numerically finding multiple critical points. Based on the local minimax characterization, a numerical local minimax algorithm is designed and successfully applied to solve many semilinear elliptic PDE for multiple solutions. Convergence results of the ....

....is designed and successfully applied to solve many semilinear elliptic PDE for multiple solutions. Convergence results of the algorithm are established in [11] The local minimax method can also be used to define an index to study local instability of saddle points [23] All the results in [10,11,23] require a peak selection p to be continuous or differentiable at a point. The first question is how to check this condition It is very difficult, since the graph of a peak selection is in general not closed and there is no explicit formula to evaluate p. The second question is that ....

[Article contains additional citation context not shown here]

Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, SIAM J. Sci. Comp., 23(2001), 840-865.

....develop some tools for local instability analysis of saddle points which can be computationally carried out. To fulfill the objective, we first have to find a method that enables us to numerically approximate an unstable solution in a stable way. This is a vary challenging task. Our previous works [20], 21] have laid a solid foundation. Then we need to find a way to measure local instabilities of 3 saddle points. Usually this is done by defining certain local instability index. Here we want to define a local instability index for a saddle point which is general enough to cover those ....

....decreasing direction space of J at u # . The definition of a local linking lacks of characterization and it is still too di#cult to compute numerically. So far no constructive method to compute such an index is available in the literature. In this paper, we use a local minimax method developed in [20], 21] to define a new local instability index which is known beforehand and can help in finding a saddle point numerically. Throughout this paper, when the Morse index is involved, we always assume that J ## (u # ) is a self adjoint Fredholm, linear operator from H # H where u # is a critical ....

[Article contains additional citation context not shown here]

Y. Li and J. Zhou, "A minimax method for finding multiple critical points and its applications to semilinear PDE", SIAMJ.Sci.Comp.,toappear.

.... Results of A Local Minimax Method for Finding Multiple Critical Points Yongxin Li # and Jianxin Zhou Abstract In [14], a new local minimax method that characterizes a saddle point as a solution to a local minimax problem is established. Based on the local characterization, a numerical minimax algorithm is designed for finding multiple saddle points. Numerical computations of many examples in semilinear elliptic ....

....in critical point theory. However, most minimax theorems in the literature (See [1] 19] 20] 21] 22, 25] such as the mountain pass, various linking and saddle point theorems, require one to solve a two level global minimax problem and therefore not for algorithm implementation. In [14], motivated by the numerical works of Choi McKenna [7] and Ding Costa Chen [12] the Morse theory and the idea to define a solution submanifold, a new local minimax theorem which characterizes a saddle point as a solution to a two level local minimax problem is developed. Based on the local ....

[Article contains additional citation context not shown here]

Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, submitted.

.... Results of A Minimax Method for Finding Multiple Critical Points Yongxin Li # and Jianxin Zhou Abstract In [12], new local minimax theorems which characterize a saddle point as a solution toatwo levellocal minimax problem are established. Based on the local characterization, a numerical minimax method is designed for finding multiple saddle points. Many numerical examples in semilinear elliptic PDE have ....

....in critical point theory. However, most minimax theorems in the literature (See [1] 16] 17] 18] 19, 22] such as the mountain pass, various linking and saddle point theorems, require one to solve a two level global optimization problem and therefore not for algorithm implementation. In [12], motivated by the numerical works of Choi McKenna [6] and Ding Costa Chen [11] the Morse theory and the idea to define a solution submanifold, new local minimax theorems which characterize a critical point as a solution to a two level local minimax problem are established. Based on the local ....

[Article contains additional citation context not shown here]

Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, submitted.

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