The Experts below are selected from a list of 84 Experts worldwide ranked by ideXlab platform
Katja Biswas - One of the best experts on this subject based on the ideXlab platform.
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An iterative aggregation and disaggregation approach to the calculation of steady state distributions of continuous processes
arXiv: Computational Physics, 2017Co-Authors: Katja BiswasAbstract:A mapping of the process on a continuous configuration space to the symbolic representation of the motion on a discrete state space will be combined with an iterative aggregation and disaggregation (IAD) procedure to obtain steady state distributions of the process. The IAD speeds up the convergence to the Unit Eigenvector, which is the steady state distribution, by forming smaller aggregated matrices whose Unit Eigenvector solutions are used to refine approximations of the steady state vector until convergence is reached. This method works very efficiently and can be used together with distributed or parallel computing methods to obtain high resolution images of the steady state distribution of complex atomistic or energy landscape type problems. The method is illustrated in two numerical examples. In the first example the transition matrix is assumed to be known. The second example represents an overdamped Brownian motion process subject to a dichotomously changing external potential.
Maria Patrizia Pera - One of the best experts on this subject based on the ideXlab platform.
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Global continuation of the eigenvalues of a perturbed linear operator
Annali di Matematica Pura ed Applicata (1923 -), 2018Co-Authors: Pierluigi Benevieri, Massimo Furi, Alessandro Calamai, Maria Patrizia PeraAbstract:Let E , F be real Banach spaces and S the Unit sphere of E . We study a nonlinear eigenvalue problem of the type $$Lx + \varepsilon N(x) = \lambda Cx$$ L x + ε N ( x ) = λ C x , where $$\varepsilon ,\lambda $$ ε , λ are real parameters, $$L:E \rightarrow F$$ L : E → F is a Fredholm linear operator of index zero, $$C:E \rightarrow F$$ C : E → F is a compact linear operator, and $$N:S \rightarrow F$$ N : S → F is a compact map. Given a solution $$(x,\varepsilon ,\lambda ) \in S \times \mathbb {R}\times \mathbb {R}$$ ( x , ε , λ ) ∈ S × R × R of this problem, we say that the first element x of the triple is a Unit Eigenvector corresponding to the eigenpair $$(\varepsilon ,\lambda )$$ ( ε , λ ) . Assuming that $$\lambda _0 \in \mathbb {R}$$ λ 0 ∈ R is such that the kernel of $$L -\lambda _0C$$ L - λ 0 C is odd dimensional and a natural transversality condition between the operators $$L -\lambda _0C$$ L - λ 0 C and C is satisfied, we prove that, in the set of all the eigenpairs, the connected component containing $$(0,\lambda _0)$$ ( 0 , λ 0 ) is either unbounded or meets an eigenpair $$(0,\lambda _1)$$ ( 0 , λ 1 ) , with $$\lambda _1 \not = \lambda _0$$ λ 1 ≠ λ 0 . Our approach is topological and based on the classical Leray–Schauder degree.
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Global continuation of the eigenvalues of a perturbed linear operator
Annali di Matematica Pura ed Applicata (1923 -), 2017Co-Authors: Pierluigi Benevieri, Massimo Furi, Alessandro Calamai, Maria Patrizia PeraAbstract:Let E, F be real Banach spaces and S the Unit sphere of E. We study a nonlinear eigenvalue problem of the type \(Lx + \varepsilon N(x) = \lambda Cx\), where \(\varepsilon ,\lambda \) are real parameters, \(L:E \rightarrow F\) is a Fredholm linear operator of index zero, \(C:E \rightarrow F\) is a compact linear operator, and \(N:S \rightarrow F\) is a compact map. Given a solution \((x,\varepsilon ,\lambda ) \in S \times \mathbb {R}\times \mathbb {R}\) of this problem, we say that the first element x of the triple is a Unit Eigenvector corresponding to the eigenpair \((\varepsilon ,\lambda )\). Assuming that \(\lambda _0 \in \mathbb {R}\) is such that the kernel of \(L -\lambda _0C\) is odd dimensional and a natural transversality condition between the operators \(L -\lambda _0C\) and C is satisfied, we prove that, in the set of all the eigenpairs, the connected component containing \((0,\lambda _0)\) is either unbounded or meets an eigenpair \((0,\lambda _1)\), with \(\lambda _1 \not = \lambda _0\). Our approach is topological and based on the classical Leray–Schauder degree.
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PERSISTENCE OF THE NORMALIZED EigenvectorS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE
Glasgow Mathematical Journal, 2013Co-Authors: Raffaele Chiappinelli, Massimo Furi, Maria Patrizia PeraAbstract:Let H be a real Hilbert space and denote by S its Unit sphere. Consider the nonlinear eigenvalue problem Ax+eB(x) = δx, where A : H → H is a bounded self-adjoint (linear) operator with nontrivial kernel KerA, and B : H → H is a (possibly) nonlinear perturbation term. A Unit Eigenvector x0 ∈ S ∩ KerA of A (thus corresponding to the eigenvalue δ = 0, that we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S ∩ KerA), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ R and e 6= 0. In this paper we prove that if B is a C1 gradient mapping and the eigenvalue δ = 0 has finite multiplicity, then the sphere S ∩KerA contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.
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PERSISTENCE OF THE NORMALIZED EigenvectorS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE
Glasgow Mathematical Journal, 2013Co-Authors: Raffaele Chiappinelli, Massimo Furi, Maria Patrizia PeraAbstract:AbstractLet H be a real Hilbert space and denote by S its Unit sphere. Consider the nonlinear eigenvalue problem Ax + ε B(x) =δ x, where A: H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B: H → H is a (possibly) nonlinear perturbation term. A Unit Eigenvector x0 ∈ S∩ Ker A of A (thus corresponding to the eigenvalue δ=0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ ℝ and ε ≠ 0. In this paper, we prove that if B is a C1 gradient mapping and the eigenvalue δ=0 has finite multiplicity, then the sphere S∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.
Kok Kiong Tan - One of the best experts on this subject based on the ideXlab platform.
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Convergence analysis of Xu's LMSER learning algorithm via deterministic discrete time system method☆
Neurocomputing, 2006Co-Authors: Kok Kiong TanAbstract:Abstract The convergence of Xu's LMSER algorithm with a constant learning rate, which is in the one Unit case, is interpreted by analyzing an associated deterministic discrete time (DDT) system. Some convergent results relating to the Xu's DDT system are obtained. An invariant set and an ultimate bound are identified so that the non-divergence of the system can be guaranteed. It is rigorously proven that all trajectories of the system from points in this invariant set will converge exponentially to a Unit Eigenvector associated with the largest eigenvalue of the correlation matrix. By comparing Xu's algorithm with Oja's algorithm, it can be observed, on the whole, the Xu's algorithm evolves faster at a cost of larger computational complexity. Extensive simulations will be carried out to illustrate the theory.
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convergence analysis of a deterministic discrete time system of oja s pca learning algorithm
IEEE Transactions on Neural Networks, 2005Co-Authors: Kok Kiong TanAbstract:The convergence of Oja's principal component analysis (PCA) learning algorithms is a difficult topic for direct study and analysis. Traditionally, the convergence of these algorithms is indirectly analyzed via certain deterministic continuous time (DCT) systems. Such a method will require the learning rate to converge to zero, which is not a reasonable requirement to impose in many practical applications. Recently, deterministic discrete time (DDT) systems have been proposed instead to indirectly interpret the dynamics of the learning algorithms. Unlike DCT systems, DDT systems allow learning rates to be constant (which can be a nonzero). This paper will provide some important results relating to the convergence of a DDT system of Oja's PCA learning algorithm. It has the following contributions: 1) A number of invariant sets are obtained, based on which we can show that any trajectory starting from a point in the invariant set will remain in the set forever. Thus, the nondivergence of the trajectories is guaranteed. 2) The convergence of the DDT system is analyzed rigorously. It is proven, in the paper, that almost all trajectories of the system starting from points in an invariant set will converge exponentially to the Unit Eigenvector associated with the largest eigenvalue of the correlation matrix. In addition, exponential convergence rate are obtained, providing useful guidelines for the selection of fast convergence learning rate. 3) Since the trajectories may diverge, the careful choice of initial vectors is an important issue. This paper suggests to use the domain of Unit hyper sphere as initial vectors to guarantee convergence. 4) Simulation results will be furnished to illustrate the theoretical results achieved.
Xining Zhang - One of the best experts on this subject based on the ideXlab platform.
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Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra
Bulletin of the American Mathematical Society, 2021Co-Authors: Peter B Denton, Stephen J Parke, Terence Tao, Xining ZhangAbstract:If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $\lambda_1(A),\dots,\lambda_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a Unit Eigenvector $v_i$ associated to the eigenvalue $\lambda_i(A)$ is related to the eigenvalues $\lambda_1(M_j),\dots,\lambda_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(\lambda_i(A)-\lambda_k(A)\right)=\prod_{k=1}^{n-1}\left(\lambda_i(A)-\lambda_k(M_j)\right)\,.$$ We refer to this identity as the \emph{Eigenvector-eigenvalue identity} and show how this identity can also be used to extract the relative phases between the components of any given Eigenvector. Despite the simple nature of this identity and the extremely mature state of development of linear algebra, this identity was not widely known until very recently. In this survey we describe the many times that this identity, or variants thereof, have been discovered and rediscovered in the literature (with the earliest precursor we know of appearing in 1834). We also provide a number of proofs and generalizations of the identity.
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Eigenvectors from eigenvalues a survey of a basic identity in linear algebra
arXiv: Rings and Algebras, 2019Co-Authors: Peter B Denton, Stephen J Parke, Xining ZhangAbstract:If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $\lambda_1(A),\dots,\lambda_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a Unit Eigenvector $v_i$ associated to the eigenvalue $\lambda_i(A)$ is related to the eigenvalues $\lambda_1(M_j),\dots,\lambda_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(\lambda_i(A)-\lambda_k(A)\right)=\prod_{k=1}^{n-1}\left(\lambda_i(A)-\lambda_k(M_j)\right)\,.$$ We refer to this identity as the \emph{Eigenvector-eigenvalue identity}. Despite the simple nature of this identity and the extremely mature state of development of linear algebra, this identity was not widely known until very recently. In this survey we describe the many times that this identity, or variants thereof, have been discovered and rediscovered in the literature (with the earliest precursor we know of appearing in 1834). We also provide a number of proofs and generalizations of the identity.
Camelia Lerintiu - One of the best experts on this subject based on the ideXlab platform.
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SUBDIFFERENTIAL OF THE LARGEST EIGENVALUE OF A SYMMETRICAL MATRIX APPLICATION OF DIRECT PROJECTION METHODS
Analysis and Applications, 2008Co-Authors: Claude Vallee, Danielle Fortune, Camelia LerintiuAbstract:In many physical circumstances, for example, in studying the linear vibrations of a mechanical or acoustical system, a key tool is to determine numerically the components of the Eigenvectors associated with the largest eigenvalue of a symmetrical matrix with real coefficients. To find out the largest eigenvalue λ1(S) of such a symmetrical n × n matrix S, the well-known Rayleigh's method consists in maximizing the quotient (VTSV)/(VTV) among all the nonvanishing vectors V of ℝn. When the eigenvalue λ1(S) is simple, the maximum is attained for vectors V colinear to a Unit Eigenvector N, and the function λ1 is differentiable with the projector NNT over the direction N as a gradient. When the largest eigenvalue is not simple, the function λ1 is no longer differentiable; it remains convex, but the subdifferential ∂λ1(S) is not reduced to a single gradient. This paper is devoted to determine the subgradients therein ∂λ1(S) by direct methods that do not require the preliminary determination of λ1(S).
