Abstract
Original language  English 

Qualification  Doctor Degree 
Awarding Institution 

Supervisors/Advisors 

Award date  22 Feb 2016 
Place of Publication  Espoo 
Publisher  
Print ISBNs  9789526066295 
Electronic ISBNs  9789526066301 
Publication status  Published  2016 
MoE publication type  G4 Doctoral dissertation (monograph) 
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Keywords
 dynamical systems
 equilibrium sets
 singular points
 tangent cones
 bifurcating directions
 nonlinear eigenproblem
 eigenspace map
 fixed rank matrix manifold
 spheres on fixed rank matrix manifold
 local sensitivity analysis
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A geometric approach to eigenanalysis : With applications to structural stability: Dissertation. / Fedoroff, Alexis.
Espoo : Aalto University, 2016. 216 p.Research output: Thesis › Dissertation › Monograph
TY  THES
T1  A geometric approach to eigenanalysis
T2  With applications to structural stability: Dissertation
AU  Fedoroff, Alexis
N1  BA2125
PY  2016
Y1  2016
N2  This thesis clarifies certain aspects of nonlinear eigenanalysis with the help of differential geometry tools. Nonlinear eigenproblems are generalizations of the linear eigenproblem that comprise higher order terms with respect to the bifurcation parameter. They arise from singularity investigations of dynamical system equilibrium sets: for a vast majority of physical problems, the eigenspace associated to the eigenproblem gives, precisely, the bifurcating direction at the singular point. The research question consists in assessing the sensitivity of the eigenspace with respect to a prescribed variation of internal parameters of the defining dynamical system. A possible source of variation comes from using the linear predictor instead of the original nonlinear eigenproblem. One of the questions an answer shall be given to, consists in assessing the difference between two eigenspaces: one given by the linear predictor and the other given by the original nonlinear eigenproblem. The tools used are taken from differential geometry. The parameter dependent Jacobian matrix evaluated at the primary equilibrium branch defines the nonlinear eigenproblem. Geometrically, it can be interpreted as a smooth curve, called the Jacobian curve, that evolutes in the ambient matrix space. At a bifurcation point, the Jacobian matrix is rank deficient, by definition. Hence, the geometric interpretation of a nonlinear eigenproblem is the intersection of the Jacobian curve with the fixed rank matrix submanifold embedded in the ambient matrix space. The intersection point constitutes, therefore, a reference point on the fixed rank matrix manifold, around which one can draw a sphere of given radius computed with respect to the Riemannian distance function. The geometric interpretation of the eigenproblem enables the eigenspace associated with the eigenproblem to be considered as a smooth map, called the eigenspace map. It is defined on the fixed rank matrix manifold and has values in the projective space. Sensitivity analysis of the eigenspace map consists, then, in computing the ratio of two distances. The first distance, which is located in the denominator of the ratio, is the Riemannian distance on the fixed rank matrix manifold between the centre point and a point on the sphere. It is, by definition, equal to the radius of the sphere. The second distance, located in the numerator of the ratio, is the distance, in the projective space, between the corresponding two image points by the eigenspace map. Numerical examples are given at the end of the work to illustrate the use of eigenspace sensitivity analysis in the context of structural stability.
AB  This thesis clarifies certain aspects of nonlinear eigenanalysis with the help of differential geometry tools. Nonlinear eigenproblems are generalizations of the linear eigenproblem that comprise higher order terms with respect to the bifurcation parameter. They arise from singularity investigations of dynamical system equilibrium sets: for a vast majority of physical problems, the eigenspace associated to the eigenproblem gives, precisely, the bifurcating direction at the singular point. The research question consists in assessing the sensitivity of the eigenspace with respect to a prescribed variation of internal parameters of the defining dynamical system. A possible source of variation comes from using the linear predictor instead of the original nonlinear eigenproblem. One of the questions an answer shall be given to, consists in assessing the difference between two eigenspaces: one given by the linear predictor and the other given by the original nonlinear eigenproblem. The tools used are taken from differential geometry. The parameter dependent Jacobian matrix evaluated at the primary equilibrium branch defines the nonlinear eigenproblem. Geometrically, it can be interpreted as a smooth curve, called the Jacobian curve, that evolutes in the ambient matrix space. At a bifurcation point, the Jacobian matrix is rank deficient, by definition. Hence, the geometric interpretation of a nonlinear eigenproblem is the intersection of the Jacobian curve with the fixed rank matrix submanifold embedded in the ambient matrix space. The intersection point constitutes, therefore, a reference point on the fixed rank matrix manifold, around which one can draw a sphere of given radius computed with respect to the Riemannian distance function. The geometric interpretation of the eigenproblem enables the eigenspace associated with the eigenproblem to be considered as a smooth map, called the eigenspace map. It is defined on the fixed rank matrix manifold and has values in the projective space. Sensitivity analysis of the eigenspace map consists, then, in computing the ratio of two distances. The first distance, which is located in the denominator of the ratio, is the Riemannian distance on the fixed rank matrix manifold between the centre point and a point on the sphere. It is, by definition, equal to the radius of the sphere. The second distance, located in the numerator of the ratio, is the distance, in the projective space, between the corresponding two image points by the eigenspace map. Numerical examples are given at the end of the work to illustrate the use of eigenspace sensitivity analysis in the context of structural stability.
KW  dynamical systems
KW  equilibrium sets
KW  singular points
KW  tangent cones
KW  bifurcating directions
KW  nonlinear eigenproblem
KW  eigenspace map
KW  fixed rank matrix manifold
KW  spheres on fixed rank matrix manifold
KW  local sensitivity analysis
M3  Dissertation
SN  9789526066295
T3  Aalto University publication series: Doctoral Dissertations
PB  Aalto University
CY  Espoo
ER 