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SUMMARY:Gradient systems: overview and recent results - Dr Pierre-Antoine 
 Absil (Department of Mathematical Engineering\, Universite catholique de L
 ouvain\, Louvain-la-Neuve\, Belgium)
DTSTART:20090501T130000Z
DTEND:20090501T140000Z
UID:TALK15000@talks.cam.ac.uk
CONTACT:Dr Guy-Bart Stan
DESCRIPTION:A continuous-time gradient system is a dynamical system of the
  form dx/dt = - grad f(x)\, where grad f denotes the gradient of the diffe
 rentiable real-valued function f\, whose domain is the Euclidean space R^n
  or more generally a (smooth) manifold M. A discrete-time gradient system 
 takes the form x_{k+1} = x_k - s grad f(x)\, where the step size s can be 
 chosen by various means.\n\nGradient systems are useful in solving various
  optimization-related problems\, e.g.\, in principal component analysis\, 
 optimal control\, balanced realizations\, ocean sampling\, noise reduction
 \, pose estimation or the Procrustes problem.\n\nIn this talk\, we present
  recent (and less recent) results pertaining to the convergence of the sol
 utions of gradient systems. In particular\, we are interested in reasonabl
 y weak conditions\, sufficient for the solution trajectories to have at mo
 st one accumulation point. \n\nIn a similar spirit\, we discuss the notion
  of "accelerated" descent methods. This notion was formalized only recentl
 y\, but the principles have been in hiding in several places\, notably the
  work of D. Bertsekas and E. Polak. The idea is as follows. If T denotes a
  descent iteration for f\, we say that a sequence {x_k} is T-accelerated i
 f the decrease of f between x_k and x_{k+1} is at least as good as the dec
 rease of f between x_k and T(x_k). We address the following question: Assu
 me that all the accumulation points of every sequence {y_k} satisfying y_{
 k+1} = T(y_k) are critical points of f\; what can be said about the accumu
 lation points of T-accelerated sequences? This question appears in the ana
 lysis of several numerical algorithms.
LOCATION:Cambridge University Engineering Department\, Lecture Room 3B
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