1,814 research outputs found
Randomness and differentiability in higher dimensions
We present two theorems concerned with algorithmic randomness and
differentiability of functions of several variables. Firstly, we prove an
effective form of the Rademacher's Theorem: we show that computable randomness
implies differentiability of computable Lipschitz functions of several
variables. Secondly, we show that weak 2-randomness is equivalent to
differentiability of computable a.e. differentiable functions of several
variables.Comment: 19 page
Projected Stochastic Gradients for Convex Constrained Problems in Hilbert Spaces
Convergence of a projected stochastic gradient algorithm is demonstrated for
convex objective functionals with convex constraint sets in Hilbert spaces. In
the convex case, the sequence of iterates converges weakly to a point
in the set of minimizers with probability one. In the strongly convex case, the
sequence converges strongly to the unique optimum with probability one. An
application to a class of PDE constrained problems with a convex objective,
convex constraint and random elliptic PDE constraints is shown. Theoretical
results are demonstrated numerically.Comment: 28 page
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