7,388 research outputs found
Shape-from-intrinsic operator
Shape-from-X is an important class of problems in the fields of geometry
processing, computer graphics, and vision, attempting to recover the structure
of a shape from some observations. In this paper, we formulate the problem of
shape-from-operator (SfO), recovering an embedding of a mesh from intrinsic
differential operators defined on the mesh. Particularly interesting instances
of our SfO problem include synthesis of shape analogies, shape-from-Laplacian
reconstruction, and shape exaggeration. Numerically, we approach the SfO
problem by splitting it into two optimization sub-problems that are applied in
an alternating scheme: metric-from-operator (reconstruction of the discrete
metric from the intrinsic operator) and embedding-from-metric (finding a shape
embedding that would realize a given metric, a setting of the multidimensional
scaling problem)
Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
The numerical solution of partial differential equations on high-dimensional
domains gives rise to computationally challenging linear systems. When using
standard discretization techniques, the size of the linear system grows
exponentially with the number of dimensions, making the use of classic
iterative solvers infeasible. During the last few years, low-rank tensor
approaches have been developed that allow to mitigate this curse of
dimensionality by exploiting the underlying structure of the linear operator.
In this work, we focus on tensors represented in the Tucker and tensor train
formats. We propose two preconditioned gradient methods on the corresponding
low-rank tensor manifolds: A Riemannian version of the preconditioned
Richardson method as well as an approximate Newton scheme based on the
Riemannian Hessian. For the latter, considerable attention is given to the
efficient solution of the resulting Newton equation. In numerical experiments,
we compare the efficiency of our Riemannian algorithms with other established
tensor-based approaches such as a truncated preconditioned Richardson method
and the alternating linear scheme. The results show that our approximate
Riemannian Newton scheme is significantly faster in cases when the application
of the linear operator is expensive.Comment: 24 pages, 8 figure
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