11,499 research outputs found
A mixed regularization approach for sparse simultaneous approximation of parameterized PDEs
We present and analyze a novel sparse polynomial technique for the
simultaneous approximation of parameterized partial differential equations
(PDEs) with deterministic and stochastic inputs. Our approach treats the
numerical solution as a jointly sparse reconstruction problem through the
reformulation of the standard basis pursuit denoising, where the set of jointly
sparse vectors is infinite. To achieve global reconstruction of sparse
solutions to parameterized elliptic PDEs over both physical and parametric
domains, we combine the standard measurement scheme developed for compressed
sensing in the context of bounded orthonormal systems with a novel mixed-norm
based regularization method that exploits both energy and sparsity. In
addition, we are able to prove that, with minimal sample complexity, error
estimates comparable to the best -term and quasi-optimal approximations are
achievable, while requiring only a priori bounds on polynomial truncation error
with respect to the energy norm. Finally, we perform extensive numerical
experiments on several high-dimensional parameterized elliptic PDE models to
demonstrate the superior recovery properties of the proposed approach.Comment: 23 pages, 4 figure
Best simultaneous approximations
summary:In this paper we study simultaneous approximation of real-valued functions in and give a generalization of some related results
Mixed spatially varying -BV regularization of inverse ill-posed problems
Several generalizations of the traditional Tikhonov-Phillips regularization
method have been proposed during the last two decades. Many of these
generalizations are based upon inducing stability throughout the use of
different penalizers which allow the capturing of diverse properties of the
exact solution (e.g. edges, discontinuities, borders, etc.). However, in some
problems in which it is known that the regularity of the exact solution is
heterogeneous and/or anisotropic, it is reasonable to think that a much better
option could be the simultaneous use of two or more penalizers of different
nature. Such is the case, for instance, in some image restoration problems in
which preservation of edges, borders or discontinuities is an important matter.
In this work we present some results on the simultaneous use of penalizers of
and of bounded variation (BV) type. For particular cases, existence and
uniqueness results are proved. Open problems are discussed and results to
signal restoration problems are presented.Comment: 18 pages, 12 figure
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