19,112 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
Small Width, Low Distortions: Quantized Random Embeddings of Low-complexity Sets
Under which conditions and with which distortions can we preserve the
pairwise-distances of low-complexity vectors, e.g., for structured sets such as
the set of sparse vectors or the one of low-rank matrices, when these are
mapped in a finite set of vectors? This work addresses this general question
through the specific use of a quantized and dithered random linear mapping
which combines, in the following order, a sub-Gaussian random projection in
of vectors in , a random translation, or "dither",
of the projected vectors and a uniform scalar quantizer of resolution
applied componentwise. Thanks to this quantized mapping we are first
able to show that, with high probability, an embedding of a bounded set
in can be achieved when
distances in the quantized and in the original domains are measured with the
- and -norm, respectively, and provided the number of quantized
observations is large before the square of the "Gaussian mean width" of
. In this case, we show that the embedding is actually
"quasi-isometric" and only suffers of both multiplicative and additive
distortions whose magnitudes decrease as for general sets, and as
for structured set, when increases. Second, when one is only
interested in characterizing the maximal distance separating two elements of
mapped to the same quantized vector, i.e., the "consistency width"
of the mapping, we show that for a similar number of measurements and with high
probability this width decays as for general sets and as for
structured ones when increases. Finally, as an important aspect of our
work, we also establish how the non-Gaussianity of the mapping impacts the
class of vectors that can be embedded or whose consistency width provably
decays when increases.Comment: Keywords: quantization, restricted isometry property, compressed
sensing, dimensionality reduction. 31 pages, 1 figur
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