85 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
Optimal approximation of infinite-dimensional holomorphic functions II: recovery from i.i.d. pointwise samples
Infinite-dimensional, holomorphic functions have been studied in detail over
the last several decades, due to their relevance to parametric differential
equations and computational uncertainty quantification. The approximation of
such functions from finitely many samples is of particular interest, due to the
practical importance of constructing surrogate models to complex mathematical
models of physical processes. In a previous work, [5] we studied the
approximation of so-called Banach-valued,
-holomorphic functions on the
infinite-dimensional hypercube from (potentially
adaptive) samples. In particular, we derived lower bounds for the adaptive
-widths for classes of such functions, which showed that certain algebraic
rates of the form are the best possible regardless of the
sampling-recovery pair. In this work, we continue this investigation by
focusing on the practical case where the samples are pointwise evaluations
drawn identically and independently from a probability measure. Specifically,
for Hilbert-valued -holomorphic functions, we
show that the same rates can be achieved (up to a small polylogarithmic or
algebraic factor) for essentially arbitrary tensor-product Jacobi
(ultraspherical) measures. Our reconstruction maps are based on least squares
and compressed sensing procedures using the corresponding orthonormal Jacobi
polynomials. In doing so, we strengthen and generalize past work that has
derived weaker nonuniform guarantees for the uniform and Chebyshev measures
(and corresponding polynomials) only. We also extend various best -term
polynomial approximation error bounds to arbitrary Jacobi polynomial
expansions. Overall, we demonstrate that i.i.d.\ pointwise samples are
near-optimal for the recovery of infinite-dimensional, holomorphic functions
Optimal approximation of infinite-dimensional holomorphic functions
Over the last decade, approximating functions in infinite dimensions from
samples has gained increasing attention in computational science and
engineering, especially in computational uncertainty quantification. This is
primarily due to the relevance of functions that are solutions to parametric
differential equations in various fields, e.g. chemistry, economics,
engineering, and physics. While acquiring accurate and reliable approximations
of such functions is inherently difficult, current benchmark methods exploit
the fact that such functions often belong to certain classes of holomorphic
functions to get algebraic convergence rates in infinite dimensions with
respect to the number of (potentially adaptive) samples . Our work focuses
on providing theoretical approximation guarantees for the class of
-holomorphic functions, demonstrating that these
algebraic rates are the best possible for Banach-valued functions in infinite
dimensions. We establish lower bounds using a reduction to a discrete problem
in combination with the theory of -widths, Gelfand widths and Kolmogorov
widths. We study two cases, known and unknown anisotropy, in which the relative
importance of the variables is known and unknown, respectively. A key
conclusion of our paper is that in the latter setting, approximation from
finite samples is impossible without some inherent ordering of the variables,
even if the samples are chosen adaptively. Finally, in both cases, we
demonstrate near-optimal, non-adaptive (random) sampling and recovery
strategies which achieve close to same rates as the lower bounds
The factors influencing employees involvement with innovation activity at national registration department of Kuching, Sarawak / Nick Dexter Anak Jeffry
Employee involvement in innovation activity plays an important role to ensure that the organization can maintain and improve their performances. Innovation brings in new ways of doing things and the development of products and services may help the organization in fulfilling the customer’s needs and wants. The purpose of this research is to examine about the factors that influencing the employees involvement with innovation activity at National Registration Department of Kuching, Sarawak. The study also explores the relationship between the factors with employee involvement with innovation activity. The independent variables involved in this research are Knowledge, Organizational Culture, Leadership and Reward while the dependent variable is employee involvement with innovation activity. A total of 150 respondents which are all from the National Registration Department of Kuching, Sarawak were participated in the survey. Those respondents involve are the full-time workers with one and above years of experience in working with the National Registration Department of Sarawak. The result belief was found to be significant with very strong correlation and there is a relationship between all the factors mentioned earlier with employee involvement with innovation activity. In addition, the findings may give potential inputs and an insight to the organization where they may know the most important factor in influencing employee involvement with innovation activity is Rewards, Organizational Culture, Leadership and Knowledge. Suggestions for future research were also provided in this research
- …