45,660 research outputs found
A Comparison between Fixed-Basis and Variable-Basis Schemes for Function Approximation and Functional Optimization
Fixed-basis and variable-basis approximation schemes are compared for the problems of function approximation and functional optimization (also known as infinite programming). Classes of problems are investigated for which variable-basis schemes with sigmoidal computational
units perform better than fixed-basis ones, in terms of the minimum number of computational units needed to achieve a desired error in function approximation or approximate optimization. Previously known bounds on the accuracy are extended, with better rates, to families o
A Comparison between Fixed-Basis and Variable-Basis Schemes for Function Approximation and Functional Optimization
Fixed-basis and variable-basis approximation schemes are compared for the problems of function approximation and functional optimization also known as infinite programming . Classes of problems are investigated for which variable-basis schemes with sigmoidal computational units perform better than fixed-basis ones, in terms of the minimum number of computational units needed to achieve a desired error in function approximation or approximate optimization. Previously known bounds on the accuracy are extended, with better rates, to families of d-variable functions whose actual dependence is on a subset of d d variables, where the indices of these d variables are not known a priori
Suboptimal solutions to network team optimization problems
Smoothness of the solutions to network team optimization problems with statistical information structure is investigated. Suboptimal solutions expressed as linear combinations of elements from sets of basis functions containing adjustable parameters are considered. Estimates of their accuracy are derived, for basis functions represented by sinusoids with variable frequencies and phases and
Gaussians with variable centers and widthss
A continuous analogue of the tensor-train decomposition
We develop new approximation algorithms and data structures for representing
and computing with multivariate functions using the functional tensor-train
(FT), a continuous extension of the tensor-train (TT) decomposition. The FT
represents functions using a tensor-train ansatz by replacing the
three-dimensional TT cores with univariate matrix-valued functions. The main
contribution of this paper is a framework to compute the FT that employs
adaptive approximations of univariate fibers, and that is not tied to any
tensorized discretization. The algorithm can be coupled with any univariate
linear or nonlinear approximation procedure. We demonstrate that this approach
can generate multivariate function approximations that are several orders of
magnitude more accurate, for the same cost, than those based on the
conventional approach of compressing the coefficient tensor of a tensor-product
basis. Our approach is in the spirit of other continuous computation packages
such as Chebfun, and yields an algorithm which requires the computation of
"continuous" matrix factorizations such as the LU and QR decompositions of
vector-valued functions. To support these developments, we describe continuous
versions of an approximate maximum-volume cross approximation algorithm and of
a rounding algorithm that re-approximates an FT by one of lower ranks. We
demonstrate that our technique improves accuracy and robustness, compared to TT
and quantics-TT approaches with fixed parameterizations, of high-dimensional
integration, differentiation, and approximation of functions with local
features such as discontinuities and other nonlinearities
A machine learning route between band mapping and band structure
The electronic band structure (BS) of solid state materials imprints the
multidimensional and multi-valued functional relations between energy and
momenta of periodically confined electrons. Photoemission spectroscopy is a
powerful tool for its comprehensive characterization. A common task in
photoemission band mapping is to recover the underlying quasiparticle
dispersion, which we call band structure reconstruction. Traditional methods
often focus on specific regions of interests yet require extensive human
oversight. To cope with the growing size and scale of photoemission data, we
develop a generic machine-learning approach leveraging the information within
electronic structure calculations for this task. We demonstrate its capability
by reconstructing all fourteen valence bands of tungsten diselenide and
validate the accuracy on various synthetic data. The reconstruction uncovers
previously inaccessible momentum-space structural information on both global
and local scales in conjunction with theory, while realizing a path towards
integrating band mapping data into materials science databases
Model order reduction approaches for infinite horizon optimal control problems via the HJB equation
We investigate feedback control for infinite horizon optimal control problems
for partial differential equations. The method is based on the coupling between
Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is
well-known that HJB equations suffer the so called curse of dimensionality and,
therefore, a reduction of the dimension of the system is mandatory. In this
report we focus on the infinite horizon optimal control problem with quadratic
cost functionals. We compare several model reduction methods such as Proper
Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati
equation based approach. Finally, we present numerical examples and discuss
several features of the different methods analyzing advantages and
disadvantages of the reduction methods
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