12,862 research outputs found
Wavelet-Galerkin solution of boundary value problems
International audienceIn this paper we review the application of wavelets to the solution of partial differential equations. We consider in detail both the single scale and the multiscale Wavelet Galerkin method. The theory of wavelets is described here using the language and mathematics of signal processing. We show a method of adapting wavelets to an interval using an extrapolation technique called Wavelet Extrapolation. Wavelets on an interval allow boundary conditions to be enforced in partial differential equations and image boundary problems to be overcome in image processing. Finally, we discuss the fast inversion of matrices arising from differential operators by preconditioning the multiscale wavelet matrix. Wavelet preconditioning is shown to limit the growth of the matrix condition number, such that Krylov subspace iteration methods can accomplish fast inversion of large matrices
How do we understand and visualize uncertainty?
Geophysicists are often concerned with reconstructing subsurface properties using observations collected at or near the surface. For example, in seismic migration, we attempt to reconstruct subsurface geometry from surface seismic recordings, and in potential field inversion, observations are used to map electrical conductivity or density variations in geologic layers. The procedure of inferring information from indirect observations is called an inverse problem by mathematicians, and such problems are common in many areas of the physical sciences. The inverse problem of inferring the subsurface using surface observations has a corresponding forward problem, which consists of determining the data that would be recorded for a given subsurface configuration. In the seismic case, forward modeling involves a method for calculating a synthetic seismogram, for gravity data it consists of a computer code to compute gravity fields from an assumed subsurface density model. Note that forward modeling often involves assumptions about the appropriate physical relationship between unknowns (at depth) and observations on the surface, and all attempts to solve the problem at hand are limited by the accuracy of those assumptions. In the broadest sense then, exploration geophysicists have been engaged in inversion since the dawn of the profession and indeed algorithms often applied in processing centers can all be viewed as procedures to invert geophysical data
Unbiased image reconstruction as an inverse problem
An unbiased method for improving the resolution of astronomical images is
presented. The strategy at the core of this method is to establish a linear
transformation between the recorded image and an improved image at some
desirable resolution. In order to establish this transformation only the actual
point spread function and a desired point spread function need be known. Any
image actually recorded is not used in establishing the linear transformation
between the recorded and improved image. This method has a number of advantages
over other methods currently in use. It is not iterative which means it is not
necessary to impose any criteria, objective or otherwise, to stop the
iterations. The method does not require an artificial separation of the image
into ``smooth'' and ``point-like'' components, and thus is unbiased with
respect to the character of structures present in the image. The method
produces a linear transformation between the recorded image and the deconvolved
image and therefore the propagation of pixel-by-pixel flux error estimates into
the deconvolved image is trivial. It is explicitly constrained to preserve
photometry.Comment: 11 pages, TeX, uses mn.tex epsf.tex, accepted for publication in
MNRA
Distributing the Kalman Filter for Large-Scale Systems
This paper derives a \emph{distributed} Kalman filter to estimate a sparsely
connected, large-scale, dimensional, dynamical system monitored by a
network of sensors. Local Kalman filters are implemented on the
(dimensional, where ) sub-systems that are obtained after
spatially decomposing the large-scale system. The resulting sub-systems
overlap, which along with an assimilation procedure on the local Kalman
filters, preserve an th order Gauss-Markovian structure of the centralized
error processes. The information loss due to the th order Gauss-Markovian
approximation is controllable as it can be characterized by a divergence that
decreases as . The order of the approximation, , leads to a lower
bound on the dimension of the sub-systems, hence, providing a criterion for
sub-system selection. The assimilation procedure is carried out on the local
error covariances with a distributed iterate collapse inversion (DICI)
algorithm that we introduce. The DICI algorithm computes the (approximated)
centralized Riccati and Lyapunov equations iteratively with only local
communication and low-order computation. We fuse the observations that are
common among the local Kalman filters using bipartite fusion graphs and
consensus averaging algorithms. The proposed algorithm achieves full
distribution of the Kalman filter that is coherent with the centralized Kalman
filter with an th order Gaussian-Markovian structure on the centralized
error processes. Nowhere storage, communication, or computation of
dimensional vectors and matrices is needed; only dimensional
vectors and matrices are communicated or used in the computation at the
sensors
Traction force microscopy on soft elastic substrates: a guide to recent computational advances
The measurement of cellular traction forces on soft elastic substrates has
become a standard tool for many labs working on mechanobiology. Here we review
the basic principles and different variants of this approach. In general, the
extraction of the substrate displacement field from image data and the
reconstruction procedure for the forces are closely linked to each other and
limited by the presence of experimental noise. We discuss different strategies
to reconstruct cellular forces as they follow from the foundations of
elasticity theory, including two- versus three-dimensional, inverse versus
direct and linear versus non-linear approaches. We also discuss how biophysical
models can improve force reconstruction and comment on practical issues like
substrate preparation, image processing and the availability of software for
traction force microscopy.Comment: Revtex, 29 pages, 3 PDF figures, 2 tables. BBA - Molecular Cell
Research, online since 27 May 2015, special issue on mechanobiolog
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