150 research outputs found
Singularities of the Radon transform
Singularities of the Radon transform of a piecewise smooth function ,
, , are calculated. If the singularities of the Radon
transform are known, then the equations of the surfaces of discontinuity of
are calculated by applying the Legendre transform to the functions,
which appear in the equations of the discontinuity surfaces of the Radon
transform of ; examples are given. Numerical aspects of the problem of
finding discontinuities of , given the discontinuities of its Radon
transform, are discussed.Comment: 7 page
The Vortex Solution in the (2+1)-Dimensional Yang-Mills-Chern-Simons Theory at High Temperature
The vortex-like solution to the non-linear field equations in a
two-dimensional SU(2) gauge theory with the Chern-Simons mass term is found at
high temperature. It is derived from the effective Lagrangian including the
leading order finite temperature corrections. The discovered field
configuration possesses the finite energy and the quantized magnetic flux. At
the centre of the vortex the point charge is located which is surrounded by the
distributed charge of the opposite sign and the vortex is neutral as a whole.
At high temperature the energy of the vortex is negative and it corresponds to
the ground state. The derived solution is considered to be a result of heating
the lattice vacuum structure formed at zero temperature.Comment: 9 pages, LaTeX, no figures, a4, cite.st
S-fraction multiscale finite-volume method for spectrally accurate wave propagation
We develop a method for numerical time-domain wave propagation based on the
model order reduction approach. The method is built with high-performance
computing (HPC) implementation in mind that implies a high level of parallelism
and greatly reduced communication requirements compared to the traditional
high-order finite-difference time-domain (FDTD) methods. The approach is
inherently multiscale, with a reference fine grid model being split into
subdomains. For each subdomain the coarse scale reduced order models (ROMs) are
precomputed off-line in a parallel manner. The ROMs approximate the
Neumann-to-Dirichlet (NtD) maps with high (spectral) accuracy and are used to
couple the adjacent subdomains on the shared boundaries. The on-line part of
the method is an explicit time stepping with the coupled ROMs. To lower the
on-line computation cost the reduced order spatial operator is sparsified by
transforming to a matrix Stieltjes continued fraction (S-fraction) form. The
on-line communication costs are also reduced due to the ROM NtD map
approximation properties. Another source of performance improvement is the time
step length. Properly chosen ROMs substantially improve the
Courant-Friedrichs-Lewy (CFL) condition. This allows the CFL time step to
approach the Nyquist limit, which is typically unattainable with traditional
schemes that have the CFL time step much smaller than the Nyquist sampling
rate.Comment: 5 pages, 3 figure
A nonlinear method for imaging with acoustic waves via reduced order model backprojection
We introduce a novel nonlinear imaging method for the acoustic wave equation
based on data-driven model order reduction. The objective is to image the
discontinuities of the acoustic velocity, a coefficient of the scalar wave
equation from the discretely sampled time domain data measured at an array of
transducers that can act as both sources and receivers. We treat the wave
equation along with transducer functionals as a dynamical system. A reduced
order model (ROM) for the propagator of such system can be computed so that it
interpolates exactly the measured time domain data. The resulting ROM is an
orthogonal projection of the propagator on the subspace of the snapshots of
solutions of the acoustic wave equation. While the wavefield snapshots are
unknown, the projection ROM can be computed entirely from the measured data,
thus we refer to such ROM as data-driven. The image is obtained by
backprojecting the ROM. Since the basis functions for the projection subspace
are not known, we replace them with the ones computed for a known smooth
kinematic velocity model. A crucial step of ROM construction is an implicit
orthogonalization of solution snapshots. It is a nonlinear procedure that
differentiates our approach from the conventional linear imaging methods
(Kirchhoff migration and reverse time migration - RTM). It resolves all
dynamical behavior captured by the data, so the error from the imperfect
knowledge of the velocity model is purely kinematic. This allows for almost
complete removal of multiple reflection artifacts, while simultaneously
improving the resolution in the range direction compared to conventional RTM.Comment: 33 pages, 8 figure
Distance preserving model order reduction of graph-Laplacians and cluster analysis
Graph-Laplacians and their spectral embeddings play an important role in
multiple areas of machine learning. This paper is focused on graph-Laplacian
dimension reduction for the spectral clustering of data as a primary
application. Spectral embedding provides a low-dimensional parametrization of
the data manifold which makes the subsequent task (e.g., clustering) much
easier. However, despite reducing the dimensionality of data, the overall
computational cost may still be prohibitive for large data sets due to two
factors. First, computing the partial eigendecomposition of the graph-Laplacian
typically requires a large Krylov subspace. Second, after the spectral
embedding is complete, one still has to operate with the same number of data
points. For example, clustering of the embedded data is typically performed
with various relaxations of k-means which computational cost scales poorly with
respect to the size of data set. In this work, we switch the focus from the
entire data set to a subset of graph vertices (target subset). We develop two
novel algorithms for such low-dimensional representation of the original graph
that preserves important global distances between the nodes of the target
subset. In particular, it allows to ensure that target subset clustering is
consistent with the spectral clustering of the full data set if one would
perform such. That is achieved by a properly parametrized reduced-order model
(ROM) of the graph-Laplacian that approximates accurately the diffusion
transfer function of the original graph for inputs and outputs restricted to
the target subset. Working with a small target subset reduces greatly the
required dimension of Krylov subspace and allows to exploit the conventional
algorithms (like approximations of k-means) in the regimes when they are most
robust and efficient.Comment: 28 pages, 10 figure
Multi-scale S-fraction reduced-order models for massive wavefield simulations
We developed a novel reduced-order multi-scale method for solving large
time-domain wavefield simulation problems. Our algorithm consists of two main
stages. During the first "off-line" stage the fine-grid operator (of the graph
Laplacian type} is partitioned on coarse cells (subdomains). Then
projection-type multi-scale reduced order models (ROMs) are computed for the
coarse cell operators. The off-line stage is embarrassingly parallel as ROM
computations for the subdomains are independent of each other. It also does not
depend on the number of simulated sources (inputs) and it is performed just
once before the entire time-domain simulation. At the second "on-line" stage
the time-domain simulation is performed within the obtained multi-scale ROM
framework. The crucial feature of our formulation is the representation of the
ROMs in terms of matrix Stieltjes continued fractions (S-fractions). The
layered structure of the S-fraction introduces several hidden layers in the ROM
representation, that results in the block-tridiagonal dynamic system within
each coarse cell. This allows us to sparsify the obtained multi-scale subdomain
operator ROMs and to reduce the communications between the adjacent subdomains
which is highly beneficial for a parallel implementation of the on-line stage.
Our approach suits perfectly the high performance computing architectures,
however in this paper we present rather promising numerical results for a
serial computing implementation only. These results include 3D acoustic and
multi-phase anisotropic elastic problems.Comment: 31 pages, 11 figure
Robust nonlinear processing of active array data in inverse scattering via truncated reduced order models
We introduce a novel algorithm for nonlinear processing of data gathered by
an active array of sensors which probes a medium with pulses and measures the
resulting waves. The algorithm is motivated by the application of array
imaging. We describe it for a generic hyperbolic system that applies to
acoustic, electromagnetic or elastic waves in a scattering medium modeled by an
unknown coefficient called the reflectivity. The goal of imaging is to invert
the nonlinear mapping from the reflectivity to the array data. Many existing
imaging methodologies ignore the nonlinearity i.e., operate under the
assumption that the Born (single scattering) approximation is accurate. This
leads to image artifacts when multiple scattering is significant. Our algorithm
seeks to transform the array data to those corresponding to the Born
approximation, so it can be used as a pre-processing step for any linear
inversion method. The nonlinear data transformation algorithm is based on a
reduced order model defined by a proxy wave propagator operator that has four
important properties. First, it is data driven, meaning that it is constructed
from the data alone, with no knowledge of the medium. Second, it can be
factorized in two operators that have an approximately affine dependence on the
unknown reflectivity. This allows the computation of the Fr\'{e}chet derivative
of the reflectivity to the data mapping which gives the Born approximation.
Third, the algorithm involves regularization which balances numerical stability
and data fitting with accuracy of the order of the standard deviation of
additive data noise. Fourth, the algebraic nature of the algorithm makes it
applicable to scalar (acoustic) and vectorial (elastic, electromagnetic) wave
data without any specific modifications.Comment: 26 pages, 6 figure
Untangling the nonlinearity in inverse scattering with data-driven reduced order models
The motivation of this work is an inverse problem for the acoustic wave
equation, where an array of sensors probes an unknown medium with pulses and
measures the scattered waves. The goal of the inversion is to determine from
these measurements the structure of the scattering medium, modeled by a
spatially varying acoustic impedance function. Many inversion algorithms assume
that the mapping from the unknown impedance to the scattered waves is
approximately linear. The linearization, known as the Born approximation, is
not accurate in strongly scattering media, where the waves undergo multiple
reflections before they reach the sensors in the array. Thus, the
reconstructions of the impedance have numerous artifacts. The main result of
the paper is a novel, linear-algebraic algorithm that uses a reduced order
model (ROM) to map the data to those corresponding to the single scattering
(Born) model. The ROM construction is based only on the measurements at the
sensors in the array. The ROM is a proxy for the wave propagator operator, that
propagates the wave in the unknown medium over the duration of the time
sampling interval. The output of the algorithm can be input into any
off-the-shelf inversion software that incorporates state of the art linear
inversion algorithms to reconstruct the unknown acoustic impedance.Comment: 27 pages, 9 figure
Direct, nonlinear inversion algorithm for hyperbolic problems via projection-based model reduction
We estimate the wave speed in the acoustic wave equation from boundary
measurements by constructing a reduced-order model (ROM) matching discrete
time-domain data. The state-variable representation of the ROM can be
equivalently viewed as a Galerkin projection onto the Krylov subspace spanned
by the snapshots of the time-domain solution. The success of our algorithm
hinges on the data-driven Gram--Schmidt orthogonalization of the snapshots that
suppresses multiple reflections and can be viewed as a discrete form of the
Marchenko--Gel'fand--Levitan--Krein algorithm. In particular, the
orthogonalized snapshots are localized functions, the (squared) norms of which
are essentially weighted averages of the wave speed. The centers of mass of the
squared orthogonalized snapshots provide us with the grid on which we
reconstruct the velocity. This grid is weakly dependent on the wave speed in
traveltime coordinates, so the grid points may be approximated by the centers
of mass of the analogous set of squared orthogonalized snapshots generated by a
known reference velocity. We present results of inversion experiments for one-
and two-dimensional synthetic models.Comment: 54 pages, 6 figures fixed typos and small errors expanded several
sections to aid in understandin
A model reduction approach to numerical inversion for a parabolic partial differential equation
We propose a novel numerical inversion algorithm for the coefficients of
parabolic partial differential equations, based on model reduction. The study
is motivated by the application of controlled source electromagnetic
exploration, where the unknown is the subsurface electrical resistivity and the
data are time resolved surface measurements of the magnetic field. The
algorithm presented in this paper considers inversion in one and two
dimensions. The reduced model is obtained with rational interpolation in the
frequency (Laplace) domain and a rational Krylov subspace projection method. It
amounts to a nonlinear mapping from the function space of the unknown
resistivity to the small dimensional space of the parameters of the reduced
model. We use this mapping as a nonlinear preconditioner for the Gauss-Newton
iterative solution of the inverse problem. The advantage of the inversion
algorithm is twofold. First, the nonlinear preconditioner resolves most of the
nonlinearity of the problem. Thus the iterations are less likely to get stuck
in local minima and the convergence is fast. Second, the inversion is
computationally efficient because it avoids repeated accurate simulations of
the time-domain response. We study the stability of the inversion algorithm for
various rational Krylov subspaces, and assess its performance with numerical
experiments.Comment: 31 pages, 9 figures, 2 table
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