5,642 research outputs found
The final publication is available at link.springer.com. An Optimal Control Approach to Find Sparse Data for Laplace Interpolation
Abstract. Finding optimal data for inpainting is a key problem in the context of partial differential equation-based image compression. We present a new model for optimising the data used for the reconstruction by the underlying homogeneous diffusion process. Our approach is based on an optimal control framework with a strictly convex cost functional containing an L1 term to enforce sparsity of the data and non-convex constraints. We propose a numerical approach that solves a series of convex optimisation problems with linear constraints. Our numerical examples show that it outperforms existing methods with respect to quality and computation time
Optimising Spatial and Tonal Data for PDE-based Inpainting
Some recent methods for lossy signal and image compression store only a few
selected pixels and fill in the missing structures by inpainting with a partial
differential equation (PDE). Suitable operators include the Laplacian, the
biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The
quality of such approaches depends substantially on the selection of the data
that is kept. Optimising this data in the domain and codomain gives rise to
challenging mathematical problems that shall be addressed in our work.
In the 1D case, we prove results that provide insights into the difficulty of
this problem, and we give evidence that a splitting into spatial and tonal
(i.e. function value) optimisation does hardly deteriorate the results. In the
2D setting, we present generic algorithms that achieve a high reconstruction
quality even if the specified data is very sparse. To optimise the spatial
data, we use a probabilistic sparsification, followed by a nonlocal pixel
exchange that avoids getting trapped in bad local optima. After this spatial
optimisation we perform a tonal optimisation that modifies the function values
in order to reduce the global reconstruction error. For homogeneous diffusion
inpainting, this comes down to a least squares problem for which we prove that
it has a unique solution. We demonstrate that it can be found efficiently with
a gradient descent approach that is accelerated with fast explicit diffusion
(FED) cycles. Our framework allows to specify the desired density of the
inpainting mask a priori. Moreover, is more generic than other data
optimisation approaches for the sparse inpainting problem, since it can also be
extended to nonlinear inpainting operators such as EED. This is exploited to
achieve reconstructions with state-of-the-art quality.
We also give an extensive literature survey on PDE-based image compression
methods
Fast Bayesian Optimal Experimental Design for Seismic Source Inversion
We develop a fast method for optimally designing experiments in the context
of statistical seismic source inversion. In particular, we efficiently compute
the optimal number and locations of the receivers or seismographs. The seismic
source is modeled by a point moment tensor multiplied by a time-dependent
function. The parameters include the source location, moment tensor components,
and start time and frequency in the time function. The forward problem is
modeled by elastodynamic wave equations. We show that the Hessian of the cost
functional, which is usually defined as the square of the weighted L2 norm of
the difference between the experimental data and the simulated data, is
proportional to the measurement time and the number of receivers. Consequently,
the posterior distribution of the parameters, in a Bayesian setting,
concentrates around the "true" parameters, and we can employ Laplace
approximation and speed up the estimation of the expected Kullback-Leibler
divergence (expected information gain), the optimality criterion in the
experimental design procedure. Since the source parameters span several
magnitudes, we use a scaling matrix for efficient control of the condition
number of the original Hessian matrix. We use a second-order accurate finite
difference method to compute the Hessian matrix and either sparse quadrature or
Monte Carlo sampling to carry out numerical integration. We demonstrate the
efficiency, accuracy, and applicability of our method on a two-dimensional
seismic source inversion problem
Inexact Solves in Interpolatory Model Reduction
We investigate the use of inexact solves for interpolatory model reduction
and consider associated perturbation effects on the underlying model reduction
problem. We give bounds on system perturbations induced by inexact solves and
relate this to termination criteria for iterative solution methods. We show
that when a Petrov-Galerkin framework is employed for the inexact solves, the
associated reduced order model is an exact interpolatory model for a nearby
full-order system; thus demonstrating backward stability. We also give evidence
that for \h2-optimal interpolation points, interpolatory model reduction is
robust with respect to perturbations due to inexact solves. Finally, we
demonstrate the effecitveness of direct use of inexact solves in optimal
approximation. The result is an effective model reduction
strategy that is applicable in realistically large-scale settings.Comment: 42 pages, 5 figure
- …