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Table Of Integrals Series And Products

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

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Inverse problems: A Bayesian perspective

Stochastic equations in infinite dimensions

We study flexible and proper smoothness priors for Bayesian
 statistical inverse problems by using Whittle-Matern Gaussian random fields.

 We review earlier results on finite-difference approximations of certain Whittle-Matern random field in $\mathbb{R}^2$. Then we derive finite-element method approximations and show that the
 discrete approximations can be expressed as solutions of sparse
 stochastic matrix equations.

 Such equations are known to be computationally efficient and
 useful in inverse problems with a large number of unknowns.

 &nbsp

 The presented construction of Whittle-Matern correlation functions
 allows both isotropic or anisotropic priors with adjustable
 parameters in correlation length and variance. These parameters can
 be used, for example, to model spatially varying structural
 information of unknowns.
 &nbsp
 As numerical examples, we apply the developed priors to
 two-dimensional electrical impedance tomography problems.

Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography

Increase in cerebral blood flow of right prefrontal cortex in man during orgasm.

Deep learning has solved many problems that are out of reach of heuristic algorithms. It has also been successfully applied in wireless communications, even though the current radio systems are well-understood and optimal algorithms exist for many tasks. While some gains have been obtained by learning individual parts of a receiver, a better approach is to jointly learn the whole receiver. This, however, often results in a challenging nonlinear problem, for which the optimal solution is infeasible to implement. To this end, we propose a deep fully convolutional neural network, DeepRx, which executes the whole receiver pipeline from frequency domain signal stream to uncoded bits in a 5G-compliant fashion. We facilitate accurate channel estimation by constructing the input of the convolutional neural network in a very specific manner using both the data and pilot symbols. Also, DeepRx outputs soft bits that are compatible with the channel coding used in 5G systems. Using 3GPP-defined channel models, we demonstrate that DeepRx outperforms traditional methods. We also show that the high performance can likely be attributed to DeepRx learning to utilize the known constellation points of the unknown data symbols, together with the local symbol distribution, for improved detection accuracy.

DeepRx: Fully Convolutional Deep Learning Receiver

Inverse problems are known to be very intolerant to both data errors
and errors in the forward model.
With several inverse problems the adequately accurate forward model can turn out to be
computationally excessively complex.
The Bayesian framework for inverse problems has recently been shown to enable
the adoption of highly approximate forward models.
This approach is based on the modelling of the associated approximation errors
that are incorporated in the construction of the computational model.
In this paper we investigate the extension of the approximation error theory to
nonstationary inverse problems.
We develop the basic framework for linear nonstationary inverse problems
that allows one to use both highly reduced states and extended time steps.
As an example we study the one dimensional heat equation.

Approximation errors in nonstationary inverse problems

Nonstationary inverse problems are usually cast in the state-space formalism. The complete statistics of linear Gaussian problems can be computed with the Kalman filters and smoothers. Nonlinear non-Gaussian problems would necessitate the adoption of particle filters or similar computationally very heavy approaches. The so-called extended Kalman filters often provide suboptimal but feasible estimates for the nonlinear problems. Several applications which lead to nonstationary inverse problems are time critical, such as process tomography and many biomedical problems. In such applications, there is typically a need to use reduced-order models which may heavily compromise the computational accuracy that is usually required for inverse problems. One approach to overcome the model reduction problem is to use the approximation error analysis. In this paper, we derive the equations for the extended Kalman filter for nonlinear state estimation problems in which the approximation error models are taken into account. We consider the approximation errors that are due to both state reduction and time stepping. As an example, we consider the identification of the coefficients of the heat equation. Our main result is that, also in nonlinear problems, approximation error analysis allows us to obtain accurate estimates and uncertainties of the parameters in reduced-order models that are suitable for fast calculation.

Janne M. J. Huttunen

Papers

Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography

Increase in cerebral blood flow of right prefrontal cortex in man during orgasm.

DeepRx: Fully Convolutional Deep Learning Receiver

Approximation errors in nonstationary inverse problems

Approximation error analysis in nonlinear state estimation with an application to state-space identification