Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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Learning regularization parameter-maps for variational image reconstruction using deep neural networks and algorithm unrolling
Get PDFWe introduce a method for fast estimation of data-adapted, spatio-temporally dependent regularization parameter-maps for variational image reconstruction, focusing on total variation (TV)-minimization. Our approach is inspired by recent developments in algorithm unrolling using deep neural networks (NNs), and relies on two distinct sub-networks. The first sub-network estimates the regularization parameter-map from the input data. The second sub-network unrolls T iterations of an iterative algorithm which approximately solves the corresponding TV-minimization problem incorporating the previously estimated regularization parameter-map. The overall network is trained end-to-end in a supervised learning fashion using pairs of clean-corrupted data but crucially without the need of having access to labels for the optimal regularization parameter-maps. We prove consistency of the unrolled scheme by showing that the unrolled energy functional used for the supervised learning Γ-converges as T tends to infinity, to the corresponding functional that incorporates the exact solution map of the TV-minimization problem. We apply and evaluate our method on a variety of large scale and dynamic imaging problems in which the automatic computation of such parameters has been so far challenging: 2D dynamic cardiac MRI reconstruction, quantitative brain MRI reconstruction, low-dose CT and dynamic image denoising. The proposed method consistently improves the TV-reconstructions using scalar parameters and the obtained parameter-maps adapt well to each imaging problem and data by leading to the preservation of detailed features. Although the choice of the regularization parameter-maps is data-driven and based on NNs, the proposed algorithm is entirely interpretable since it inherits the properties of the respective iterative reconstruction method from which the network is implicitly defined
A porous-media model for reactive fluid-rock interaction in a dehydrating rock
Get PDFWe study the GENERIC structure of models for reactive two-phase flows and their connection to a porous-media model for reactive fluid-rock interaction used in Geosciences. For this we discuss the equilibration of fast dissipative processes in the GENERIC framework. Mathematical properties of the porous-media model and first results on its mathematical analysis are provided. The mathematical assumptions imposed for the analysis are critically validated with the thermodynamical rock data sets
Temporal cavity soliton interaction in passively mode-locked semiconductor lasers
Get PDFWeak interaction due to gain saturation and recovery of temporal cavity solitons in a delay differential model of a long cavity semiconductor laser is studied numerically and analytically using an asymptotic approach. It is shown that apart from usual soliton repulsion leading to a harmonic mode-locking regime a soliton attraction is also possible in a laser with nonzero linewidth enhancement factor. It is shown numerically that the attraction can lead either to a soliton merging or to a pulse bound state formation
Gradient flows for coupling order parameters and mechanics
No full textWe construct a formal gradient flow structure for phase-field evolution coupled to mechanics in Lagrangian coordinates, present common ways to couple the evolution, and provide an incremental minimization strategy. While the usual presentation of continuum mechanics is intentionally brief, we construct an extensible functional analytical framework and a discretization approach that preserves the underlying variational structure. We consider phase separation and swelling of gels and then study stationary states of multiphase systems with surface tension and contact lines and show the robustness of the general approach for large deformations. We highlight differences between compressible and incompressible models and discuss issues of the sharp-interface limit for different magnitudes of the Cahn–Hilliard mobility
Optimal beam forming for laser materials processing
Get PDFWe investigate an optimal control problem related to laser material treatments such as welding, remelting, hardening, or the 3D printing of metal components. The mathematical model leads to the investigation of a quasilinear elliptic state system with additional non-monotone lower-oder terms. We analyze the state system, derive first order optimality conditions and show first results for beam shaping
Pressure-robust approximation of the incompressible Navier--Stokes equations in a rotating frame of reference
Get PDFA pressure-robust space discretization of the incompressible Navier--Stokes equations in a rotating frame of reference is considered. The discretization employs divergence-free, H1-conforming mixed finite element methods like Scott--Vogelius pairs. An error estimate for the velocity is derived that tracks the dependency of the error bound on the coefficients of the problem, in particular on the angular velocity. Numerical examples illustrate the theoretical results
Augmenting the grad-div stabilization for Taylor--Hood finite elements with a vorticity stabilization
Get PDFThe least squares vorticity stabilization (LSVS), proposed in Ahmed et al. for the Scott--Vogelius finite element discretization of the Oseen equations, is studied as an augmentation of the popular grad-div stabilized Taylor--Hood pair of spaces. An error analysis is presented which exploits the situation that the velocity spaces of Scott--Vogelius and Taylor--Hood are identical. Convection-robust error bounds are derived under the assumption that the Scott--Vogelius discretization is well posed on the considered grid. Numerical studies support the analytic results and they show that the LSVS-grad-div method might lead to notable error reductions compared with the standard grad-div method
An Approximate Two-Point Dirichlet Flux for Quasilinear Convection Diffusion Equations
No full textModeling and simulation of ion transport in electrolytes is an important tool to investigate electrochemical devices as well as biological systems at the cell scale. Well designed models follow first principles of non-equilibrium thermodynamics and include the fact that ions have a finite size. It is highly desirable that these properties are valid as well for discretized models. In this contribution, we present two numerical fluxes for two-point flux finite volume schemes which fulfill these requirements. We review recent results on entropic behavior and convergence. Concluding, we present first simulation results for biological ion channels
Estimation and inference for deep neuronal networks
Get PDFNonlinear regression problem is one of the most popular and important statistical tasks. The first methods like least squares estimation go back to Gauss and Legendre. Recent models and developments in statistics and machine learning like Deep Neuronal Networks (DNN) or nonlinear PDE stimulate new research in this direction which has to address the important issues and challenges of modern statistical inference such as huge complexity and parameter dimension of the model, limited sample size, lack of convexity and identifiability, among many others. Classical results of nonparametric statistics in terms of rate of convergence do not really address the mentioned issues. This paper offers a general approach to studying a nonlinear regression problem based on the notion of effective dimension. First, a special case of models with stochastically linear structure (SLS) is studied. The results provide finite sample expansions for the loss of the penalized maximum likelihood estimation (MLE). The leading term of such expansions as well as the corresponding remainder are given via the effective dimension and the effective sample size. The obtained expansions can be used to obtain sharp risk bounds and for statistical inference. Despite generality, all the presented bounds are nearly sharp and the classical asymptotic results can be obtained as simple corollaries. Although the basic SLS assumptions are not fulfilled for nonlinear smooth regression, we explain how the stochastic linearity can be achieved by extending the parameter space. The obtained general results are specified to nonlinear smooth regression and to a DNN with one hidden layer