pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems

pde2path is a free and easy to use Matlab continuation/bifurcation package for elliptic systems of PDEs with arbitrary many components, on general two dimensional domains, and with rather general boundary conditions. The package is based on the FEM of the Matlab pdetoolbox, and is explained by a number of examples, including Bratu's problem, the Schnakenberg model, Rayleigh-Benard convection, and von Karman plate equations. These serve as templates to study new problems, for which the user has to provide, via Matlab function files, a description of the geometry, the boundary conditions, the coefficients of the PDE, and a rough initial guess of a solution. The basic algorithm is a one parameter arclength continuation with optional bifurcation detection and branch-switching. Stability calculations, error control and mesh-handling, and some elementary time-integration for the associated parabolic problem are also supported. The continuation, branch-switching, plotting etc are performed via Matlab command-line function calls guided by the AUTO style. The software can be downloaded from www.staff.uni-oldenburg.de/hannes.uecker/pde2path, where also an online documentation of the software is provided such that in this paper we focus more on the mathematics and the example systems

pde2path - version 2.0: faster FEM, multi-parameter continuation, nonlinear boundary conditions, and periodic domains - a short manual

pdepath 2.0 is an upgrade of the continuation/bifurcation package pde2path for elliptic systems of PDEs over bounded 2D domains, based on the Matlab pdetoolbox. The new features include a more efficient use of FEM, easier switching between different single parameter continuations, genuine multi-parameter continuation (e.g., fold continuation), more efficient implementation of nonlinear boundary conditions, cylinder and torus geometries (i.e., periodic boundary conditions), and a general interface for adding auxiliary equations like mass conservation or phase equations for continuation of traveling waves. The package (library, demos, manuals) can be downloaded at www.staff.uni-oldenburg.de/hannes.uecker/pde2pat

Real-time magnetic resonance imaging: Radial gradient-echo sequences with nonlinear inverse reconstruction.

Objective The aim of this study is to evaluate a real-time magnetic resonance imaging (MRI) method that not only promises high spatiotemporal resolution but also practical robustness in a wide range of scientific and clinical applications. Materials and Methods The proposed method relies on highly undersampled gradient-echo sequences with radial encoding schemes. The serial image reconstruction process solves the true mathematical task that emerges as a nonlinear inverse problem with the complex image and all coil sensitivity maps as unknowns. Extensions to model-based reconstructions for quantitative parametric mapping further increase the number of unknowns, for example, by adding parameters for phase-contrast flow or T1 relaxation. In all cases, an iterative numerical solution that minimizes a respective cost function is achieved with use of the iteratively regularized Gauss-Newton method. Convergence is supported by regularization, for example, to the preceding frame, whereas temporal fidelity is ensured by downsizing the regularization strength in comparison to the data consistency term in each iterative step. Practical implementations of highly parallelized algorithms are realized on a computer with multiple graphical processing units. It is "invisibly" integrated into a commercial 3-T MRI system to allow for conventional usage and to provide online reconstruction, display, and storage of regular DICOM image series. Results Depending on the application, the proposed method offers serial imaging, that is, the recording of MRI movies, with variable spatial resolution and up to 100 frames per second (fps)-corresponding to 10 milliseconds image acquisition times. For example, movements of the temporomandibular joint during opening and closing of the mouth are visualized with use of simultaneous dual-slice movies of both joints at 2 x 10 fps (50 milliseconds per frame). Cardiac function may be studied at 30 to 50 fps (33.3 to 20 milliseconds), whereas articulation processes typically require 50 fps (20 milliseconds) or orthogonal dual-slice acquisitions at 2 x 25 fps (20 milliseconds). Methodological extensions to model-based reconstructions achieve improved quantitative mapping of flow velocities and T1 relaxation times in a variety of clinical scenarios. Conclusions Real-time gradient-echo MRI with extreme radial undersampling and nonlinear inverse reconstruction allows for direct monitoring of arbitrary physiological processes and body functions. In many cases, pertinent applications offer hitherto impossible clinical studies (eg, of high-resolution swallowing dynamics) or bear the potential to replace existing MRI procedures (eg, electrocardiogram-gated cardiac examinations). As a consequence, many novel opportunities will require a change of paradigm in MRI-based radiology. At this stage, extended clinical trials are needed

APIR-Net: Autocalibrated Parallel Imaging Reconstruction using a Neural Network

Deep learning has been successfully demonstrated in MRI reconstruction of accelerated acquisitions. However, its dependence on representative training data limits the application across different contrasts, anatomies, or image sizes. To address this limitation, we propose an unsupervised, auto-calibrated k-space completion method, based on a uniquely designed neural network that reconstructs the full k-space from an undersampled k-space, exploiting the redundancy among the multiple channels in the receive coil in a parallel imaging acquisition. To achieve this, contrary to common convolutional network approaches, the proposed network has a decreasing number of feature maps of constant size. In contrast to conventional parallel imaging methods such as GRAPPA that estimate the prediction kernel from the fully sampled autocalibration signals in a linear way, our method is able to learn nonlinear relations between sampled and unsampled positions in k-space. The proposed method was compared to the start-of-the-art ESPIRiT and RAKI methods in terms of noise amplification and visual image quality in both phantom and in-vivo experiments. The experiments indicate that APIR-Net provides a promising alternative to the conventional parallel imaging methods, and results in improved image quality especially for low SNR acquisitions.Comment: To appear in the proceedings of MICCAI 2019 Workshop Machine Learning for Medical Image Reconstructio

Phase diagram analysis and crystal growth of solid solutions Ca_{1-x}Sr_xF_2

The binary phase diagram CaF$_2$--SrF$_2$ was investigated by differential thermal analysis (DTA). Both substances exhibit unlimited mutual solubility with an azeotropic point showing a minimum melting temperature of T_\mathrm{min}=1373^{\circ}$C for the composition Ca$_{0.582}$Sr$_{0.418}$F$_2$. Close to this composition, homogeneous single crystals up to 30 mm diameter without remarkable segregation could be grown by the Czochralski method.Comment: accepted for publication in J. Crystal Growt