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We consider the numerical approximation of second-order semi-linear parabolic stochastic partial differential equations interpreted in the mild sense which we solve on general two-dimensional domains with a C² boundary with homogeneous Dirichlet boundary conditions. The equations are driven by Gaussian additive noise, and several Lipschitz-like conditions are imposed on the nonlinear function. We discretize in space with a spectral Galerkin method and in time using an explicit Euler-like scheme. For irregular shapes, the necessary Dirichlet eigenvalues and eigenfunctions are obtained from a boundary integral equation method. This yields a nonlinear eigenvalue problem, which is discretized using a boundary element collocation method and is solved with the Beyn contour integral algorithm. We present an error analysis as well as numerical results on an exemplary asymmetric shape, and point out limitations of the approach.
Analysis and computation of the transmission eigenvalues with a conductive boundary condition
(2022)
We provide a new analytical and computational study of the transmission eigenvalues with a conductive boundary condition. These eigenvalues are derived from the scalar inverse scattering problem for an inhomogeneous material with a conductive boundary condition. The goal is to study how these eigenvalues depend on the material parameters in order to estimate the refractive index. The analytical questions we study are: deriving Faber–Krahn type lower bounds, the discreteness and limiting behavior of the transmission eigenvalues as the conductivity tends to infinity for a sign changing contrast. We also provide a numerical study of a new boundary integral equation for computing the eigenvalues. Lastly, using the limiting behavior we will numerically estimate the refractive index from the eigenvalues provided the conductivity is sufficiently large but unknown.
Fields of asymmetric tensors play an important role in many applications such as medical imaging (diffusion tensor magnetic resonance imaging), physics, and civil engineering (for example Cauchy-Green-deformation tensor, strain tensor with local rotations, etc.). However, such asymmetric tensors are usually symmetrized and then further processed. Using this procedure results in a loss of information. A new method for the processing of asymmetric tensor fields is proposed restricting our attention to tensors of second-order given by a 2x2 array or matrix with real entries. This is achieved by a transformation resulting in Hermitian matrices that have an eigendecomposition similar to symmetric matrices. With this new idea numerical results for real-world data arising from a deformation of an object by external forces are given. It is shown that the asymmetric part indeed contains valuable information.
An alternative method is presented to numerically compute interior elastic transmission eigenvalues for various domains in two dimensions. This is achieved by discretizing the resulting system of boundary integral equations in combination with a nonlinear eigenvalue solver. Numerical results are given to show that this new approach can provide better results than the finite element method when dealing with general domains.
The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10− 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.
There is a very large number of very important situations which can be modeled with nonlinear parabolic partial differential equations (PDEs) in several dimensions. In general, these PDEs can be solved by discretizing in the spatial variables and transforming them into huge systems of ordinary differential equations (ODEs), which are very stiff. Therefore, standard explicit methods require a large number of iterations to solve stiff problems. But implicit schemes are computationally very expensive when solving huge systems of nonlinear ODEs. Several families of Extrapolated Stabilized Explicit Runge-Kutta schemes (ESERK) with different order of accuracy (3 to 6) are derived and analyzed in this work. They are explicit methods, with stability regions extended, along the negative real semi-axis, quadratically with respect to the number of stages s, hence they can be considered to solve stiff problems much faster than traditional explicit schemes. Additionally, they allow the adaptation of the step length easily with a very small cost.
Two new families of ESERK schemes (ESERK3 and ESERK6) are derived, and analyzed, in this work. Each family has more than 50 new schemes, with up to 84.000 stages in the case of ESERK6. For the first time, we also parallelized all these new variable step length and variable number of stages algorithms (ESERK3, ESERK4, ESERK5, and ESERK6). These parallelized strategies allow to decrease times significantly, as it is discussed and also shown numerically in two problems. Thus, the new codes provide very good results compared to other well-known ODE solvers. Finally, a new strategy is proposed to increase the efficiency of these schemes, and it is discussed the idea of combining ESERK families in one code, because typically, stiff problems have different zones and according to them and the requested tolerance the optimum order of convergence is different.
Interior transmission eigenvalue problems for the Helmholtz equation play an important role in inverse wave scattering. Some distribution properties of those eigenvalues in the complex plane are reviewed. Further, a new scattering model for the interior transmission eigenvalue problem with mixed boundary conditions is described and an efficient algorithm for computing the interior transmission eigenvalues is proposed. Finally, extensive numerical results for a variety of two-dimensional scatterers are presented to show the validity of the proposed scheme.
A second-order L-stable exponential time-differencing (ETD) method is developed by combining an ETD scheme with approximating the matrix exponentials by rational functions having real distinct poles (RDP), together with a dimensional splitting integrating factor technique. A variety of non-linear reaction-diffusion equations in two and three dimensions with either Dirichlet, Neumann, or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second-order implicit-explicit schemes. An additional performance boost is gained through further use of basic parallelization techniques.
In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests.
The inverse scattering problem for a conductive boundary condition and transmission eigenvalues
(2018)
In this paper, we consider the inverse scattering problem associated with an inhomogeneous media with a conductive boundary. In particular, we are interested in two problems that arise from this inverse problem: the inverse conductivity problem and the corresponding interior transmission eigenvalue problem. The inverse conductivity problem is to recover the conductive boundary parameter from the measured scattering data. We prove that the measured scatted data uniquely determine the conductivity parameter as well as describe a direct algorithm to recover the conductivity. The interior transmission eigenvalue problem is an eigenvalue problem associated with the inverse scattering of such materials. We investigate the convergence of the eigenvalues as the conductivity parameter tends to zero as well as prove existence and discreteness for the case of an absorbing media. Lastly, several numerical and analytical results support the theory and we show that the inside–outside duality method can be used to reconstruct the interior conductive eigenvalues.
The aim of the current study was to investigate the performance of integrated RF
transmit arrays with high channel count consisting of meander microstrip antennas
for body imaging at 7 T and to optimize the position and number of transmit ele-
ments. RF simulations using multiring antenna arrays placed behind the bore liner
were performed for realistic exposure conditions for body imaging. Simulations were
performed for arrays with as few as eight elements and for arrays with high channel
counts of up to 48 elements. The B1+ field was evaluated regarding the degrees of
freedom for RF shimming in the abdomen. Worst-case specific absorption rate
(SARwc ), SAR overestimation in the matrix compression, the number of virtual obser-
vation points (VOPs) and SAR efficiency were evaluated. Constrained RF shimming
was performed in differently oriented regions of interest in the body, and the devia-
tion from a target B1+ field was evaluated. Results show that integrated multiring
arrays are able to generate homogeneous B1+ field distributions for large FOVs, espe-
cially for coronal/sagittal slices, and thus enable body imaging at 7 T with a clinical
workflow; however, a low duty cycle or a high SAR is required to achieve homoge-
neous B1+ distributions and to exploit the full potential. In conclusion, integrated
arrays allow for high element counts that have high degrees of freedom for the pulse
optimization but also produce high SARwc , which reduces the SAR accuracy in the
VOP compression for low-SAR protocols, leading to a potential reduction in array
performance. Smaller SAR overestimations can increase SAR accuracy, but lead to a
high number of VOPs, which increases the computational cost for VOP evaluation
and makes online SAR monitoring or pulse optimization challenging. Arrays with
interleaved rings showed the best results in the study.
After a brief introduction of conventional laboratory structures, this work focuses on an innovative and universal approach for a setup of a training laboratory for electric machines and drive systems. The novel approach employs a central 48 V DC bus, which forms the backbone of the structure. Several sets of DC machine, asynchronous machine and synchronous machine are connected to this bus. The advantages of the novel system structure are manifold, both from a didactic and a technical point of view: Student groups can work on their own performance level in a highly parallelized and at the same time individualized way. Additional training setups (similar or different) can easily be added. Only the total power dissipation has to be provided, i.e. the DC bus balances the power flow between the student groups. Comparative results of course evaluations of several cohorts of students are shown.
The Inverted Rotary Pendulum: Facilitating Practical Teaching in Advanced Control Engineering
(2024)
This paper outlines a practical approach to teach control engineering principles, with an inverted rotary pendulum, serving as an illustrative example. It shows how the pendulum is embedded in an advanced course of control engineering. This approach is incorporated into a flipped-classroom concept, as well as classical teaching concepts, offering students practical experience in control engineering. In addition, the design of the pendulum is shown, using a Raspberry Pi as the target platform for Matlab Simulink. This pendulum can be used in the classroom to evaluate the controller design mentioned above. It is analysed if the use of the pendulum generates a deeper understanding of the learning contents.
This paper serves as an introduction to the ECTS monitoring system and its potential applications in higher education. It also emphasizes the potential for ECTS monitoring to become a proactive system, supporting students by predicting academic success and identifying groups of potential dropouts for tailored support services. The use of the nearest neighbor analysis is suggested for improving data analysis and prediction accuracy.
4CH TX/RX Surface Coil for 7T: Design, Optimization and Application for Cardiac Function Imaging
(2010)
Practical impediments of ultra high field cardiovascular MR (CVMR) can be catalogued in exacerbated magnetic field and radio frequency (RF) inhomogeneities, susceptibility and off-resonance effects, conductive and dielectric effects in tissue, and RF power deposition constraints, which all bear the potential to spoil the benefit of CVMR at 7T. Therefore, a four element cardiac transceive surface coil array was developed. Cardiac imaging provided clinically acceptable signal homogeneity with an excellent blood myocardium contrast. Subtle anatomic structures, such as pericardium, mitral and tricuspid valves and their apparatus, papillary muscles, and trabecles were accurately delineated.
As high-field cardiac MRI (CMR) becomes more widespread the propensity of ECG to distortions and mistriggering increases and with it the motivation for a cardiac triggering alternative. Hence, this study explores the suitability of acoustic cardiac triggering (ACT) for left ventricular (LV) function assessment in healthy subjects at 1.5T and 3.0T.
High Spatial Resolution 3D MRI of the Larynx Using a Dedicated TX/RX Phased Array Coil at 7.0T
(2010)
MRI holds great potential for elucidating laryngeal and vocal fold anatomy together with the assessment of physiological processes associated in human phonation. However, MRI of human phonation remains very challenging due to the small size of the targeted structures, interfering signal from fat, air between the vocal folds and surrounding muscles and physiological motion. These anatomical/physiological constraints translate into stringent technical requirements in balancing, scan time, image contrast, immunity to physiological motion, temporal resolution and spatial resolution. Motivated by these challenges and limitations this study is aiming at translating the sensitivity gain at ultra-high magnetic fields for enhanced high spatial resolution 3D imaging of the larynx and vocal tract. To approach this goal a dedicated two channel TX/RX larynx coil is being proposed.