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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.
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.
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.
In this work, we present a compact, bifunctional chip-based sensor setup that measures the temperature and electrical conductivity of water samples, including specimens from rivers and channels, aquaculture, and the Atlantic Ocean. For conductivity measurements, we utilize the impedance amplitude recorded via interdigitated electrode structures at a single triggering frequency. The results are well in line with data obtained using a calibrated reference instrument. The new setup holds for conductivity values spanning almost two orders of magnitude (river versus ocean water) without the need for equivalent circuit modelling. Temperature measurements were performed in four-point geometry with an on-chip platinum RTD (resistance temperature detector) in the temperature range between 2 °C and 40 °C, showing no hysteresis effects between warming and cooling cycles. Although the meander was not shielded against the liquid, the temperature calibration provided equivalent results to low conductive Milli-Q and highly conductive ocean water. The sensor is therefore suitable for inline and online monitoring purposes in recirculating aquaculture systems.
Magnetic Resonance Imaging (MRI) of moving organs requires synchronization with physiological motion or flow, which dictate the viable window for data acquisition. To meet this challenge, this study proposes an acoustic gating device (ACG) that employs acquisition and processing of acoustic signals for synchronization while providing MRI compatibility, immunity to interferences with electro-magnetic and acoustic fields and suitability for MRI at high magnetic field strengths. The applicability and robustness of the acoustic gating approach is examined in a pilot study, where it substitutes conventional ECG-gating for cardiovascular MR. The merits and limitations of the ACG approach are discussed. Implications for MR imaging in the presence of physiological motion are considered including synchronization with other structure- or motion borne sounds.
This study demonstrates the feasibility of applying free-breathing, cardiac-gated, susceptibility-weighted fast spin-echo imaging together with black blood preparation and navigator-gated respiratory motion compensation for anatomically accurate T₂ mapping of the heart. First, T₂ maps are presented for oil phantoms without and with respiratory motion emulation (T₂ = (22.1 ± 1.7) ms at 1.5 T and T₂ = (22.65 ± 0.89) ms at 3.0 T). T₂ relaxometry of a ferrofluid revealed relaxivities of R2 = (477.9 ± 17) mM⁻¹s⁻¹ and R2 = (449.6 ± 13) mM⁻¹s⁻¹ for UFLARE and multiecho gradient-echo imaging at 1.5 T. For inferoseptal myocardial regions mean T₂ values of 29.9 ± 6.6 ms (1.5 T) and 22.3 ± 4.8 ms (3.0 T) were estimated. For posterior myocardial areas close to the vena cava T₂-values of 24.0 ± 6.4 ms (1.5 T) and 15.4 ± 1.8 ms (3.0 T) were observed. The merits and limitations of the proposed approach are discussed and its implications for cardiac and vascular T₂-mapping are considered.
The simultaneous assessment of glottal dynamics and larynx position can be beneficial for the diagnosis of disordered voice or speech production and swallowing. Up to now, methods either concentrate on assessment of the glottis opening using optical, acoustical or electrical (electroglottography, EGG) methods, or on visualisation of the larynx position using ultrasound, computer tomography or magnetic resonance imaging techniques.
The method presented here makes use of a time-multiplex measurement approach of space-resolved transfer impedances through the larynx. The fast sequence of measurements allows a quasi simultaneous assessment of both larynx position and EGG signal using up to 32 transmit–receive signal paths. The system assesses the dynamic opening status of the glottis as well as the vertical and back/forward motion of the larynx.
Two electrode-arrays are used for the measurement of the electrical transfer impedance through the neck in different directions. From the acquired data the global and individual conductivity is calculated as well as a 2D point spatial representation of the minimum impedance.
The position information is shown together with classical EGG signals allowing a synchronous visual assessment of glottal area and larynx position. A first application to singing voice analysis is presented that indicate a high potential of the method for use as a non-invasive tool in the diagnosis of voice, speech, and swallowing disorders.
Objectives
Interest in cardiovascular magnetic resonance (CMR) at 7 T is motivated by the expected increase in spatial and temporal resolution, but the method is technically challenging. We examined the feasibility of cardiac chamber quantification at 7 T.
Methods
A stack of short axes covering the left ventricle was obtained in nine healthy male volunteers. At 1.5 T, steady-state free precession (SSFP) and fast gradient echo (FGRE) cine imaging with 7 mm slice thickness (STH) were used. At 7 T, FGRE with 7 mm and 4 mm STH were applied. End-diastolic volume, end-systolic volume, ejection fraction and mass were calculated.
Results
All 7 T examinations provided excellent blood/myocardium contrast for all slice directions. No significant difference was found regarding ejection fraction and cardiac volumes between SSFP at 1.5 T and FGRE at 7 T, while volumes obtained from FGRE at 1.5 T were underestimated. Cardiac mass derived from FGRE at 1.5 and 7 T was larger than obtained from SSFP at 1.5 T. Agreement of volumes and mass between SSFP at 1.5 T and FGRE improved for FGRE at 7 T when combined with an STH reduction to 4 mm.
Conclusions
This pilot study demonstrates that cardiac chamber quantification at 7 T using FGRE is feasible and agrees closely with SSFP at 1.5 T.
Background
To demonstrate the applicability of acoustic cardiac triggering (ACT) for imaging of the heart at ultrahigh magnetic fields (7.0 T) by comparing phonocardiogram, conventional vector electrocardiogram (ECG) and traditional pulse oximetry (POX) triggered 2D CINE acquisitions together with (i) a qualitative image quality analysis, (ii) an assessment of the left ventricular function parameter and (iii) an examination of trigger reliability and trigger detection variance derived from the signal waveforms.
Results
ECG was susceptible to severe distortions at 7.0 T. POX and ACT provided waveforms free of interferences from electromagnetic fields or from magneto-hydrodynamic effects. Frequent R-wave mis-registration occurred in ECG-triggered acquisitions with a failure rate of up to 30% resulting in cardiac motion induced artifacts. ACT and POX triggering produced images free of cardiac motion artefacts. ECG showed a severe jitter in the R-wave detection. POX also showed a trigger jitter of approximately Δt = 72 ms which is equivalent to two cardiac phases. ACT showed a jitter of approximately Δt = 5 ms only. ECG waveforms revealed a standard deviation for the cardiac trigger offset larger than that observed for ACT or POX waveforms.
Image quality assessment showed that ACT substantially improved image quality as compared to ECG (image quality score at end-diastole: ECG = 1.7 ± 0.5, ACT = 2.4 ± 0.5, p = 0.04) while the comparison between ECG vs. POX gated acquisitions showed no significant differences in image quality (image quality score: ECG = 1.7 ± 0.5, POX = 2.0 ± 0.5, p = 0.34).
Conclusions
The applicability of acoustic triggering for cardiac CINE imaging at 7.0 T was demonstrated. ACT's trigger reliability and fidelity are superior to that of ECG and POX. ACT promises to be beneficial for cardiovascular magnetic resonance at ultra-high field strengths including 7.0 T.
Purpose
To design and evaluate a four-channel cardiac transceiver coil array for functional cardiac imaging at 7T.
Materials and Methods
A four-element cardiac transceiver surface coil array was developed with two rectangular loops mounted on an anterior former and two rectangular loops on a posterior former. specific absorption rate (SAR) simulations were performed and a Burn:x-wiley:10531807:media:JMRI22451:tex2gif-stack-1 calibration method was applied prior to obtain 2D FLASH CINE (mSENSE, R = 2) images from nine healthy volunteers with a spatial resolution of up to 1 × 1 × 2.5 mm3.
Results
Tuning and matching was found to be better than 10 dB for all subjects. The decoupling (S21) was measured to be >18 dB between neighboring loops, >20 dB for opposite loops, and >30 dB for other loop combinations. SAR values were well within the limits provided by the IEC. Imaging provided clinically acceptable signal homogeneity with an excellent blood-myocardium contrast applying the Burn:x-wiley:10531807:media:JMRI22451:tex2gif-stack-2 calibration approach.
Conclusion
A four-channel cardiac transceiver coil array for 7T was built, allowing for cardiac imaging with clinically acceptable signal homogeneity and an excellent blood-myocardium contrast. Minor anatomic structures, such as pericardium, mitral, and tricuspid valves and their apparatus, as well as trabeculae, were accurately delineated.
Objective
The purpose of this study is to (i) design a small and mobile Magnetic field ALert SEnsor (MALSE), (ii) to carefully evaluate its sensors to their consistency of activation/deactivation and sensitivity to magnetic fields, and (iii) to demonstrate the applicability of MALSE in 1.5 T, 3.0 T and 7.0 T MR fringe field environments.
Methods
MALSE comprises a set of reed sensors, which activate in response to their exposure to a magnetic field. The activation/deactivation of reed sensors was examined by moving them in/out of the fringe field generated by 7TMR.
Results
The consistency with which individual reed sensors would activate at the same field strength was found to be 100% for the setup used. All of the reed switches investigated required a substantial drop in ambient magnetic field strength before they deactivated.
Conclusions
MALSE is a simple concept for alerting MRI staff to a ferromagnetic object being brought into fringe magnetic fields which exceeds MALSEs activation magnetic field. MALSE can easily be attached to ferromagnetic objects within the vicinity of a scanner, thus creating a barrier for hazardous situations induced by ferromagnetic parts which should not enter the vicinity of an MR-system to occur.
Spontaneous language has rarely been subjected to neuroimaging studies. This study therefore introduces a newly developed method for the analysis of linguistic phenomena observed in continuous language production during fMRI.
Most neuroimaging studies investigating language have so far focussed on single word or — to a smaller extent — sentence processing, mostly due to methodological considerations. Natural language production, however, is far more than the mere combination of words to larger units. Therefore, the present study aimed at relating brain activation to linguistic phenomena like word-finding difficulties or syntactic completeness in a continuous language fMRI paradigm. A picture description task with special constraints was used to provoke hesitation phenomena and speech errors. The transcribed speech sample was segmented into events of one second and each event was assigned to one category of a complex schema especially developed for this purpose. The main results were: conceptual planning engages bilateral activation of the precuneus. Successful lexical retrieval is accompanied – particularly in comparison to unsolved word-finding difficulties – by the left middle and superior temporal gyrus. Syntactic completeness is reflected in activation of the left inferior frontal gyrus (IFG) (area 44). In sum, the method has proven to be useful for investigating the neural correlates of lexical and syntactic phenomena in an overt picture description task. This opens up new prospects for the analysis of spontaneous language production during fMRI.
Purpose:
To investigate the feasibility of using magnetohydrodynamic (MHD) effects for synchronization of magnetic resonance imaging (MRI) with the cardiac cycle.
Materials and Methods:
The MHD effect was scrutinized using a pulsatile flow phantom at B0 = 7.0 T. MHD effects were examined in vivo in healthy volunteers (n = 10) for B0 ranging from 0.05–7.0 T. Noncontrast-enhanced MR angiography (MRA) of the carotids was performed using a gated steady-state free-precession (SSFP) imaging technique in conjunction with electrocardiogram (ECG) and MHD synchronization.
Results:
The MHD potential correlates with flow velocities derived from phase contrast MRI. MHD voltages depend on the orientation between B0 and the flow of a conductive fluid. An increase in the interelectrode spacing along the flow increases the MHD potential. In vivo measurement of the MHD effect provides peak voltages of 1.5 mV for surface areas close to the common carotid artery at B0 = 7.0 T. Synchronization of MRI with the cardiac cycle using MHD triggering is feasible. MHD triggered MRA of the carotids at 3.0 T showed an overall image quality and richness of anatomic detail, which is comparable to ECG-triggered MRAs.
Conclusion:
This feasibility study demonstrates the use of MHD effects for synchronization of MR acquisitions with the cardiac cycle. J. Magn. Reson. Imaging 2012;36:364–372. © 2012 Wiley Periodicals, Inc.