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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.
Magnetic nanoparticles (MNP) are investigated with great interest for biomedical applications in diagnostics (e.g. imaging: magnetic particle imaging (MPI)), therapeutics (e.g. hyperthermia: magnetic fluid hyperthermia (MFH)) and multi-purpose biosensing (e.g. magnetic immunoassays (MIA)). What all of these applications have in common is that they are based on the unique magnetic relaxation mechanisms of MNP in an alternating magnetic field (AMF). While MFH and MPI are currently the most prominent examples of biomedical applications, here we present results on the relatively new biosensing application of frequency mixing magnetic detection (FMMD) from a simulation perspective. In general, we ask how the key parameters of MNP (core size and magnetic anisotropy) affect the FMMD signal: by varying the core size, we investigate the effect of the magnetic volume per MNP; and by changing the effective magnetic anisotropy, we study the MNPs’ flexibility to leave its preferred magnetization direction. From this, we predict the most effective combination of MNP core size and magnetic anisotropy for maximum signal generation.
Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary
(2024)
In this paper, we consider the inverse shape problem of recovering isotropic scatterers with a conductive boundary condition. Here, we assume that the measured far-field data is known at a fixed wave number. Motivated by recent work, we study a new direct sampling indicator based on the Landweber iteration and the factorization method. Therefore, we prove the connection between these reconstruction methods. The method studied here falls under the category of qualitative reconstruction methods where an imaging function is used to recover the absorbing scatterer. We prove stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.
Analyzing electroencephalographic (EEG) time series can be challenging, especially with deep neural networks, due to the large variability among human subjects and often small datasets. To address these challenges, various strategies, such as self-supervised learning, have been suggested, but they typically rely on extensive empirical datasets. Inspired by recent advances in computer vision, we propose a pretraining task termed "frequency pretraining" to pretrain a neural network for sleep staging by predicting the frequency content of randomly generated synthetic time series. Our experiments demonstrate that our method surpasses fully supervised learning in scenarios with limited data and few subjects, and matches its performance in regimes with many subjects. Furthermore, our results underline the relevance of frequency information for sleep stage scoring, while also demonstrating that deep neural networks utilize information beyond frequencies to enhance sleep staging performance, which is consistent with previous research. We anticipate that our approach will be advantageous across a broad spectrum of applications where EEG data is limited or derived from a small number of subjects, including the domain of brain-computer interfaces.
The artificial olfactory image was proposed by Lundström et al. in 1991 as a new strategy for an electronic nose system which generated a two-dimensional mapping to be interpreted as a fingerprint of the detected gas species. The potential distribution generated by the catalytic metals integrated into a semiconductor field-effect structure was read as a photocurrent signal generated by scanning light pulses. The impact of the proposed technology spread beyond gas sensing, inspiring the development of various imaging modalities based on the light addressing of field-effect structures to obtain spatial maps of pH distribution, ions, molecules, and impedance, and these modalities have been applied in both biological and non-biological systems. These light-addressing technologies have been further developed to realize the position control of a faradaic current on the electrode surface for localized electrochemical reactions and amperometric measurements, as well as the actuation of liquids in microfluidic devices.
Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations.
In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric 2x2 matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.
Frequency mixing magnetic detection (FMMD) is a sensitive and selective technique to detect magnetic nanoparticles (MNPs) serving as probes for binding biological targets. Its principle relies on the nonlinear magnetic relaxation dynamics of a particle ensemble interacting with a dual frequency external magnetic field. In order to increase its sensitivity, lower its limit of detection and overall improve its applicability in biosensing, matching combinations of external field parameters and internal particle properties are being sought to advance FMMD. In this study, we systematically probe the aforementioned interaction with coupled Néel–Brownian dynamic relaxation simulations to examine how key MNP properties as well as applied field parameters affect the frequency mixing signal generation. It is found that the core size of MNPs dominates their nonlinear magnetic response, with the strongest contributions from the largest particles. The drive field amplitude dominates the shape of the field-dependent response, whereas effective anisotropy and hydrodynamic size of the particles only weakly influence the signal generation in FMMD. For tailoring the MNP properties and parameters of the setup towards optimal FMMD signal generation, our findings suggest choosing large particles of core sizes dc > 25 nm nm with narrow size distributions (σ < 0.1) to minimize the required drive field amplitude. This allows potential improvements of FMMD as a stand-alone application, as well as advances in magnetic particle imaging, hyperthermia and magnetic immunoassays.
Methane is a valuable energy source helping to mitigate the growing energy demand worldwide. However, as a potent greenhouse gas, it has also gained additional attention due to its environmental impacts. The biological production of methane is performed primarily hydrogenotrophically from H2 and CO2 by methanogenic archaea. Hydrogenotrophic methanogenesis also represents a great interest with respect to carbon re-cycling and H2 storage. The most significant carbon source, extremely rich in complex organic matter for microbial degradation and biogenic methane production, is coal. Although interest in enhanced microbial coalbed methane production is continuously increasing globally, limited knowledge exists regarding the exact origins of the coalbed methane and the associated microbial communities, including hydrogenotrophic methanogens. Here, we give an overview of hydrogenotrophic methanogens in coal beds and related environments in terms of their energy production mechanisms, unique metabolic pathways, and associated ecological functions.
This paper investigates the interior transmission problem for homogeneous media via eigenvalue trajectories parameterized by the magnitude of the refractive index. In the case that the scatterer is the unit disk, we prove that there is a one-to-one correspondence between complex-valued interior transmission eigenvalue trajectories and Dirichlet eigenvalues of the Laplacian which turn out to be exactly the trajectorial limit points as the refractive index tends to infinity. For general simply-connected scatterers in two or three dimensions, a corresponding relation is still open, but further theoretical results and numerical studies indicate a similar connection.