08 Fakultät Mathematik und Physik

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Institute: search.filters.UBSInstitut.Institut für Angewandte Analysis und numerische SimulationAuthor: search.filters.author.Barth, Andrea

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Now showing 1 - 4 of 4
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    On some distributional properties of subordinated Gaussian random fields
    (2022) Merkle, Robin; Barth, Andrea
    Motivated by the subordinated Brownian motion, we define a new class of (in general discontinuous) random fields on higher-dimensional parameter domains: the subordinated Gaussian random field. We investigate the pointwise marginal distribution of the constructed random fields, derive a Lévy-Khinchin-type formula and semi-explicit formulas for the covariance function. Further, we study the pointwise stochastic regularity and present various numerical examples.
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    Subordinated Gaussian random fields in elliptic partial differential equations
    (2022) Merkle, Robin; Barth, Andrea
    To model subsurface flow in uncertain heterogeneous or fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient - also called random field - may be used. In case of a one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities [see for example Barth and Stein (Stoch Part Differ Equ Anal Comput 6(2):286-334, 2018)]. In this paper a new subordination approach is employed [see also Barth and Merkle (Subordinated gaussian random fields. ArXiv e-prints, arXiv:2012.06353 [math.PR], 2020)] to generate Lévy-type discontinuous random fields on a two-dimensional spatial parameter domain. Existence and uniqueness of a (pathwise) solution to a general elliptic partial differential equation is proved and an approximation theory for the diffusion coefficient and the corresponding solution provided. Further, numerical examples using a Monte Carlo approach on a Finite Element discretization validate our theoretical results.
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    Multilevel Monte Carlo estimators for elliptic PDEs with Lévy-type diffusion coefficient
    (2022) Merkle, Robin; Barth, Andrea
    General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often taken into account by a randomization of the diffusion coefficient of the elliptic equation which reveals the necessity of the construction of flexible, spatially discontinuous random fields. Subordinated Gaussian random fields are random functions on higher dimensional parameter domains with discontinuous sample paths and great distributional flexibility. In the present work, we consider a random elliptic partial differential equation (PDE) where the discontinuous subordinated Gaussian random fields occur in the diffusion coefficient. Problem specific multilevel Monte Carlo (MLMC) Finite Element methods are constructed to approximate the mean of the solution to the random elliptic PDE. We prove a-priori convergence of a standard MLMC estimator and a modified MLMC-control variate estimator and validate our results in various numerical examples.
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    Integrative medicine during the intensive phase of chemotherapy in pediatric oncology in Germany : a randomized controlled trial with 5-year follow up
    (2022) Seifert, Georg; Blakeslee, Sarah B.; Calaminus, Gabriele; Kandil, Farid I.; Barth, Andrea; Bernig, Toralf; Classen, Carl Friedrich; Corbacioglu, Selim; Föll, Jürgen; Gottschling, Sven; Gruhn, Bernd; vom Hoff-Heise, Claudia; Lode, Holger N.; Martin, David; Nathrath, Michaela; Neunhoeffer, Felix; Pekrun, Arnulf; Wulff, Beate; Zuzak, Tycho; Henze, Günter; Längler, Alfred
    Background. Integrative medicine is used frequently alongside chemotherapy treatment in pediatric oncology, but little is known about the influence on toxicity. This German, multi-center, open-label, randomized controlled trial assessed the effects of complementary treatments on toxicity related to intensive-phase chemotherapy treatment in children aged 1-18 with the primary outcome of the toxicity sum score. Secondary outcomes were chemotherapy-related toxicity, overall and event-free survival after 5 years in study patients. Methods. Intervention and control were given standard chemotherapy according to malignancy & tumor type. The intervention arm was provided with anthroposophic supportive treatment (AST); given as anthroposophic base medication (AMP), as a base medication for all patients and additional on-demand treatment tailored to the intervention malignancy groups. The control was given no AMP. The toxicity sum score (TSS) was assessed using NCI-CTC scales. Results. Data of 288 patients could be analyzed. Analysis did not reveal any statistically significant differences between the AST and the control group for the primary endpoint or the toxicity measures (secondary endpoints). Furthermore, groups did not differ significantly in the five-year overall and event-free survival follow up. Discussion. In this trial findings showed that AST was able to be safely administered in a clinical setting, although no beneficial effects of AST between group toxicity scores, overall or event-free survival were shown.