@article{HellmuthKlingenberg2022,
  author    = {Hellmuth, Kathrin and Klingenberg, Christian},
  title     = {Computing Black Scholes with uncertain volatility — a machine learning approach},
  series = {Mathematics},
  volume    = {10},
  journal   = {Mathematics},
  number    = {3},
  issn      = {2227-7390},
  doi       = {10.3390/math10030489},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-262280},
  year      = {2022},
  abstract  = {In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.},
  language  = {en}
}
@article{HellmuthKlingenbergLietal.2021,
  author    = {Hellmuth, Kathrin and Klingenberg, Christian and Li, Qin and Tang, Min},
  title     = {Multiscale convergence of the inverse problem for chemotaxis in the Bayesian setting},
  series = {Computation},
  volume    = {9},
  journal   = {Computation},
  number    = {11},
  issn      = {2079-3197},
  doi       = {10.3390/computation9110119},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-250216},
  year      = {2021},
  abstract  = {Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller-Segel equation and a chemotaxis kinetic equation. These two equations describe the organism's movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller-Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms' population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions.},
  language  = {en}
}
@article{FeireislKlingenbergMarkfelder2020,
  author    = {Feireisl, Eduard and Klingenberg, Christian and Markfelder, Simon},
  title     = {On the density of "wild" initial data for the compressible Euler system},
  series = {Calculus of Variations and Partial Differential Equations},
  volume    = {59},
  journal   = {Calculus of Variations and Partial Differential Equations},
  issn      = {0944-2669},
  doi       = {10.1007/s00526-020-01806-5},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232607},
  year      = {2020},
  abstract  = {We consider a class of "wild" initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural L1-topology and show that its complement is rather large, specifically it is an open dense set.},
  language  = {en}
}
@article{LuKlingenbergRendonetal.2014,
  author    = {Lu, Yun-guang and Klingenberg, Christian and Rendon, Leonardo and Zheng, De-Yin},
  title     = {Global Solutions for a Simplified Shallow Elastic Fluids Model},
  series = {Abstract and Applied Analytics},
  journal   = {Abstract and Applied Analytics},
  number    = {920248},
  issn      = {1687-0409},
  doi       = {10.1155/2014/920248},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-117978},
  year      = {2014},
  abstract  = {The Cauchy problem for a simplified shallow elastic fluids model, one 3 x 3 system of Temple's type, is studied and a global weak solution is obtained by using the compensated compactness theorem coupled with the total variation estimates on the first and third Riemann invariants, where the second Riemann invariant is singular near the zero layer depth (rho - 0). This work extends in some sense the previous works, (Serre, 1987) and (Leveque and Temple, 1985), which provided the global existence of weak solutions for 2 x 2 strictly hyperbolic system and (Heibig, 1994) for n x n strictly hyperbolic system with smooth Riemann invariants.},
  language  = {en}
}