A5089341038- Stochastic processes and financial applications
- Numerical methods in inverse problems
- Advanced Harmonic Analysis Research
- Advanced Mathematical Modeling in Engineering
- Advanced Numerical Methods in Computational Mathematics
- Nonlinear Partial Differential Equations
- Differential Equations and Boundary Problems
- Advanced Mathematical Physics Problems
- Image and Signal Denoising Methods
- Stability and Controllability of Differential Equations
- Navier-Stokes equation solutions
- Mathematical Approximation and Integration
- Evolutionary Game Theory and Cooperation
- Evolution and Genetic Dynamics
- Philosophy and History of Science
- advanced mathematical theories
- Advanced Banach Space Theory
- Model Reduction and Neural Networks
University of Kassel
2022
University of Duisburg-Essen
2019-2020
University of Otago
2016-2019
Philipps University of Marburg
2010-2016
We study the spatial regularity of semilinear parabolic stochastic partial differential equations on bounded Lipschitz domains 𝒪⊆ ℝ d in scale , 1/τ=α/d+1/p, p≥2 fixed. The Besov smoothness this determines order convergence that can be achieved by adaptive numerical algorithms and other nonlinear approximation schemes. proofs are performed establishing weighted Sobolev estimates combining them with wavelet characterizations spaces.
We use the scale of Besov spaces $B^\alpha_{\tau,\tau}(\mathcal{O})$, $1/\tau=\alpha/d+1/p$, $\alpha>0$, $p$ fixed, to study spatial regularity solutions linear parabolic stochastic partial differential equations on bounded Lipschitz domains $\
We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $\mathcal{O}\subset \mathbb{R}^d$ both theoretical and numerical purpose. use N.V. Krylov’s framework weighted Sobolev spaces $\mathfrak{H}^{\gamma,q}_{p,\theta} (\mathcal{O},T)$ The summability parameters $p$ $q$ in space time may differ. Existence uniqueness solutions these is established Hölder analysed. Moreover, we prove a general embedding...
We use the scale , to study regularity of stationary Stokes equation on bounded Lipschitz domains with connected boundary. The in these Besov spaces determines order convergence nonlinear approximation schemes. Our proofs rely a combination weighted Sobolev estimates and wavelet characterizations spaces. Using Banach’s fixed point theorem, we extend this analysis Navier–Stokes suitable Reynolds number data, respectively.
We introduce and analyse a class of weighted Sobolev spaces with mixed weights on angular domains. The are based both the distance to boundary one vertex domain. Moreover, we show how regularity Poisson equation can be analysed in framework these by means Mellin transform, provided integrability parameter equals two. Our main motivation comes from study stochastic partial differential equations associated degenerate deterministic parabolic equations.
We prove that deep neural networks are capable of approximating solutions semilinear Kolmogorov PDE in the case gradient-independent, Lipschitz-continuous nonlinearities, while required number parameters grow at most polynomially both dimension $d \in \mathbb{N}$ and prescribed reciprocal accuracy $\varepsilon$. Previously, this has only been proven heat equations.
We develop a stochastic integration theory for processes with values in quasi-Banach space. The integrator is cylindrical Brownian motion. main results give sufficient conditions integrability. They are natural extensions of known r
In this paper we develop a stochastic integration theory for processes with values in quasi-Banach space. The integrator is cylindrical Brownian motion. main results give sufficient conditions integrability. They are natural extensions of known the Banach space setting. We apply our to heat equation where forcing terms assumed have Besov regularity variable integrability exponent $p\in (0,1]$. latter consider its potential application adaptive wavelet methods partial differential equations.
We establish a refined $L_p$-estimate ($p\geq 2$) for the stochastic heat equation on angular domains in $\mathbb{R}^2$ with mixed weights based both, distance to boundary and vertex. This way we can capture both causes singularities of solution: incompatibility noise condition one hand influence (here, vertex) other hand. Higher order $L_p$-Sobolev regularity is also established.
We use the scale $B^s_{\tau}(L_\tau(\Omega))$, $1/\tau=s/d+1/2$, $s>0$, to study regularity of stationary Stokes equation on bounded Lipschitz domains $\Omega\subset\mathbb{R}^d$, $d\geq 3$, with connected boundary. The in these Besov spaces determines order convergence nonlinear approximation schemes. Our proofs rely a combination weighted Sobolev estimates and wavelet characterizations spaces. By using Banach's fixed point theorem, we extend this analysis Navier-Stokes suitable Reynolds...
We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study spatial regularity solutions linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The smoothness determines order convergence that can be achieved by nonlinear approximation schemes. proofs are based a combination weighted Sobolev estimates and characterizations wavelet expansions.
We define a set inner product to be function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue Cauchy-Schwartz inequality (which not implied by other conditions). show that any can embedded into an space associated support functions, thereby extending fundamental results Hormander Radstrom. The provides geometry bodies. explore some properties geometry, discuss application these ideas reconstruction...