A5046617182- Probabilistic and Robust Engineering Design
- Advanced Numerical Methods in Computational Mathematics
- Model Reduction and Neural Networks
- Tensor decomposition and applications
- Advanced Mathematical Modeling in Engineering
- Structural Health Monitoring Techniques
- Topology Optimization in Engineering
- Mathematical Approximation and Integration
- Wind and Air Flow Studies
- Numerical methods in engineering
- Computational Fluid Dynamics and Aerodynamics
- Numerical methods for differential equations
- Advanced Multi-Objective Optimization Algorithms
- Advanced Numerical Analysis Techniques
- Gaussian Processes and Bayesian Inference
- Markov Chains and Monte Carlo Methods
- Differential Equations and Numerical Methods
- Image and Signal Denoising Methods
- Electromagnetic Simulation and Numerical Methods
- Composite Material Mechanics
- Gene Regulatory Network Analysis
- Metaheuristic Optimization Algorithms Research
- Contact Mechanics and Variational Inequalities
- Sparse and Compressive Sensing Techniques
- Fatigue and fracture mechanics
Weierstrass Institute for Applied Analysis and Stochastics
2015-2024
Università della Svizzera italiana
2020
Humboldt-Universität zu Berlin
2011
University of Warwick
2008-2009
German Cancer Research Center
2005
Heidelberg University
2005
We analyze a posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods countably-parametric, elliptic boundary value problems. A residual estimator which separates the effects of gpc-Galerkin discretization in parameter space physical energy norm is established. It proved that algorithm converges. To this end, contraction property its iterates proved. shown sequences triangulations are produced by FE active gpc coefficients asymptotically...
In silico experiments bear the potential for further understanding of biological transport processes by allowing a systematic modification any spatial property and providing immediate simulation results. Cell polarization reorganization membrane proteins are fundamental cell division, chemotaxis morphogenesis. We chose yeast Saccharomyces cerevisiae as an exemplary model system which entails shuttling small Rho GTPases such Cdc42 Rho, between active membrane-bound form inactive cytosolic...
.An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It shown that the "curse of dimensionality" can be alleviated computation Bermudan prices with Monte Carlo least-squares approach as well dual martingale method, both using high-dimensional tensorized polynomial expansions. Complexity estimates are provided a description optimization procedures in tensor train format. Numerical experiments illustrate favorable accuracy proposed...
Understanding the precise interaction mechanisms between quantum systems and their environment is crucial for advancing stable technologies, designing reliable experimental frameworks, building accurate models of real-world phenomena. However, simulating open systems, which feature complex non-unitary dynamics, poses significant computational challenges that require innovative methods to overcome. In this work, we introduce tensor jump method (TJM), a scalable, embarrassingly parallel...
The focus of this work is the introduction some computable a posteriori error control to popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications where quantity interest should be estimated accurately. Based on spatial discretization by finite element method, goal functional defined which encodes interest. devised goal-oriented estimator enables one determine guaranteed path-wise bounds quantity. An adaptive algorithm proposed employs...
Convergence of an adaptive collocation method for the parametric stationary diffusion equation with finite-dimensional affine coefficient is shown. The algorithm relies on a recently introduced residual-based reliable posteriori error estimator. For convergence proof, strategy used stochastic Galerkin hierarchical estimator transferred to setting. Extensions other variants methods (including now classical approach proposed in [T. Gerstner and M. Griebel, Computing, 71 (2003), pp. 65--87])...
Equilibration error estimators have been shown to commonly lead very accurate guaranteed bounds in the a posteriori control of finite element methods (FEMs) for second order elliptic equations. Here, we revisit equilibrated fluxes higher-order FEMs with nonconstant coefficients and illustrate favorable performance different variants estimator within two deterministic benchmark settings. After introduction respective parametric problem stochastic Galerkin FEM discretization, novel energy norm...
In this paper we investigate and compare different gradient algorithms designed for the domain expression of shape derivative. Our main focus is to examine usefulness kernel reproducing Hilbert spaces PDE-constrained optimization problems. We show that radial kernels provide convenient formulas can be efficiently used in numerical simulations. The gradients associated with depend on a so-called smoothing parameter allows smoothness adjustment during process. Besides, modify movement shape....
Engineers are faced with the challenge of supporting decision making under uncertainty. Engineering decisions often depend on model‐based predictions performance engineering system interest. Input uncertainties models can be categorized into two distinct types: aleatory (random/irreducible) or epistemic (reducible). Polymorphic uncertainty quantification (UQ) used to treat and in a unified framework. The polymorphic UQ framework employs probability theory model variables alternative...
Abstract This paper presents a novel method for the accurate functional approximation of possibly highly concentrated probability densities. It is based on combination several modern techniques such as transport maps and low-rank approximations via nonintrusive tensor train reconstruction. The central idea to carry out computations statistical quantities interest moments convenient representation reference density which numerical methods can be employed. Since from target usually not...
Abstract This paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank solutions high-dimensional parametric random PDEs, which have become an area intensive research in Uncertainty Quantification (UQ). In order to obtain generalized polynomial chaos representation approximate stochastic solution, novel black-box rank-adapted tensor reconstruction procedure is proposed. The performance described approach illustrated with several numerical...
A sampling-free approach to Bayesian inversion with an explicit polynomial representation of the parameter densities is developed, based on affine-parametric a linear forward model. This becomes feasible due complete treatment in function spaces, which requires efficient model reduction technique for numerical computations.
Abstract Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive FE method linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces ensure solvability the variational formulation. The inherently high computational complexity operator made tractable by using hierarchical tensor...
We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology the efficient numerical solution of high-dimensional parametric PDEs. An in-depth theoretical analysis shows that proposed architecture is able to approximate multigrid V-cycles arbitrary precision number weights only depending logarithmically on resolution finest mesh. As consequence, approximation bounds PDEs by networks are independent...
A simulation based method for the numerical solution of PDEs with random coefficients is presented. By Feynman--Kac formula, can be represented as conditional expectation a functional corresponding stochastic differential equation (SDE) driven by independent noise. time discretization SDE set points in domain and subsequent Monte Carlo regression lead to an approximation global PDE. We provide initial error complexity analysis proposed along examples illustrating its behavior.
We consider best approximation problems in a nonlinear subset ℳ of Banach space functions (𝒱,∥•∥). The norm is assumed to be generalization the L 2 -norm for which only weighted Monte Carlo estimate ∥•∥ n can computed. objective obtain an v ∈ unknown function u 𝒱 by minimizing empirical ∥ − . this problem general subsets and establish error bounds error. Our results are based on restricted isometry property (RIP) holds probability independent specified least squares setting. Several model...
We analyze a posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods countably-parametric, elliptic boundary value problems. A resid- ual estimator which separates the effects of gpc-Galerkin discretization in parameter space physical energy norm is established. It proved that algorithm converges. To this end, contraction property its iterates proved. shown sequences triangulations are produced by FE active gpc coefficients asymptotically...