TY  - JOUR
A1  - Weiss, Andrea Y.
A1  - Huisinga, Wilhelm
T1  - Error-controlled global sensitivity analysis of ordinary differential equations
JF  - Journal of computational physics
N2  - We propose a novel strategy for global sensitivity analysis of ordinary differential equations. It is based on an error-controlled solution of the partial differential equation (PDE) that describes the evolution of the probability density function associated with the input uncertainty/variability. The density yields a more accurate estimate of the output uncertainty/variability, where not only some observables (such as mean and variance) but also structural properties (e.g., skewness, heavy tails, bi-modality) can be resolved up to a selected accuracy. For the adaptive solution of the PDE Cauchy problem we use the Rothe method with multiplicative error correction, which was originally developed for the solution of parabolic PDEs. We show that, unlike in parabolic problems, conservation properties necessitate a coupling of temporal and spatial accuracy to avoid accumulation of spatial approximation errors over time. We provide convergence conditions for the numerical scheme and suggest an implementation using approximate approximations for spatial discretization to efficiently resolve the coupling of temporal and spatial accuracy. The performance of the method is studied by means of low-dimensional case studies. The favorable properties of the spatial discretization technique suggest that this may be the starting point for an error-controlled sensitivity analysis in higher dimensions.
KW  - ODE with random initial conditions
KW  - Global sensitivity analysis
KW  - Cauchy problem
KW  - Error control/adaptivity
KW  - Rothe method
KW  - Approximate approximations
Y1  - 2011
U6  - https://doi.org/10.1016/j.jcp.2011.05.011
SN  - 0021-9991
VL  - 230
IS  - 17
SP  - 6824
EP  - 6842
PB  - Elsevier
CY  - San Diego
ER  - 
TY  - JOUR
A1  - Makhmudo, K. O.
A1  - Makhmudov, O. I.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Equations of Maxwell type
JF  - Journal of mathematical analysis and applications
N2  - For an elliptic complex of first order differential operators on a smooth manifold X, we define a system of two equations which can be thought of as abstract Maxwell equations. The formal theory of this system proves to be very similar to that of classical Maxwell's equations. The paper focuses on boundary value problems for the abstract Maxwell equations, especially on the Cauchy problem.
KW  - Electromagnetic waves
KW  - Scattering
KW  - Elliptic complex
KW  - Green formulas
KW  - Stratton-Chu formulas
KW  - Cauchy problem
Y1  - 2011
U6  - https://doi.org/10.1016/j.jmaa.2011.01.012
SN  - 0022-247X
VL  - 378
IS  - 1
SP  - 64
EP  - 75
PB  - Elsevier
CY  - San Diego
ER  -