813 research outputs found
Weak order for the discretization of the stochastic heat equation driven by impulsive noise
Considering a linear parabolic stochastic partial differential equation
driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H,
t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an
impulsive cylindrical process and Q describes the spatial covariance structure
of the noise; Tr(A^{-\alpha})0 and A^\beta Q is bounded
for some \beta\in(\alpha-1,\alpha]. A discretization
(X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space
(parameter h>0) and a \theta-method in time (parameter \Delta t=T/N). For
\phi\in C^2_b(H;R) we show an integral representation for the error
|E\phi(X^N_h)-E\phi(X_T)| and prove that
|E\phi(X^N_h)-E\phi(X_T)|=O(h^{2\gamma}+(\Delta t)^{\gamma}) where
\gamma<1-\alpha+\beta.Comment: 29 pages; Section 1 extended, new results in Appendix
On the Alekseev-Gr\"obner formula in Banach spaces
The Alekseev-Gr\"obner formula is a well known tool in numerical analysis for
describing the effect that a perturbation of an ordinary differential equation
(ODE) has on its solution. In this article we provide an extension of the
Alekseev-Gr\"obner formula for Banach space valued ODEs under, loosely
speaking, mild conditions on the perturbation of the considered ODEs.Comment: 36 page
Stochastic fiber dynamics in a spatially semi-discrete setting
We investigate a spatially discrete surrogate model for the dynamics of a
slender, elastic, inextensible fiber in turbulent flows. Deduced from a
continuous space-time beam model for which no solution theory is available, it
consists of a high-dimensional second order stochastic differential equation in
time with a nonlinear algebraic constraint and an associated Lagrange
multiplier term. We establish a suitable framework for the rigorous formulation
and analysis of the semi-discrete model and prove existence and uniqueness of a
global strong solution. The proof is based on an explicit representation of the
Lagrange multiplier and on the observation that the obtained explicit drift
term in the equation satisfies a one-sided linear growth condition on the
constraint manifold. The theoretical analysis is complemented by numerical
studies concerning the time discretization of our model. The performance of
implicit Euler-type methods can be improved when using the explicit
representation of the Lagrange multiplier to compute refined initial estimates
for the Newton method applied in each time step.Comment: 20 pages; typos removed, references adde
A Formalization of Kant's Second Formulation of the Categorical Imperative
We present a formalization and computational implementation of the second
formulation of Kant's categorical imperative. This ethical principle requires
an agent to never treat someone merely as a means but always also as an end.
Here we interpret this principle in terms of how persons are causally affected
by actions. We introduce Kantian causal agency models in which moral patients,
actions, goals, and causal influence are represented, and we show how to
formalize several readings of Kant's categorical imperative that correspond to
Kant's concept of strict and wide duties towards oneself and others. Stricter
versions handle cases where an action directly causally affects oneself or
others, whereas the wide version maximizes the number of persons being treated
as an end. We discuss limitations of our formalization by pointing to one of
Kant's cases that the machinery cannot handle in a satisfying way
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