Felix Lindner

Matteo Beccari, Martin Hutzenthaler, Arnulf Jentzen et al.

The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing nonlinearities. It remained an open question whether such a divergence phenomenon also holds in the case of stochastic partial differential equations with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. In this work we solve this problem by proving that full-discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations. This article also contains a short literature overview on existing numerical approximation results for stochastic differential equations with superlinearly growing nonlinearities.

Petru A. Cioica, Stephan Dahlke, Stefan Kinzel et al.

We use the scale of Besov spaces B^α_{τ,τ}(O), α>0, 1/τ=α/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

Mihály Kovács, Felix Lindner, René L. Schilling

We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive Lévy noise. We first derive a representation formula for the error which we then apply to study space-time discretizations of the stochastic heat equation, a Volterra-type integro-differential equation, and the wave equation as examples. For twice continuously differentiable test functions with bounded second derivative (with an additional condition on the second derivative for the wave equation) the weak rate of convergence is found to be twice the strong rate. The results extend earlier work by two of the authors as we consider general square-integrable infinite-dimensional Lévy processes and do not require boundedness of the test functions and their first derivative. Furthermore, the present framework is applicable to both hyperbolic and parabolic equations, and even to stochastic Volterra integro-differential equations.

Petru A. Cioica, Kyeong-Hun Kim, Kijung Lee et al.

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $O \subset R^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces $\mathfrak{H}^{γ,q}_{p,θ}(O;T)$. The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the Hölder regularity in time is analysed. Moreover, we prove a general embedding of weighted Lp(O)-Sobolev spaces into the scale of Besov spaces $B^α_{τ,τ}(O), 1/τ=α/d+1/p, α> 0$. This leads to a Hölder-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.

Benjamin Krarup, Felix Lindner, Senka Krivic et al.

The continued development of robots has enabled their wider usage in human surroundings. Robots are more trusted to make increasingly important decisions with potentially critical outcomes. Therefore, it is essential to consider the ethical principles under which robots operate. In this paper we examine how contrastive and non-contrastive explanations can be used in understanding the ethics of robot action plans. We build upon an existing ethical framework to allow users to make suggestions about plans and receive automatically generated contrastive explanations. Results of a user study indicate that the generated explanations help humans to understand the ethical principles that underlie a robot's plan.

Jakob Karalus, Felix Lindner

The capability to interactively learn from human feedback would enable agents in new settings. For example, even novice users could train service robots in new tasks naturally and interactively. Human-in-the-loop Reinforcement Learning (HRL) combines human feedback and Reinforcement Learning (RL) techniques. State-of-the-art interactive learning techniques suffer from slow learning speed, thus leading to a frustrating experience for the human. We approach this problem by extending the HRL framework TAMER for evaluative feedback with the possibility to enhance human feedback with two different types of counterfactual explanations (action and state based). We experimentally show that our extensions improve the speed of learning.

Benjamin Krarup, Senka Krivic, Felix Lindner et al.

The development of robotics and AI agents has enabled their wider usage in human surroundings. AI agents are more trusted to make increasingly important decisions with potentially critical outcomes. It is essential to consider the ethical consequences of the decisions made by these systems. In this paper, we present how contrastive explanations can be used for comparing the ethics of plans. We build upon an existing ethical framework to allow users to make suggestions to plans and receive contrastive explanations.

Felix Lindner, Martin Mose Bentzen

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.

Felix Lindner, René L. Schilling

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^{-α})<\infty for some α>0 and A^βQ is bounded for some β\in(α-1,α]. A discretization (X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space (parameter h>0) and a θ-method in time (parameter Δt=T/N). For ϕ\in C^2_b(H;R) we show an integral representation for the error |Eϕ(X^N_h)-Eϕ(X_T)| and prove that |Eϕ(X^N_h)-Eϕ(X_T)|=O(h^{2γ}+(Δt)^γ) where γ<1-α+β.