This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family of smooth solutions. Conditions on the nonlinear terms, relating to the order of the derivatives of the delta function part, are established so that the regularized solutions converge to a limiting distribution.

In this paper, a theory is developed of generalized oscillatory integrals (OIs) whose phase functions and amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need for a general framework for partial differential operators with non-smooth coefficients and distribution dataffi The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wavefront sets.

Simulation tools for gravitational mass flows (e.g., avalanches, debris flows) are commonly used for research and applications in hazard assessment or mitigation planning. As a basis for a transparent and reproducible decision making process, associated uncertainties need to be identified in order to quantify and eventually communicate the associated variabilities of the results. Main sources of variabilities in the simulation results are associated with parameter variations arising from observation and model uncertainties. These are connected to the measurement inaccuracies or poor process understanding and the numerical model implementation. Probabilistic approaches provide various theoretical concepts to treat these uncertainties, but their direct application is not straightforward. To provide a comprehensive tool, introducing conditional runout probabilities for the decision making process we (i) introduce a mathematical framework based on well-established Bayesian concepts, (ii) develop a work flow that couples this framework to the existing simulation tool r.avaflow, and (iii) apply the work flow to two case studies, highlighting its application potential and limitations. The presented approach allows for back, forward and predictive calculations. Back calculations are used to determine parameter distributions, identifying and mapping the model, implementation and data uncertainties. These parameter distributions serve as a base for forward and predictive calculations, embedded in the probabilistic framework. The result variability is quantified in terms of conditional probabilities with respect to the observed data and the associated simulation and data uncertainties. To communicate the result variability the conditional probabilities are visualized, allowing to identify areas with large or small result variability. The conditional probabilities are particularly interesting for predictive avalanche simulations at locations with no prior information where visualization explicitly shows the result variabilities based on parameter distributions derived through back calculations from locations with well-documented observations.