John Driscoll

John Driscoll, Sigal Gottlieb, Zachary J. Grant et al.

A mixed accuracy framework for Runge--Kutta methods presented in [Grant, JSC 2022] has been shown to speed up the computation in diagonally implicit Runge--Kutta (DIRK) methods by using less expensive low accuracy approaches for the implicit stages. This theory included both smooth and nonsmooth perturbations, and subsequent work focused primarily on the case of nonsmooth perturbations that arise from mixed precision simulations. In this work the focus is on smooth perturbations that arise from using less accurate models or under-resolved iterative solvers to simplify the implicit computations. We develop an accuracy and stability analysis based on the framework in [Grant, JSC 2022] to design methods that strategically replace the original operator by a lower accuracy operator to reduce computational cost while mitigating the effect of the perturbations. In particular, we focus on designing novel methods that are high order for smooth perturbations that satisfy additional local consistency conditions. Finally, we verify the performance of the novel perturbed DIRK methods designed in this work and numerically study the impact of different types of smooth perturbations on the accuracy and stability of the methods.

John Driscoll, Sigal Gottlieb, Zachary J. Grant et al.

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive low accuracy approaches such as lower precision computation of the implicit solve, under-resolved iterative solvers, or simpler, less accurate models for the implicit stages. Understanding the effect of the perturbation errors introduced by the low accuracy computations enables the design of stable and accurate mixed accuracy DIRK methods where the errors from the low-accuracy computation are damped out by multiplication by \dt at multiple points in the simulation, resulting in a more accurate simulation than if low-accuracy was used for all computation. To improve upon this, explicit corrections were previously proposed and analyzed for accuracy, and their performance was tested in related work. Explicit corrections work well when the time-step is sufficiently small, but may introduce instabilities when the time-step is larger. In this work, the stability of the mixed accuracy approach is carefully studied, and used to design novel stabilized correction approaches.

John Driscoll, Yulin Chen, Viki Shi et al.

This paper studies how parents want to moderate children's interactions with Generative AI chatbots, with the goal of informing the design of future GenAI parental control tools. We first used an LLM to generate synthetic child-GenAI chatbot interaction scenarios and worked with four parents to validate their realism. From this dataset, we carefully selected 12 diverse examples that evoked varying levels of concern and were rated the most realistic. Each example included a prompt and a GenAI chatbot response. We presented these to parents (N=24) and asked whether they found them concerning, why, and how they would prefer the responses to be modified and communicated. Our findings reveal three key insights: (1) parents express concern about interactions that current GenAI chatbot parental controls neglect; (2) parents want fine-grained transparency and moderation at the conversation level; and (3) parents need personalized controls that adapt to their desired strategies and children's ages.