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Abstract We present high-resolution, three-dimensional hydrodynamic simulations of the fueling of supermassive black holes in elliptical galaxies from a turbulent medium on galactic scales, taking M87* as a typical case. The simulations use a new GPU-accelerated version of the Athena++ AMR code, and they span more than six orders of magnitude in radius, reaching scales similar to that of the black hole horizon. The key physical ingredients are radiative cooling and a phenomenological heating model. We find that the accretion flow takes the form of multiphase gas at radii less than about a kpc. The cold gas accretion includes two dynamically distinct stages: the typical disk stage in which the cold gas resides in a rotationally supported disk, and relatively rare chaotic stages (≲10% of the time) in which the cold gas inflows via chaotic streams. Though cold gas accretion dominates the time-averaged accretion rate at intermediate radii, accretion at the smallest radii is dominated by hot virialized gas at most times. The accretion rate scales with radius as M ̇ ∝ r 1 / 2 when hot gas dominates, and we obtain M ̇ ≃ 10 − 4 – 10 − 3 M ⊙ yr − 1 near the event horizon, similar to what is inferred from EHT observations. The orientation of the cold gas disk can differ significantly on different spatial scales. We propose a subgrid model for accretion in lower-resolution simulations in which the hot gas accretion rate is suppressed relative to the Bondi rate by ∼ ( r g / r Bondi ) 1 / 2 . Our results can also provide more realistic initial conditions for simulations of black hole accretion at the event horizon scale. 

Computation of injective (or inversion-free) maps is a key task in geometry processing, physical simulation, and shape optimization. Despite being a longstanding problem, it remains challenging due to its highly nonconvex and combinatoric nature. We propose computation ofvariational quasi-harmonic mapsto obtain smooth inversion-free maps. Our work is built on a key observation about inversion-free maps: A planar map is a diffeomorphism if and only if it is quasi-harmonic and satisfies a special Cauchy boundary condition. We hence equate the inversion-free mapping problem to an optimal control problem derived from our theoretical result, in which we search in the space of parameters that define an elliptic PDE. We show that this problem can be solved by minimizing within a family of functionals. Similarly, our discretized functionals admit exactly injective maps as the minimizers, empirically producing inversion-free discrete maps of triangle meshes. We design efficient numerical procedures for our problem that prioritize robust convergence paths. Experiments show that on challenging examples our methods can achieve up to orders of magnitude improvement over state-of-the-art, in terms of speed or quality. Moreover, we demonstrate how to optimize a generic energy in our framework while restricting to quasi-harmonic maps. 

Manufactured parts are meticulously engineered to perform well with respect to several conflicting metrics, like weight, stress, and cost. The best achievable trade-offs reside on the Pareto front , which can be discovered via performance-driven optimization. The objectives that define this Pareto front often incorporate assumptions about the context in which a part will be used, including loading conditions, environmental influences, material properties, or regions that must be preserved to interface with a surrounding assembly. Existing multi-objective optimization tools are only equipped to study one context at a time, so engineers must run independent optimizations for each context of interest. However, engineered parts frequently appear in many contexts: wind turbines must perform well in many wind speeds, and a bracket might be optimized several times with its bolt-holes fixed in different locations on each run. In this paper, we formulate a framework for variable-context multi-objective optimization. We introduce the Pareto gamut , which captures Pareto fronts over a range of contexts. We develop a global/local optimization algorithm to discover the Pareto gamut directly, rather than discovering a single fixed-context "slice" at a time. To validate our method, we adapt existing multi-objective optimization benchmarks to contextual scenarios. We also demonstrate the practical utility of Pareto gamut exploration for several engineering design problems.