We study the rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales, where the Hamiltonian
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Rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales
Abstract depends on both of the spatial variable and the oscillatory variable. In particular, we show that for the Cauchy problem, the rate of convergence is by optimal control formulas, scale separations and curve cutting techniques. We also show the rate of homogenization for the static problem based on the same idea. Additionally, we provide examples that illustrate the rate of convergence for the Cauchy problem is optimal for and . -
null (Ed.)Despite the potential of spatial displays for supporting teachers’ classroom orchestration through real-time classroom analytics, the process to design these displays is a challenging and under-explored topic in the learning analytics (LA) community. This paper proposes a mid-fidelity Virtual Prototyping method (VPM), which involves simulating a classroom environment and candidate designs in virtual space to address these challenges. VPM allows for rapid prototyping of spatial features, requires no specialized hardware, and enables teams to conduct remote evaluation sessions. We report observations and findings from an initial exploration with five potential users through a design process utilizing VPM to validate designs for an AR-based spatial display in the context of middle-school orchestration tools. We found that designs created using virtual prototyping sufficiently conveyed a sense of three-dimensionality to address subtle design issues like occlusion and depth perception. We discuss the opportunities and limitations of applying virtual prototyping, particularly its potential to allow for more robust co-design with stakeholders earlier in the design process.more » « less