We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain subadditive quantities derived from a variational interpretation of parabolic equations. These subadditive quantities are intimately connected to spatial averages of the fluxes and gradients of solutions. We implement a renormalization-type scheme to obtain an algebraic rate for their convergence, which is essentially a quantification of the weak convergence of the gradients and fluxes of solutions to their homogenized limits. As a consequence, we obtain estimates of the homogenization error for the Cauchy-Dirichlet problem which are optimal in stochastic integrability. We also develop a higher regularity theory for solutions of the heterogeneous equation, including a uniform C 0,1 -type estimate and a Liouville theorem of every finite order.
Traditional grain size determination in materials characterization involves microscopy images and a laborious process requiring significant manual input and human expertise. In recent years, the development of computer vision (CV) has provided an alternative approach to microstructural characterization with preliminary implementations greatly simplifying the grain size determination process. Here, an end-to-end workflow to measure grain size in microscopy images without any manual input is presented. Following the ASTM standards for grain size determination, results from the line intercept (Heyn’s method) and planimetric (Saltykov’s method) approaches are used as the baseline. A pre-trained holistically nested edge detection (HED) model is used for CV-based edge detection, and the results are further compared to the classic Canny edge detection method. Post-processing was performed using open-source image processing packages to extract the grain size. In optical microscope images, the pre-trained HED model achieves much higher accuracy than the Canny edge detection method while reducing the image processing time by one to two orders of magnitude compared to traditional methods. The effects of morphological operations on the predicted grain size accuracy are also explored. Overall, the proposed end-to-end convolutional neural network (CNN)-based workflow can significantly reduce the processing time while maintaining the same accuracy as the traditional manual method.
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