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Abstract. We consider one-dimensional hyperbolic PDEs, linear and nonlinear, with random initial data. Our focus is the pointwise statistics, i.e., the probability measure of the solution at any fixed point in space and time. For linear hyperbolic equations, the probability density function (PDF) of these statistics satisfies the same linear PDE. For nonlinear hyperbolic PDEs, we derive a linear transport equation for the cumulative distribution function (CDF) and a nonlocal linear PDE for the PDF. Both results are valid only as long as no shocks have formed, a limitation which is inherent to the problem, as demonstrated by a counterexample. For systems of linear hyperbolic equations, we introduce the multi-point statistics and derive their evolution equations. In all of the settings we consider, the resulting PDEs for the statistics are of practical significance: they enable efficient evaluation of the random dynamics, without requiring an ensemble of solutions of the underlying PDE, and their cost is not affected by the dimension of the random parameter space. Additionally, the evolution equations for the statistics lead to a priori statistical error bounds for Monte Carlo methods (in particular, kernel density estimators) when applied to hyperbolic PDEs with random data. 

Abstract. For nanomaterials, the concept of heat conductivity needs to be revisited. At this fine scale, heat is propagated using the phonon transport equation, an ab initio model derived from first principles. In this problem, a material’s thermal property is coded in a coefficient termed the relaxation time [Formula: see text], depending on the phonon frequency. We propose a novel inverse problem in this paper that infers the relaxation time using a material’s temperature response upon heat injection. This inverse problem is formulated in a PDE-constrained optimization and is numerically solved by the stochastic gradient descent method (SGD) and its variants. In the execution of SGD, the Fréchet derivative is computed and Lipschitz continuity is proved. This approach, in comparison to the earlier studies, enhances the ab initio concept of heat conduction in nanostructures associated to nanomaterials. 

Abstract CRISPR-Cas9 has established itself as a robust tool for conducting loss-of-function gene research in emerging model species, including the freshwater zooplankton Daphnia. However, sensitive detection of mutations, especially in genetic mosaic and pooled samples, remains a challenge. In this study, we evaluate 2 of the most widely used mutation screening techniques, the T7 Endonuclease I assay and fragment analysis for their sensitivity, accuracy, and practical use in detecting CRISPR-induced indels in 4 targeted genes, DNMT3A, DNMT3B, PERIOD2, and DMRT1 in Daphnia magna. Here, we show that T7 Endonuclease I, although it offers a quick and cost-effective screening method, often produces false positives, especially when examining pooled samples. Conversely, fragment analysis facilitates detecting allele size differences at a fine resolution, reproducibility in detecting indels, and distinguishing zygosity, and is more reliable as a method to detect mutation. Our comparative analyses convey the importance of carefully selecting the appropriate screening methods depending on research questions.