Search for: All records

This study focuses on addressing the inverse source problem associated with the parabolic equation. We rely on sparse boundary flux data as our measurements, which are acquired from a restricted section of the boundary. While it has been established that utilizing sparse boundary flux data can enable source recovery, the presence of a limited number of observation sensors poses a challenge for accurately tracing the inverse quantity of interest. To overcome this limitation, we introduce a sampling algorithm grounded in Langevin dynamics that incorporates dynamic sensors to capture the flux information. Furthermore, we propose and discuss two distinct dynamic sensor migration strategies. Remarkably, our findings demonstrate that even with only two observation sensors at our disposal, it remains feasible to successfully reconstruct the high-dimensional unknown parameters.

 

Parallel tempering (PT), also known as replica exchange, is the go-to workhorse for simulations of multi-modal distributions. The key to the success of PT is to adopt efficient swap schemes. The popular deterministic even-odd (DEO) scheme exploits the non-reversibility property and has successfully reduced the communication cost from O(P 2) to O(P) given sufficient many P chains. However, such an innovation largely disappears in big data problems due to the limited chains and extremely few bias-corrected swaps. To handle this issue, we generalize the DEO scheme to promote the non-reversibility and obtain an appealing communication cost O(P log P) based on the optimal window size. In addition, we also analyze the bias when we adopt stochastic gradient descent (SGD) with large and constant learning rates as exploration kernels. Such a user-friendly nature enables us to conduct large-scale uncertainty approximation tasks without much tuning costs. 

Lattice thermal conductivity is important for many applications, but experimental measurements or first principles calculations including three-phonon and four-phonon scattering are expensive or even unaffordable. Machine learning approaches that can achieve similar accuracy have been a long-standing open question. Despite recent progress, machine learning models using structural information as descriptors fall short of experimental or first principles accuracy. This study presents a machine learning approach that predicts phonon scattering rates and thermal conductivity with experimental and first principles accuracy. The success of our approach is enabled by mitigating computational challenges associated with the high skewness of phonon scattering rates and their complex contributions to the total thermal resistance. Transfer learning between different orders of phonon scattering can further improve the model performance. Our surrogates offer up to two orders of magnitude acceleration compared to first principles calculations and would enable large-scale thermal transport informatics.

 

Abstract We consider the inverse source problem in the parabolic equation, where the unknown source possesses the semi-discrete formulation. Theoretically, we prove that the flux data from any nonempty open subset of the boundary can uniquely determine the semi-discrete source. This means the observed area can be extremely small, and that is the reason we call it sparse boundary data. For the numerical reconstruction, we formulate the problem from the Bayesian sequential prediction perspective and conduct the numerical examples which estimate the space-time-dependent source state by state. To better demonstrate the method’s performance, we solve two common multiscale problems from two models with a long source sequence. The numerical results illustrate that the inversion is accurate and efficient. 

We introduce a proper notion of two-dimensional signature for images. This object is inspired by the so-called rough paths theory, and it captures many essential features of a two-dimensional object such as an image. It thus serves as a low-dimensional feature for pattern classification. Here, we implement a simple procedure for texture classification. In this context, we show that a low-dimensional set of features based on signatures produces an excellent accuracy.