Junxian Wang

Hyea Hyun Kim, Eric Chung, Junxian Wang

BDDC and FETI-DP algorithms are developed for three-dimensional elliptic problems with adaptively enriched coarse components. It is known that these enriched components are necessary in the development of robust preconditioners. To form the adaptive coarse components, carefully designed generalized eigenvalue problems are introduced for each faces and edges, and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance $λ_{TOL}$. Upper bounds for condition numbers of the preconditioned systems are shown to be $C λ_{TOL}$, with the constant $C$ depending only on the maximum number of edges and faces per subdomain, and the maximum number of subdomains sharing an edge. Numerical results are presented to test the robustness of the proposed approach.

Jie Peng, Junxian Wang, Shi Shu

Balancing domain decomposition by constraints (BDDC) algorithms with adaptive primal constraints are developed in a concise variational framework for the weighted plane wave least-squares (PWLS) discritization of Helmholtz equations with high and various wave numbers. The unknowns to be solved in this preconditioned system are defined on elements rather than vertices or edges, which are different from the well-known discritizations such as the classical finite element method. Through choosing suitable "interface" and appropriate primal constraints with complex coefficients and introducing some local techniques, we developed a two-level adaptive BDDC algorithm for the PWLS discretization, and the condition number of the preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is only dependent on the maximum number of interfaces per subdomain. A multilevel algorithm is also attempted to resolve the bottleneck in large scale coarse problem. Numerical results are carried out to confirm the theoretical results and illustrate the efficiency of the proposed algorithms.

Jie Peng, Shi Shu, Junxian Wang

A balancing domain decomposition by constraints (BDDC) algorithm with adaptive primal constraints in variational form is introduced and analyzed for high-order mortar discretization of two-dimensional elliptic problems with high varying and random coefficients. Some vector-valued auxiliary spaces and operators with essential properties are defined to describe the variational algorithm, and the coarse space is formed by using a transformation operator on each interface. Compared with the adaptive BDDC algorithms for conforming Galerkin approximations, our algorithm is more simple, because there is not any continuity constraints at subdomain vertices in the mortar method involved in this paper. The condition number of the preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of interfaces per subdomain, and independent of the mesh size and the contrast of the given coefficients. Numerical results show the robustness and efficiency of the algorithm for various model problems.

Junxian Wang, Wesley P. Chan, Pamela Carreno-Medrano et al.

Recent protocols and metrics for training and evaluating autonomous robot navigation through crowds are inconsistent due to diversified definitions of "social behavior". This makes it difficult, if not impossible, to effectively compare published navigation algorithms. Furthermore, with the lack of a good evaluation protocol, resulting algorithms may fail to generalize, due to lack of diversity in training. To address these gaps, this paper facilitates a more comprehensive evaluation and objective comparison of crowd navigation algorithms by proposing a consistent set of metrics that accounts for both efficiency and social conformity, and a systematic protocol comprising multiple crowd navigation scenarios of varying complexity for evaluation. We tested four state-of-the-art algorithms under this protocol. Results revealed that some state-of-the-art algorithms have much challenge in generalizing, and using our protocol for training, we were able to improve the algorithm's performance. We demonstrate that the set of proposed metrics provides more insight and effectively differentiates the performance of these algorithms with respect to efficiency and social conformity.