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This work presents inGRASS, a novel algorithm designed for incremental spectral sparsification of large undirected graphs. The proposed inGRASS algorithm is highly scalable and parallel-friendly, having a nearly linear time complexity for the setup phase and the ability to update the spectral sparsifier in O(logN) time for each incremental change made to the original graph with N nodes. A key component in the setup phase of inGRASS is a multilevel resistance embedding framework introduced for efficiently identifying spectrally-critical edges and effectively detecting redundant ones, which is achieved by decomposing the initial sparsifier into many node clusters with bounded effective-resistance diameters leveraging a low-resistance-diameter decomposition (LRD) scheme. The update phase of inGRASS exploits low-dimensional node embedding vectors for efficiently estimating the importance and uniqueness of each newly added edge. As demonstrated through extensive experiments, inGRASS achieves up to over 200× speedups while retaining comparable solution quality in incremental spectral sparsification of graphs obtained from various datasets, such as circuit simulations, finite element analysis, and social networks. 

Recent spectral graph sparsificationresearch aims to construct ultra-sparse subgraphs for preserving the original graph spectral (structural) properties, such as the first few Laplacian eigenvalues and eigenvectors, which has led to the development of a variety of nearly linear time numerical and graph algorithms. However, there is very limited progress in the spectral sparsification of directed graphs. In this work, we prove the existence of nearly linear-sized spectral sparsifiers for directed graphs under certain conditions. Furthermore, we introduce a practically efficient spectral algorithm (diGRASS) for sparsifying real-world, large-scale directed graphs leveraging spectral matrix perturbation analysis. The proposed method has been evaluated using a variety of directed graphs obtained from real-world applications, showing promising results for solving directed graph Laplacians, spectral partitioning of directed graphs, and approximately computing (personalized) PageRank vectors. 

SGM-PINN is a graph-based importance sampling framework to improve the training efficacy of Physics-Informed Neural Networks (PINNs) on parameterized problems. By applying a graph decomposition scheme to an undirected Probabilistic Graphical Model (PGM) built from the training dataset, our method generates node clusters encoding conditional dependence between training samples. Biasing sampling towards more important clusters allows smaller mini-batches and training datasets, improving training speed and accuracy. We additionally fuse an efficient robustness metric with residual losses to determine regions requiring additional sampling. Experiments demonstrate the advantages of the proposed framework, achieving 3× faster convergence compared to prior state-of-the-art sampling methods. 

Vectorless integrity verification is becoming increasingly critical to the robust design of nanoscale integrated circuits. This article introduces a general vectorless integrity verification framework that allows computing the worst-case voltage drops or temperature (gradient) distributions across the entire chip under a set of local and global workload (power density) constraints. To address the computational challenges introduced by the large power grids and three-dimensional mesh-structured thermal grids, we propose a novel spectral approach for highly scalable vectorless verification of large chip designs by leveraging a hierarchy of almost linear-sized spectral sparsifiers of input grids that can well retain effective resistances between nodes. As a result, the vectorless integrity verification solution obtained on coarse-level problems can effectively help compute the solution of the original problem. Our approach is based on emerging spectral graph theory and graph signal processing techniques, which consists of a graph topology sparsification and graph coarsening phase, an edge weight scaling phase, as well as a solution refinement procedure. Extensive experimental results show that the proposed vectorless verification framework can efficiently and accurately obtain worst-case scenarios in even very large designs.