Integration factor combined with level set method for reaction-diffusion systems with free boundary in high spatial dimensions

For reaction-diffusion equations in irregular domain with moving boundaries, the numerical stability constraints from the reaction and diffusion terms often require very restricted time step size, while complex geometries may lead to difficulties in accuracy when discretizing the high-order derivatives on grid points near the boundary. It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties. Applying an implicit scheme may be able to remove the stability constraints on the time step, however, it usually requires solving a large global system of nonlinear equations for each time step, and the computational cost could be significant. Integration factor (IF) or exponential differencing time (ETD) methods are one of the popular methods for temporal partial differential equations (PDEs) among many other methods. In our paper, we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries. In particular, we rewrite all ETD schemes into a linear combination of specific {\phi}-functions and apply one start-of-the-art algorithm to compute the matrix-vector multiplications, which offers significant computational advantages with adaptive Krylov subspaces. In addition, we extend this method by incorporating the level set method to solve the free boundary problem. The accuracy, stability, and efficiency of the developed method are demonstrated by numerical examples.Comment: 20 pages, 6 figures, 2 table

Numerical Methods for a Two-Species Competition-Diffusion Model With Free Boundaries

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce a front tracking method coupled with an implicit solver for the 1D model. For the general 2D model, we use a level set approach to handle the moving boundaries to efficiently treat complicated topological changes. Several numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches

Beyond Universal Transformer: block reusing with adaptor in Transformer for automatic speech recognition

Transformer-based models have recently made significant achievements in the application of end-to-end (E2E) automatic speech recognition (ASR). It is possible to deploy the E2E ASR system on smart devices with the help of Transformer-based models. While these models still have the disadvantage of requiring a large number of model parameters. To overcome the drawback of universal Transformer models for the application of ASR on edge devices, we propose a solution that can reuse the block in Transformer models for the occasion of the small footprint ASR system, which meets the objective of accommodating resource limitations without compromising recognition accuracy. Specifically, we design a novel block-reusing strategy for speech Transformer (BRST) to enhance the effectiveness of parameters and propose an adapter module (ADM) that can produce a compact and adaptable model with only a few additional trainable parameters accompanying each reusing block. We conducted an experiment with the proposed method on the public AISHELL-1 corpus, and the results show that the proposed approach achieves the character error rate (CER) of 9.3%/6.63% with only 7.6M/8.3M parameters without and with the ADM, respectively. In addition, we also make a deeper analysis to show the effect of ADM in the general block-reusing method

Expansion of Breast Cancer Stem Cells with Fibrous Scaffolds

Cancer stem cells (CSCs) are hypothesized as tumor-initiating cells within tumors and main contributors of tumor growth, metastasis and recurrence. Mammary cancer cells, MCF-7 cells, were cultured on 3D polycaprolactone (PCL) fibrous scaffolds, showing an increased proportion of CSCs. The expression of stem cell markers, including OCT3/4 and SOX2, and breast CSC-specific markers, SOX4 and CD49f, was significantly upregulated, and the mammosphere-forming capability in cells cultured on PCL fibrous scaffolds increased. The fibrous scaffolds also induced the elongation of MCF-7 cells and extended cell proliferation. The increase of CSC properties after being cultured on fibrous scaffolds was further confirmed with two luminal-type mammary cell lines, T47D and SK-BR-3, and a basal-type cell line, MDA-MB-231, by ALDEFLUOR assay and mammosphere formation assay. Moreover, we observed the upregulation of epithelial to mesenchymal transition and increased invasive capability in cells cultured on PCL fibrous scaffolds. These data suggest that the increase of CSC proportion in a 3D culture system may account for the enhanced malignancy. Therefore, our PCL fibrous scaffolds can potentially be used for CSCs enrichment and anti-cancer drug screening