Numerical simulation of microfluidics on complex surfaces holds significant industrial and scientific value. However, existing numerical methods still face challenges in terms of both physical consistency and computational efficiency. This paper proposes a physically rigorous, computationally robust, and efficient smoothed particle hydrodynamics (SPH) framework. Specifically, we introduce a nonlocal representation of attractive and repulsive forces...
In symmetric block eigenvalue algorithms, such as the subspace iteration algorithm and the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm, a large block size is often employed to achieve robustness and rapid convergence. However, using a large block size also increases the computational cost. Traditionally, the block size is typically reduced after convergence of...
Owing to its simplicity and efficiency, the Sherman-Morrison (SM) formula has seen widespread use across various scientific and engineering applications for solving rank-one perturbed linear systems of the form (A + uv')x = b. Although the formula dates back at least to 1944, its numerical stability properties have remained an open question and continue to be a...
We studied mixed-precision preconditioned Jacobi algorithms for computing eigenvalues and singular values. The main idea was to construct a preconditioner in low precision and apply it in higher precision to improve the accuracy and convergence of the Jacobi algorithm. We presented a relative forward error analysis showing that the proposed approaches can achieve significantly smaller...
Airfoil aerodynamic optimization is critical for improving lift-to-drag performance, reducing fuel consumption, and mitigating environmental impact in aerospace engineering. Recent advances in data-driven surrogate modeling offer efficient alternatives to costly CFD simulations. In this work, we introduce CoPiT (Conditional Position-induced Transformer), a novel neural operator that augments the Position-induced Transformer (PiT) with Feature-wise Linear Modulation...
In its 30 or so years existence the Material Point Method (MPM) has proved to be outstanding in solving challenging modeling problems in engineering and in animation. The method models objects as a collection of particles that move on a fixed background grid. The method is used in a range of applications from modeling snow,...
This presentation focuses on the numerical analysis of a diffusion problem with a boundary condition involving a surface Laplacian using a high-order finite element method. In order to define this surface operator on the boundary, the domain is assumed to be smooth: hence, the meshed domain does not coincide with the initial physical domain, resulting...
The Generalized Minimal Residual methods (GMRES) for the solution of general square linear systems is often combined with a preconditioner to improve the convergence speed and the overall computing performance of the method. Successful mixed precision implementations for the application of the preconditioner inside GMRES have been previously proposed: certain strategy prescribes to apply the...
We develop a mixed-precision iterative refinement framework for solving low-rank Lyapunov matrix equations , where or *. Via rounding error analysis of the algorithms we derive sufficient conditions for the attainable normwise residuals in different precision settings and show how the algorithmic parameters should be chosen. Using the sign function Newton iteration as the solver, we show...
Polynomial matrices arise naturally in a variety of signal processing, control, communication, and MIMO system applications. Similar to conventional matrices, they require structured decompositions—such as eigenvalue decomposition (EVD), singular value decomposition (SVD), and QR decomposition—to enable optimal solutions in many practical problems. These factorizations play a critical role in broadband data compaction, MIMO-OFDM system design,...
