More Like this

1. A new approach on the stability and convergence of a time-space nonuniform finite difference approximation of a degenerate Kawarada problem

   https://doi.org/10.1080/00207160.2024.2342472  

   Torres, Eduardo Servin; Sheng, Qin (April 2024, International Journal of Computer Mathematics)

   Nonlinear Kawarada equations have been used to model solid fuel combustion processes in the oil industry. An effective way to approximate solutions of such equations is to take advantage of the finite difference configurations. Traditionally, the nonlinear term of the equation is linearized while the numerical stability of a difference scheme is investigated. This leaves certain ambiguity and uncertainty in the analysis. Based on nonuniform grids generated through a quenching-seeking moving mesh method in space and adaptation in time, this paper introduces a completely new stability analysis of the approximation without freezing the nonlinearity involved. Pointwise orders of convergence are investigated numerically. Simulation experiments are carried out to accompany the mathematical analysis to strengthen our conclusions. 

   more » « less
2. Scalar Auxiliary Variable (SAV) Stabilization of Implicit‐Explicit (IMEX) Time Integration Schemes for Non‐Linear Structural Dynamics

   https://doi.org/10.1002/nme.7660  

   Kwon, Sun‐Beom; Prakash, Arun (January 2025, International Journal for Numerical Methods in Engineering)

   Implicit-explicit (IMEX) time integration schemes are well suited for non-linear structural dynamics because of their low computational cost and high accuracy. However, the stability of IMEX schemes cannot be guaranteed for general non-linear problems. In this article, we present a scalar auxiliary variable (SAV) stabilization of high-order IMEX time integration schemes that leads to unconditional stability. The proposed IMEX-BDFk-SAV schemes treat linear terms implicitly using kth-order backward difference formulas (BDFk) and non-linear terms explicitly. This eliminates the need for iterations in non-linear problems and leads to low computational costs. Truncation error analysis of the proposed IMEX-BDFk-SAV schemes confirms that up to kth-order accuracy can be achieved and this is verified through a series of convergence tests. Unlike existing SAV schemes for first-order ordinary differential equations (ODEs), we introduce a novel SAV for the proposed schemes that allows direct solution of the second-order ODEs without transforming them into a system of first-order ODEs. Finally, we demonstrate the performance of the proposed schemes by solving several non-linear problems in structural dynamics and show that the proposed schemes can achieve high accuracy at a low computational cost while maintaining unconditional stability. 

   more » « less
3. An Efficient Quasi-Newton Method with Tensor Product Implementation for Solving Quasi-Linear Elliptic Equations and Systems

   https://doi.org/10.1007/s10915-025-02897-y  

   Hao, Wenrui; Lee, Sun; Zhang, Xiangxiong (April 2025, Journal of Scientific Computing)

   Abstract In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of linear Laplacian and simplified nonlinear terms, our method reduces the computational overhead typical of traditional Newton methods while handling the large, sparse matrices generated from discretized PDEs. We also provide a convergence analysis demonstrating local convergence to the exact solution under optimal choices for the regularization parameter, ensuring stability and efficiency in each iteration. Numerical experiments in two- and three-dimensional domains validate the proposed method’s robustness and computational gains with tensor product implementation. This approach offers a promising pathway for accelerating quasi-linear elliptic equations and systems solvers, expanding the feasibility of complex simulations in physics, engineering, and other fields leveraging advanced hardware capabilities. 

   more » « less
4. Asymptotic Network Independence and Step-Size for a Distributed Subgradient Method

   Alex Olshevsky (January 2022, Journal of machine learning research)

   We consider whether distributed subgradient methods can achieve a linear speedup over a centralized subgradient method. While it might be hoped that distributed network of n nodes that can compute n times more subgradients in parallel compared to a single node might, as a result, be n times faster, existing bounds for distributed optimization methods are often consistent with a slowdown rather than speedup compared to a single node. We show that a distributed subgradient method has this “linear speedup” property when using a class of square-summable-but-not-summable step-sizes which include 1/t^β when β ∈ (1/2,1); for such step-sizes, we show that after a transient period whose size depends on the spectral gap of the network, the method achieves a performance guarantee that does not depend on the network or the number of nodes. We also show that the same method can fail to have this “asymptotic network independence” property under the optimally decaying step-size 1/t^{1/2} and, as a consequence, can fail to provide a linear speedup compared to a single node with 1/t^{1/2} step-size. 

   more » « less
5. Tracking control by the Newton–Raphson method with output prediction and controller speedup

   https://doi.org/10.1002/rnc.6976  

   Wardi, Y.; Seatzu, C.; Cortés, J.; Egerstedt, M.; Shivam, S.; Buckley, I. (September 2023, International Journal of Robust and Nonlinear Control)

   Summary This paper presents a control technique for output tracking of reference signals in continuous‐time dynamical systems. The technique is comprised of the following three elements: (i) a fluid‐flow version of the Newton–Raphson method for solving algebraic equations, (ii) a system‐output prediction which has to track the future reference signal, and (iii) a speedup of the control action for enhancing the tracker's accuracy and, in some cases, stabilizing the closed‐loop system. The technique can be suitable for linear and nonlinear systems, implementable by simple algorithms, and can track reference points as well as time‐dependent reference signals. Though inherently local, the tracking controller is proven to have a global convergence for a class of linear systems. The derived theoretical results of the paper include convergence of the tracking controller and error analysis, and are supported by illustrative simulation and laboratory experiments. 

   more » « less