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1. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems III: Nonlinear Theory

   https://doi.org/10.1007/s10884-022-10143-4  

   Hupkes, H. J. ; Van Vleck, E. S. ( April 2022 , Journal of Dynamics and Differential Equations)

   In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE and establish the existence of travelling waves. In particular, we consider the time dependent spatial mesh adaptation method that aims to equidistribute the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021). Using the Fredholm theory developed in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021) we setup a fixed point procedure that enables the travelling PDE waves to be lifted to our spatially discrete setting.

    

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2. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems I: Well-Posedness

   https://doi.org/10.1007/s10884-021-10013-5  

   Hupkes, H. J. ; Van Vleck, E. S. ( June 2022 , Journal of Dynamics and Differential Equations)

   Abstract In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations. 

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3. Accuracy of slender body theory in approximating force exerted by thin fiber on viscous fluid

   https://doi.org/10.1111/sapm.12380  

   Mori, Yoichiro ; Ohm, Laurel ( March 2021 , Studies in Applied Mathematics)

   We consider the mapping properties of the integral operator arising in nonlocal slender body theory (SBT) for the model geometry of a straight, periodic filament. It is well known that the classical singular SBT integral operator suffers from high wavenumber instabilities, making it unsuitable for approximating theslender body inverse problem, where the fiber velocity is prescribed and the integral operator must be inverted to find the force density along the fiber. Regularizations of the integral operator must therefore be used instead. Here, we consider two regularization methods: spectral truncation and the‐regularization of Tornberg and Shelley (2004). We compare the mapping properties of these approximations to the underlying partial differential equation (PDE) solution, which for the inverse problem is simply the Stokes Dirichlet problem with data constrained to be constant on cross sections. For the straight‐but‐periodic fiber with constant radius, we explicitly calculate the spectrum of the operator mapping fiber velocity to force for both the PDE and the approximations. We prove that the spectrum of the original SBT operator agrees closely with the PDE operator at low wavenumbers but differs at high frequencies, allowing us to define a truncated approximation with a wavenumber cutoff. For both the truncated and‐regularized approximations, we obtain rigorous‐based convergence to the PDE solution as: A fiber velocity withregularity givesconvergence, while a fiber velocity with at leastregularity yieldsconvergence. Moreover, we determine the dependence of the‐regularized error estimate on the regularization parameter.

    

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4. Bringing physics into the coarse‐grid selection: Approximate diffusion distance/effective resistance measures for network analysis and algebraic multigrid for graph Laplacians and systems of elliptic partial differential equations

   https://doi.org/10.1002/nla.2539  

   Lee, Barry ( November 2023 , Numerical Linear Algebra with Applications)

   Vassilevski, Panayot (Ed.)

   In a recent paper, the author examined a correlation affinity measure for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (C nodes) in systems of elliptic partial differential equations (PDEs). This measure was applied to a set of relaxed vectors, which exposed the near‐nullspace components of the PDE operator. Selecting the CDOFs using this affinity measure and constructing the interpolation operators using a least‐squares procedure, an algebraic multigrid (AMG) method was developed. However, there are several noted issues with this AMG solver. First, to capture strong anisotropies, a large number of test vectors may be needed; and second, the solver's performance can be sensitive to the initial set of random test vectors. Both issues reflect the sensitive statistical nature of the measure. In this article, we derive several other statistical measures that ameliorate these issues and lead to better AMG performance. These measures are related to a Markov process, which the PDE itself may model. Specifically, the measures are based on the diffusion distance/effective resistance for such process, and hence, these measures incorporate physics into the CDOF selection. Moreover, because the diffusion distance/effective resistance can be used to analyze graph networks, these measures also provide a very economical scheme for analyzing large‐scale networks. In this article, the derivations of these measures are given, and numerical experiments for analyzing networks and for AMG performance on weighted‐graph Laplacians and systems of elliptic boundary‐value problems are presented.

    

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5. Computing Optimal Upper Bounds on the H2-norm of ODE-PDE Systems using Linear Partial Inequalities

   https://doi.org/10.1016/j.ifacol.2023.10.538  

   Braghini, Danilo ; Peet, Matthew M. ( January 2023 , IFAC-PapersOnLine)

   Recently, a broad class of linear delayed and ODE-PDEs systems was shown to have an equivalent representation using Partial Integral Equations (PIEs). In this paper, we use this PIE representation, combined with algorithms for convex optimization of Partial Integral (PI) operators to bound the H2-norm for input-output systems of this class. Specifically, the methods proposed here apply to delayed and ODE-PDE systems (including delayed PDE systems) in one or two spatial variables where the disturbance does not enter through the boundary. For such systems, we define a notion of H2-norm using an initial state-to-output framework and show that this notion reduces to more traditional concepts under the assumption of existence of a strongly continuous semigroup. Next, we consider input-output systems for which there exists a PIE representation and for such systems show that computing a minimal upper bound on the H2-norm of delayed and PDE systems can be equivalently formulated as a convex optimization problem subject to linear PI operator inequalities (LPIs). We convert, then, these optimization problems to Semi-Definite Programming (SDP) problems using the PIETOOLS toolbox. Finally, we apply the results to several numerical examples – focusing on time-delay systems (TDS) for which comparable H2 approximation results are available in the literature. The numerical results demonstrate the accuracy of the computed upper bound on the H2-norm. 

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