Related papers: Assembly of multiscale linear PDE operators
Membrane particles such as proteins and lipids organize into zones that perform unique functions. Here, I introduce a topological and category-theoretic framework to represent particle and zone intra-scale interactions and inter-scale…
In this paper, we investigate the combination of multigrid methods and neural networks, starting from a Finite Element discretization of an elliptic PDE. Multigrid methods use interpolation operators to transfer information between…
We provide novel linear combination of unitaries decompositions for a class of discrete elliptic differential operators. Specifically, Poisson problems augmented with periodic, Dirichlet, Neumann, Robin, and mixed boundary conditions are…
This paper presents a new algorithm KIOPS for computing linear combinations of $\varphi$-functions that appear in exponential integrators. This algorithm is suitable for large-scale problems in computational physics where little or no…
Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for nonlinear algebraic…
We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of…
A numerical framework is developed to solve various types of PDEs on complicated domains, including steady and time-dependent, non-linear and non-local PDEs, with different boundary conditions that can also include non-linear and non-local…
A new numerical approach is proposed for the simulation of coupled three-dimensional and one-dimensional elliptic equations (3D-1D coupling) arising from dimensionality reduction of 3D-3D problems with thin inclusions. The method is based…
We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode-Decomposition type…
Historically, analysis for multiscale PDEs is largely unified while numerical schemes tend to be equation-specific. In this paper, we propose a unified framework for computing multiscale problems through random sampling. This is achieved by…
The solution of matrices with $2\times 2$ block structure arises in numerous areas of computational mathematics, such as PDE discretizations based on mixed-finite element methods, constrained optimization problems, or the implicit or steady…
In an exploration paper, {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (I)}, we designed algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete…
We present a new framework for expressing finite element methods on multiple intersecting meshes: multimesh finite element methods. The framework enables the use of separate meshes to discretize parts of a computational domain that are…
We present a general framework in which both Krylov state and operator complexities can be put on the same footing. In our formalism, the Krylov complexity is defined in terms of the density matrix of the associated state which, for the…
Many problems in computational science and engineering involve partial differential equations and thus require the numerical solution of large, sparse (non)linear systems of equations. Multigrid is known to be one of the most efficient…
This paper deals with the definition and optimization of augmentation spaces for faster convergence of the conjugate gradient method in the resolution of sequences of linear systems. Using advanced convergence results from the literature,…
Multiscale modeling is essential for understanding the complex behavior of materials. However, accurately transferring all relevant information from one scale to another has remained an outstanding challenge. Neural operators,…
The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally…
Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…
This paper investigates the notion of Krylov complexity, a measure of operator growth, within the framework of 1-matrix quantum mechanics (1-MQM). Krylov complexity quantifies how an operator evolves over time by expanding it in a series of…