Related papers: A reduced basis method for the nonlinear Poisson-B…
Model reduction attempts to guarantee a desired "model quality", e.g. given in terms of accuracy requirements, with as small a model size as possible. This article highlights some recent developments concerning this issue for the so called…
We propose and test the first Reduced Radial Basis Function Method (R$^2$BFM) for solving parametric partial differential equations on irregular domains. The two major ingredients are a stable Radial Basis Function (RBF) solver that has an…
We develop an efficient and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the regularization technique of Chen, Holst, and Xu; this technique made possible the first a…
Recent advances in both theoretical and computational methods have enabled large-scale, precision calculations of the properties of atomic nuclei. With the growing complexity of modern nuclear theory, however, also comes the need for novel…
We introduce a hyperreduced reduced basis element method for model reduction of parameterized, component-based systems in continuum mechanics governed by nonlinear partial differential equations. In the offline phase, the method constructs,…
A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs et al., we construct an appropriate constrained free energy functional, such that its…
In this paper, we focus on the reduced basis methodology in the context of non-linear non-affinely parametrized partial differential equations in which affine decomposition necessary for the reduced basis methodology are not obtained [4,…
A classical reduced order model for dynamical problems involves spatial reduction of the problem size. However, temporal reduction accompanied by the spatial reduction can further reduce the problem size without losing accuracy much, which…
In this paper we have derived explicitly computable bounds on the error in energy norms for the fully nonlinear Poisson-Boltzmann equation. Together with the computable bounds, we have also obtained efficient error indicators which can…
Restricted Boltzmann Machines (RBMs) offer a versatile architecture for unsupervised machine learning that can in principle approximate any target probability distribution with arbitrary accuracy. However, the RBM model is usually not…
This paper proposes and tests the first-ever reduced basis warm-start iterative method for the parametrized linear systems, exemplified by those discretizing the parametric partial differential equations. Traditional iterative methods are…
Kinetic transport equations are notoriously difficult to simulate because of their complex multiscale behaviors and the need to numerically resolve a high dimensional probability density function. Past literature has focused on building…
Restricted Boltzmann machines (RBMs) are energy-based models analogous to the Ising model and are widely applied in statistical machine learning. The standard inverse Ising problem with a complete dataset requires computing both data and…
The Restricted Boltzmann Machine (RBM) is a stochastic neural network capable of solving a variety of difficult tasks such as NP-Hard combinatorial optimization problems and integer factorization. The RBM architecture is also very compact;…
The use of model-based numerical simulation of wave propagation in rooms for engineering applications requires that acoustic conditions for multiple parameters are evaluated iteratively and this is computationally expensive. We present a…
The Poisson-Boltzmann equation is a widely used model to study the electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate…
The reduced basis method is a powerful model reduction technique designed to speed up the computation of multiple numerical solutions of parametrized partial differential equations. We consider a quantity of interest, which is a linear…
This paper is concerned with the Taylor-reduced basis method (Taylor-RBM) for the efficient approximation of eigenspaces of large scale parametric Hermitian matrices. The Taylor-RBM is a local model order reduction method, which constructs…
This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin…
Random Batch Methods (RBM) for mean-field interacting particle systems enable the reduction of the quadratic computational cost associated with particle interactions to a near-linear cost. The essence of these algorithms lies in the random…