Related papers: Multilevel spectral coarsening for graph Laplacian…
We consider the problem of jointly estimating multiple related zero-mean Gaussian distributions from data. We propose to jointly estimate these covariance matrices using Laplacian regularized stratified model fitting, which includes loss…
Monte Carlo methods play important part in modern statistical physics. The application of these methods suffer from two main difficulties.The first is caused by the relatively small number of particles that can participate in any numerical…
Graph-based variational methods have recently shown to be highly competitive for various classification problems of high-dimensional data, but are inherently difficult to handle from an optimization perspective. This paper proposes a convex…
Multilevel modeling extends traditional modeling techniques with a potentially unlimited number of abstraction levels. Multilevel models can be formally represented by multilevel typed graphs whose manipulation and transformation are…
We consider the problem of minimizing a block separable convex function (possibly nondifferentiable, and including constraints) plus Laplacian regularization, a problem that arises in applications including model fitting, regularizing…
We study the convergence properties of the Gibbs Sampler in the context of posterior distributions arising from Bayesian analysis of conditionally Gaussian hierarchical models. We develop a multigrid approach to derive analytic expressions…
Reservoir computing has been successfully applied to graphs as a preprocessing method to improve the training efficiency of Graph Neural Networks (GNNs). However, a common issue that arises when repeatedly applying layer operators on graphs…
We introduce new multilevel methods for solving large-scale unconstrained optimization problems. Specifically, the philosophy of multilevel methods is applied to Newton-type methods that regularize the Newton sub-problem using second order…
We present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that are modeled using the Richards' equation. We propose a stochastic extension for the empirical models that are…
The multi-level method for discrete state systems, first introduced by Anderson and Higham [Multiscale Model. Simul. 10:146--179, 2012], is a highly efficient simulation technique that can be used to elucidate statistical characteristics of…
Many problems in nonlinear and statistical physics are formulated through represented flows, including physical-space vector fields, phase-space drift fields, and truncated renormalization-group beta functions. We introduce a complementary…
We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we…
Graph Semi-Supervised learning is an important data analysis tool, where given a graph and a set of labeled nodes, the aim is to infer the labels to the remaining unlabeled nodes. In this paper, we start by considering an optimization-based…
Smooth random Gaussian functions play an important role in mathematical physics, a main example being the random plane wave model conjectured by Berry to give a universal description of high-energy eigenfunctions of the Laplacian on generic…
We tensorize the Faber spline system from [14] to prove sequence space isomorphisms for multivariate function spaces with higher mixed regularity. The respective basis coefficients are local linear combinations of discrete function values…
The objective of this paper is to introduce and demonstrate a robust methodology for solving multi-constrained 3D topology optimization problems. The proposed methodology is a combination of the topological level-set formulation, augmented…
In this paper we develop adaptive numerical schemes for certain nonlinear variational problems. The discretization of the variational problems is done by representing the solution as a suitable frame decomposition, i.e., a complete, stable,…
A novel coupled level-set lattice Boltzmann method on adaptive Cartesian grids for simulating liquid-gas multiphase flows is presented. The approach addresses the inherent challenges of accurately modeling multiphase systems characterized…
We introduce a graph renormalization procedure based on the coarse-grained Laplacian, which generates reduced-complexity representations for characteristic scales identified through the spectral gap. This method retains both diffusion…
Multilevel quadrature methods for parametric operator equations such as the multilevel (quasi-) Monte Carlo method are closely related to the sparse tensor product approximation between the spatial variable and the parameter. In this…