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- research-articleNovember 2023
- short-paperOctober 2022
CAPER: Coarsen, Align, Project, Refine - A General Multilevel Framework for Network Alignment
CIKM '22: Proceedings of the 31st ACM International Conference on Information & Knowledge ManagementPages 4747–4751https://doi.org/10.1145/3511808.3557563Network alignment, or the task of finding corresponding nodes in different networks, is an important problem formulation in many application domains. We propose CAPER, a multilevel alignment framework that Coarsens the input graphs, Aligns the coarsened ...
High performance uncertainty quantification with parallelized multilevel Markov chain Monte Carlo
SC '21: Proceedings of the International Conference for High Performance Computing, Networking, Storage and AnalysisArticle No.: 75, Pages 1–15https://doi.org/10.1145/3458817.3476150Numerical models of complex real-world phenomena often necessitate High Performance Computing (HPC). Uncertainties increase problem dimensionality further and pose even greater challenges.
We present a parallelization strategy for multilevel Markov chain ...
- research-articleJanuary 2021
Optimal and Low-Memory Near-Optimal Preconditioning of Fully Implicit Runge--Kutta Schemes for Parabolic PDEs
SIAM Journal on Scientific Computing (SISC), Volume 43, Issue 5Pages A3527–A3551https://doi.org/10.1137/20M1387985Runge--Kutta (RK) schemes, especially Gauss--Legendre and some other fully implicit RK (FIRK) schemes, are desirable for the time integration of parabolic partial differential equations due to their A-stability and high-order accuracy. However, it is ...
- research-articleJanuary 2021
On High-Order Multilevel Optimization Strategies
SIAM Journal on Optimization (SIOPT), Volume 31, Issue 1Pages 307–330https://doi.org/10.1137/19M1255355We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on $q$th-order Taylor models (with $q\geq 1$) that have been recently proposed in the ...
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- research-articleJanuary 2019
Efficient Adaptive Multilevel Stochastic Galerkin Approximation Using Implicit A Posteriori Error Estimation
SIAM Journal on Scientific Computing (SISC), Volume 41, Issue 3Pages A1681–A1705https://doi.org/10.1137/18M1194420Partial differential equations (PDEs) with inputs that depend on infinitely many parameters pose serious theoretical and computational challenges. Sophisticated numerical algorithms that automatically determine which parameters need to be activated in the ...
- research-articleOctober 2017
A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields
SIAM Journal on Scientific Computing (SISC), Volume 39, Issue 5Pages S543–S562https://doi.org/10.1137/16M1082688We propose an alternative method to generate samples of a spatially correlated random field with applications to large-scale problems for forward propagation of uncertainty. A classical approach for generating these samples is the Karhunen--Loève (KL) ...
- research-articleOctober 2016
A Multilevel Framework for Sparse Optimization with Application to Inverse Covariance Estimation and Logistic Regression
SIAM Journal on Scientific Computing (SISC), Volume 38, Issue 5Pages S566–S592https://doi.org/10.1137/15M102469XSolving $l_1$ regularized optimization problems is common in the fields of computational biology, signal processing, and machine learning. Such $l_1$ regularization is utilized to find sparse minimizers of convex functions. A well-known example is the ...
- research-articleJanuary 2016
Multilevel Higher Order QMC Petrov--Galerkin Discretization for Affine Parametric Operator Equations
SIAM Journal on Numerical Analysis (SINUM), Volume 54, Issue 4Pages 2541–2568https://doi.org/10.1137/16M1078690We develop a convergence analysis of a multilevel algorithm combining higher order quasi--Monte Carlo (QMC) quadratures with general Petrov--Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic types, extending ...
- research-articleJanuary 2014
Spatial Two-Level Interacting Particle Simulations and Information Theory--Based Error Quantification
SIAM Journal on Scientific Computing (SISC), Volume 36, Issue 2Pages A634–A667https://doi.org/10.1137/120887060We propose a hierarchy of two-level kinetic Monte Carlo methods for sampling high-dimensional, stochastic lattice particle dynamics with complex interactions. The method is based on the efficient coupling of different spatial resolution levels, taking ...
- articleOctober 2011
Assessing the performance of an optimization-based multilevel method
Optimization Methods & Software (OPMS), Volume 26, Issue 4-5Pages 693–717https://doi.org/10.1080/10556788.2010.525637Many large nonlinear optimization problems are based upon a hierarchy of models, corresponding to levels of discretization or detail in the problem. Optimization-based multilevel methods-that is, multilevel methods based on solving coarser versions of ...
- research-articleOctober 2010
Multilevel manifold learning with application to spectral clustering
CIKM '10: Proceedings of the 19th ACM international conference on Information and knowledge managementPages 419–428https://doi.org/10.1145/1871437.1871493In the past decade, a number of nonlinear dimensionality reduction methods using an affinity graph have been developed for manifold learning. This paper explores a multilevel framework with the goal of reducing the cost of unsupervised manifold learning ...
- articleAugust 2010
Two-Level Newton and Hybrid Schwarz Preconditioners for Fluid-Structure Interaction
SIAM Journal on Scientific Computing (SISC), Volume 32, Issue 4Pages 2395–2417https://doi.org/10.1137/090779425We introduce and study numerically a two-level Schwarz preconditioner for Newton-Krylov methods for fluid-structure interaction, with special consideration of the application area of simulating blood flow. Our approach monolithically couples the fluid ...
- articleAugust 2010
Multigrid Algorithm for High Order Denoising
SIAM Journal on Imaging Sciences (SJISBI), Volume 3, Issue 3Pages 363–389https://doi.org/10.1137/080737903Image denoising has been a research topic deeply investigated within the last two decades. Excellent results have been obtained by using such models as the total variation (TV) minimization by Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259-268], ...
- research-articleJune 2010
On high-order denoising models and fast algorithms for vector-valued images
IEEE Transactions on Image Processing (TIP), Volume 19, Issue 6Pages 1518–1527https://doi.org/10.1109/TIP.2010.2042655Variational techniques for gray-scale image denoising have been deeply investigated for many years; however, little research has been done for the vector-valued denoising case and the very few existent works are all based on total-variation ...
- articleMarch 2010
A Multilevel Algorithm for Simultaneously Denoising and Deblurring Images
In this paper, we develop a fast multilevel algorithm for simultaneously denoising and deblurring images under the total variation regularization. Although much effort has been devoted to developing fast algorithms for the numerical solution and the ...
- articleFebruary 2010
Local Multilevel Methods for Adaptive Finite Element Methods for Nonsymmetric and Indefinite Elliptic Boundary Value Problems
In this paper, we propose some local multilevel algorithms for solving linear systems arising from adaptive finite element approximations of nonsymmetric and indefinite elliptic boundary value problems. Two types of local smoothers are constructed. One ...
- ArticleSeptember 2009
Analyzing Large Sparse Markov Chains of Kronecker Products
QEST '09: Proceedings of the 2009 Sixth International Conference on the Quantitative Evaluation of SystemsPage 5https://doi.org/10.1109/QEST.2009.12Kronecker products are used to define the underlying Markov chain (MC)in various modeling formalisms, including compositional Markovian models,hierarchical Markovian models, and stochastic process algebras. The motivation behind using a Kronecker ...
- articleJuly 2009
Preconditioning of Boundary Value Problems Using Elementwise Schur Complements
SIAM Journal on Matrix Analysis and Applications (SIMAX), Volume 31, Issue 2Pages 767–789https://doi.org/10.1137/070679673This paper deals with an efficient technique for computing high-quality approximations of Schur complement matrices to be used in various preconditioners for the iterative solution of finite element discretizations of elliptic boundary value problems. ...
- articleApril 2009
A multilevel algorithm for solving the trust-region subproblem
Optimization Methods & Software (OPMS), Volume 24, Issue 2Pages 299–311https://doi.org/10.1080/10556780802571467We present a multilevel numerical algorithm for the exact solution of the Euclidean trust-region subproblem. This particular subproblem typically arises when optimizing a nonlinear (possibly non-convex) objective function whose variables are discretized ...