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Stochastic sampling methods are arguably the most direct and least intrusive means of incorporating parametric uncertainty into numerical simulations of partial differential equations with random inputs. However, to achieve an overall error…
Stochastic Approximation has been a prominent set of tools for solving problems with noise and uncertainty. Increasingly, it becomes important to solve optimization problems wherein there is noise in both a set of constraints that a…
Sample-based Bayesian inference provides a route to uncertainty quantification in the geosciences, and inverse problems in general, though is very computationally demanding in the naive form that requires simulating an accurate computer…
With the rapid evolution of the Internet, the vast amount of data has created opportunities for fostering the development of steganographic techniques. However, traditional steganographic techniques encounter challenges due to distortions…
Stochastic versions of proximal methods have gained much attention in statistics and machine learning. These algorithms tend to admit simple, scalable forms, and enjoy numerical stability via implicit updates. In this work, we propose and…
Domain decomposition methods are widely used for the numerical solution of partial differential equations on high performance computers. We develop an adjoint-based a posteriori error analysis for both multiplicative and additive…
In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated using parametrized, single-objective sub-problems which are solved simultaneously. To this end a consensus based…
We propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry adapted coordinates system where the given SDE has notable invariant properties. An approximation scheme preserving the…
This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC…
Supercomputing systems today often come in the form of large numbers of commodity systems linked together into a computing cluster. These systems, like any distributed system, can have large numbers of independent hardware components…
The idle computers on a local area, campus area, or even wide area network represent a significant computational resource---one that is, however, also unreliable, heterogeneous, and opportunistic. This type of resource has been used…
An algorithm is proposed, analyzed, and tested experimentally for solving stochastic optimization problems in which the decision variables are constrained to satisfy equations defined by deterministic, smooth, and nonlinear functions. It is…
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic…
Stochastic optimization methods have been hugely successful in making large-scale optimization problems feasible when computing the full gradient is computationally prohibitive. Using the theory of modified equations for numerical…
Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth…
Decentralized stochastic optimization methods have gained a lot of attention recently, mainly because of their cheap per iteration cost, data locality, and their communication-efficiency. In this paper we introduce a unified convergence…
Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of…
We study the stochastic $p$-Laplace system in a bounded domain. We propose two new space-time discretizations based on the approximation of time-averaged values. We establish linear convergence in space and $1/2$ convergence in time.…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…