Related papers: Krylov Methods for Adjoint-Free Singular Vector Ba…

Data representation techniques have made a substantial contribution to advancing data processing and machine learning (ML). Improving predictive power was the focus of previous representation techniques, which unfortunately perform rather…

We provide a novel accelerated first-order method that achieves the asymptotically optimal convergence rate for smooth functions in the first-order oracle model. To this day, Nesterov's Accelerated Gradient Descent (AGD) and variations…

Flexible Krylov methods are a common standpoint for inverse problems. In particular, they are used to address the challenges associated with explicit variational regularization when it goes beyond the two-norm, for example involving an…

Recent works have established the utility of sparsity-promoting norms for extracting spatially-localized instability mechanisms in fluid flows, with possible implications for flow control. However, these prior works have focused on linear…

Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov subspace methods in which a Krylov subspace is augmented with a fixed subspace spanned by vectors deemed to be helpful in…

This thesis deals with the investigation of a H(div)-conforming hybrid discontinuous Galerkin discretization for incompressible turbulent flows. The discretization method provides many physical and solving-oriented properties, which may be…

We present a general framework to predict precursors to extreme events in turbulent dynamical systems. The approach combines phase-space reconstruction techniques with recurrence matrices and convolutional neural networks to identify…

Krylov subspace methods in quantum dynamics identify the minimal subspace in which a process unfolds. To date, their use is restricted to time evolutions governed by time-independent generators. We introduce a generalization valid for…

Karlin's (1982) Theorem 5.2 shows that linear systems alternating between growth and mixing phases have lower asymptotic growth with greater mixing. Here this result is extended to linear differential equations that combine site-specific…

We propose a new method for computing Dynamic Mode Decomposition (DMD) evolution matrices, which we use to analyze dynamical systems. Unlike the majority of existing methods, our approach is based on a variational formulation consisting of…

Many of the new developments in machine learning are connected with gradient-based optimization methods. Recently, these methods have been studied using a variational perspective. This has opened up the possibility of introducing…

Convergence analysis of accelerated first-order methods for convex optimization problems are presented from the point of view of ordinary differential equation solvers. A new dynamical system, called Nesterov accelerated gradient flow, has…

In this paper, by introducing a class of relaxed filtered Krylov subspaces, we propose the relaxed filtered Krylov subspace method for computing the eigenvalues with the largest real parts and the corresponding eigenvectors of non-symmetric…

This paper considers the problem of estimating the principal eigenvector of a covariance matrix from independent and identically distributed data samples in streaming settings. The streaming rate of data in many contemporary applications…

We present a sampling strategy suitable for optimization problems characterized by high-dimensional design spaces and noisy outputs. Such outputs can arise, for example, in time-averaged objectives that depend on chaotic states. The…

In this paper we develop randomized Krylov subspace methods for efficiently computing regularized solutions to large-scale linear inverse problems. Building on the recently developed randomized Gram-Schmidt process, where sketched inner…

This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in…

Matrix exponential discriminant analysis (EDA) is a generalized discriminant analysis method based on matrix exponential. It can essentially overcome the intrinsic difficulty of small sample size problem that exists in the classical linear…

A compact version of the variation evolving method (VEM) is developed in the primal variable space for optimal control computation. Following the idea that originates from the Lyapunov continuous-time dynamics stability theory in the…

Krylov methods rely on iterated matrix-vector products $A^k u_j$ for an $n\times n$ matrix $A$ and vectors $u_1,\ldots,u_m$. The space spanned by all iterates $A^k u_j$ admits a particular basis -- the \emph{maximal Krylov basis} -- which…