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In many applications, projection-based reduced-order models (ROMs) have demonstrated the ability to provide rapid approximate solutions to high-fidelity full-order models (FOMs). However, there is no a priori assurance that these…
Rational Krylov subspaces have become a reference tool in dimension reduction procedures for several application problems. When data matrices are symmetric, a short-term recurrence can be used to generate an associated orthonormal basis. In…
Computational modeling of the brain has become a key part of understanding how the brain clears metabolic waste, but patient-specific modeling on a significant scale is still out of reach with current methods. We introduce a novel approach…
A model order reduction algorithm is presented that generates a reduced-order model of the original high-order model, which ensures high-fidelity within the desired time interval. The reduced model satisfies a subset of the first-order…
Mathematical morphology is a theory concerned with non-linear operators for image processing and analysis. The underlying framework for mathematical morphology is a partially ordered set with well-defined supremum and infimum operations.…
The modified Gram-Schmidt (MGS) orthogonalization is one of the most well-used algorithms for computing the thin QR factorization. MGS can be straightforwardly extended to a non-standard inner product with respect to a symmetric positive…
This paper considers structure-preserving model order reduction (MOR) techniques for port-Hamiltonian (pH) systems, which are typically derived from energy-based modelling. To keep favorable properties of pH systems such as stability and…
In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation…
An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by…
This work investigates a two-stage method for constructing projection-based reduced-order models (ROMs) of parameterized partial differential equations (PDEs). Based on established tensorial ROM methodology, the proposed approach reduces…
We develop a novel deep learning technique, termed Deep Orthogonal Decomposition (DOD), for dimensionality reduction and reduced order modeling of parameter dependent partial differential equations. The approach consists in the construction…
Vector set orthogonal normalization and matrix QR decomposition are fundamental problems in matrix analysis with important applications in many fields. We know that Gram-Schmidt process is a widely used method to solve these two problems.…
We examine nonlinear dynamical systems of ordinary differential equations or differential algebraic equations. In an uncertainty quantification, physical parameters are replaced by random variables. The inner variables as well as a quantity…
Reduced order modeling has gained considerable attention in recent decades owing to the advantages offered in reduced computational times and multiple solutions for parametric problems. The focus of this manuscript is the application of…
This thesis presents recent advances in model order reduction methods with the primary aim to construct online-efficient reduced surrogate models for parameterized multiscale phenomena and accelerate large-scale PDE-constrained parameter…
Inverse problems are pervasive mathematical methods in inferring knowledge from observational and experimental data by leveraging simulations and models. Unlike direct inference methods, inverse problem approaches typically require many…
Iterative majorize-minimize (MM) (also called optimization transfer) algorithms solve challenging numerical optimization problems by solving a series of "easier" optimization problems that are constructed to guarantee monotonic descent of…
The goal of image ordinal estimation is to estimate the ordinal label of a given image with a convolutional neural network. Existing methods are mainly based on ordinal regression and particularly focus on modeling the ordinal mapping from…
Estimating parameters of Partial Differential Equations (PDEs) is of interest in a number of applications such as geophysical and medical imaging. Parameter estimation is commonly phrased as a PDE-constrained optimization problem that can…
Classical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. For problems with slowly decaying Kolmogorov-n-widths such as certain transport-dominated problems, however,…