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In this chapter, we investigate recently proposed nonlinear conjugate gradient (NCG) methods for shape optimization problems. We briefly introduce the methods as well as the corresponding theoretical background and investigate their…
In this paper, a restricted memory quasi-Newton bundle method for minimizing a locally Lipschitz continuous function over a Riemannian manifold is proposed. The curvature information of the objective function is approximated by applying a…
The paper addresses the stabilization of nonlinear systems with semi-quadratic cost: quadratic with respect to controls and nonlinear for state variables. Paper presents the effective new feedback synthesis procedure. The novel feedback…
Model order reduction in high-dimensional, nonlinear dynamical systems if often enabled through fast-slow timescale separation. One such approach involves identifying a low-dimensional slow manifold to which the state rapidly converges and…
We present a data-driven method for separating complex, multiscale systems into their constituent time-scale components using a recursive implementation of dynamic mode decomposition (DMD). Local linear models are built from windowed…
We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex,…
For nonlinear reduced-order models, especially for those with non-polynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. As a result, the reduced-order model loses its…
Nonlinear dynamics has an important role when designing modern synchrotron lattices. In this letter, we introduce a new method of using a square matrix to analyze periodic nonlinear dynamical systems [1, 2]. Applying the method to the…
This work presents a method for constructing online-efficient reduced models of large-scale systems governed by parametrized nonlinear scalar conservation laws. The solution manifolds induced by transport-dominated problems such as…
Nonlinear extensions to the active subspaces method have brought remarkable results for dimension reduction in the parameter space and response surface design. We further develop a kernel-based nonlinear method. In particular we introduce…
Nonlinear dynamic models are widely used for characterizing functional forms of processes that govern complex biological pathway systems. Over the past decade, validation and further development of these models became possible due to data…
In dynamical systems theory, a fixed point of the dynamics is called nonhyperbolic if the linearization of the system around the fixed point has at least one eigenvalue with zero real part. The center manifold existence theorem guarantees…
This paper presents a structure-preserving model reduction approach applicable to large-scale, nonlinear port-Hamiltonian systems. Structure preservation in the reduction step ensures the retention of port-Hamiltonian structure which, in…
We use the recent theory of Spectral Submanifolds (SSM) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization…
The Barzilai-Borwein (BB) gradient method is efficient for solving large-scale unconstrained problems to the modest accuracy and has a great advantage of being easily extended to solve a wide class of constrained optimization problems. In…
This paper proposes a constructive approach to safety control of nonlinear cascade systems subject to multiple state constraints. New design ingredients include a unified characterization of safety and stability for systematic designs of…
We propose a non-intrusive, Autoencoder-based framework for reduced-order modeling in continuum mechanics. Our method integrates three stages: (i) an unsupervised Autoencoder compresses high-dimensional finite element solutions into a…
Equations governing the nonlinear dynamics of complex systems are usually unknown and indirect methods are used to reconstruct their manifolds. In turn, they depend on embedding parameters requiring other methods and long temporal sequences…
A nonlinear-manifold reduced order model (NM-ROM) is a great way of incorporating underlying physics principles into a neural network-based data-driven approach. We combine NM-ROMs with domain decomposition (DD) for efficient computation.…
This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with…