Gaussian processes are a versatile tool in statistics and machine learning. They are suitable candidates to generate surrogate models that come with integrated uncertainty quantification of the reduced model. The quality of the model heavily depends on the choice of the hyperparameters and to train these on the given data many linear algebra challenges arise. In this talk we will briefly...
Finding the unique stabilizing solution $X = X^H$ of a large-scale continuous-time algebraic Riccati equation (CARE) $0 = R(X) := A^HX + XA + C^HC - XBB^HX$ with a large, sparse n-x-n matrix $A$, an $n\times m$ matrix $B$ and a $p\times n$ matrix $C$ is of interest in a number of applications. It is assumed, that $B$ and $C^H$ have full column and row rank, respectively, with $m$, $p << n$....
The solution of Sylvester-like matrix equations still constitutes a core task in systems and control theory, with their solution also being required in the extensive field of eigenvalue analysis. In addition to numerous iterative algorithms, Bartels and Stewart presented an algorithm for dense matrices in the 1970s. This algorithm has been improved over the last two decades by the introduction...
Reduced rank extrapolation (RRE) is an acceleration method typically used to accelerate the iterative solution of nonlinear systems of equations using a fixed-point process. In this context, the iterates are vectors generated from a fixed-point mapping function. However, when considering the iterative solution of large-scale matrix equations, the iterates are low-rank matrices generated from a...
Let $G, H \in \mathbb{C}^{n,n}$ be Hermitian and $S \in \mathbb{C}^{n,n}$ be a symmetric matrix. We consider the problem of maximizing the Rayleigh quotient of $G$ with respect to constraints involving symmetric matrix $S$ and Hermitian matrix $H$. More precisely, we compute
$$
\begin{align}
m(G,H,S):=\sup\bigg{ \frac{v^Gv}{v^v} :~v\in \mathbb{C}^{n} \setminus {0},\,v^TSv=0,\,...
We present a novel tensor empirical interpolation method operating within the tensor t-product framework. Compared to traditional empirical interpolation, our approach significantly enhances the reconstruction accuracy of tensor-valued datasets by using optimally placed sparse sensor measurements along the dimension of interest. This improvement is achieved by preserving the native tensor...
In this talk, we introduce a mesh-free two-level hybrid Tucker tensor format for the approximation of multivariate functions. This new format combines the product Chebyshev interpolation with the ALS-based Tucker decomposition of the coefficients tensor. The benefits of this tensor approximation are two-fold. On the one hand, it allows to avoid the rank-structured approximation of functional...
In this talk, we revisit the L2 optimal approximation problem through various formulations and applications, exploring its rich mathematical structure and diverse implications. We begin with the classical case where the optimal approximant is a rational function, highlighting how Hermite interpolation at specific reflected points emerges as the necessary condition for optimality. Building on...
We consider an elastic solid, e.g. a ball, which sinks towards the ground in a container filled with a liquid. The ball bounces off the bottom.
From a mathematical point of view, it is not clear what exactly happens. Assuming that the Navier-Stokes equations describe the situation, contact should not occur. The common hypothesis is that if the surface of the sphere is assumed to be...
spatially localized features characterized by slow singular value decay. The proposed approach builds upon two existing methodologies for reduced and full-order non-intrusive modeling, namely Operator Inference (OpInf) and sparse Full-Order Model (sFOM) inference. We decompose the domain into two complementary subdomains which exhibit fast and slow singular value decay, accordingly. The...
Metastablility is a phenomenon which often inhibits the efficient simulation of dynamical systems, or the generation of samples from high-dimensional probability measures. In particular, metastability is frequently encountered in computer simulations of biological macromolecules using molecular dynamics. It is well-known that metastable transitions and their time scales are encoded in the...
Stochastic dynamics with metastability are a recurring theme in many scientific disciplines, for instance, in simulations of macro-molecules, in climate systems, and in applications of uncertainty quantification. Metastability describes the existence of long-lived macro-states in a dynamical system's state space, such that transitions between these macro-states are rare events. It is thus...
Reconstructing nonlinear parametric functions based on sparse measurements or partially observed sensor data is an important problem arising in a number of practical applications such as climate science, fluid dynamics, and biology. To achieve accurate approximation, two crucial aspects need to be considered: (a) choosing the optimal location for data sampling and (b) identifying a suitable...
With the capability of modern computers for processing large amounts of data, machine learning (ML) is being more and more applied in computational science. Many of the ML learning methods aim to accurately predict the whole solution vector of dynamical systems via data compression, such as convolutional autoencoder. These works focus on predicting only the quantities of interests (QoIs) or...
Parametrized nonlinear dynamical systems often arise upon discretizing parametrized partial differential equations (PDEs). Typically, the former tends to be of a large scale, i.e., the number of state variables could be significantly high (the state vector is denoted with x(t, p), p is the vector of parameters, the nonlinearity is f(x(t, p), p), while the matrices corresponding to the linear...
Accurate simulations of gas flow within pipeline networks provide crucial insights for transmission and design operators. This importance intensifies with the growing integration of hydrogen-blended fuels in hydrogen-based energy systems. Existing models predominantly address single-component, one-dimensional flow within individual pipe segments, interconnected at network junctions. Recently,...
In the usual one-precision environment the Bartels--Stewart algorithm is the standard algorithm for solving small dense Sylvester matrix equations AX+XB=C. their algorithm consists of three steps, including the computation of the Schur decompositions, the substitution for solving the Sylvester equation with quasi-triangular coefficients, and the final recovering transformation.
In the new...
Numerical algorithms and computational tools are essential for managing and analyzing complex data processing tasks. As meta-data and parameter-driven simulations have become more prevalent, the need for automated workflows to reproduce computational experiments across platforms has significantly increased. In general, a computational workflow is defined as a step-by-step description for...
The class of bilinear dynamical systems often arises in the description and approximation of large-scale nonlinear systems with analytic nonlinearities. This bilinear structure allows the extension of various established model order reduction (MOR) approaches, from their linear counterparts. In this work, our aim is to introduce a realization-free, data-based MOR method, inspired by the...
The increasing complexity in design and manufacturing has driven the need for advanced techniques for fast modeling problems with high-dimensional parameter spaces. Avoiding high-fidelity finite element models while achieving fast and accurate simulations in such contexts is challenging. Projection-based parametric model order reduction (pMOR) has drawn significant attention in recent years....
pyMOR, a Python toolbox for building reduced order models, supports a variety of solver backends, such as dune-gdt, FEniCS, NGSolve, NumPy and potentially MPI distributed variants. An abstract interface, the VectorArray, is used to wrap vectors and matrices from these backends on which pyMOR's algorithms work. For compatibility reasons direct access of entries has to be avoided, which...
Solving the Schrödinger equation is a central task in quantum chemistry. Diffusion Monte Carlo (DMC) is a stochastic method that utilizes the connection between this equation and those governing diffusion phenomena to obtain accurate estimates of the eigenvalues and eigenfunctions involved.
This is made possible by the well-known Feynman-Kac formula, which allows the Schrödinger equation to...
For the standard algebraic generalized Riccati equation
$C^TQC+ A^TXE + E^TXA -E^TXBR^{-1}B^TXE = 0$
different algorithms inspired by the low-rank alternating directions implicit (LR-ADI) iteration and Krylov subspace projection have successfully been applied, based on the $ZZ^T$ factorization of the iterates, when $Q, R$ are both symmetric, and $Q\geq0$ and $R>0$.
In this...
We study the problem of constructing solutions [1, 2] $X \in \mathbb{R}^{\alpha \times \beta}$ of a linear matrix equation
$LX = F, F \in \mathbb{R}^{\gamma \times \delta},$ (1)
where $L : \mathbb{R}^{\alpha \times \beta} \rightarrow \mathbb{R}^{\gamma \times \delta}$ is a linear bounded matrix functional. A partial case of the matrix algebraic equation (1) is the well-known Sylvester and...
Biomolecular processes often involve transitions between metastable states governing molecular function. Extracting this information is challenging due to the high dimensionality of simulation data. Kernel methods are a powerful tool for automated analysis of complex systems, yet their need for pairwise kernel evaluations leads to scalability issues, particularly in large datasets.
To address...
We analyze two classes of mathematical models describing chemical reactions of the type "$A \to$ product," conducted in tubular reactors. The first class involves nonlinear hyperbolic partial differential equations modeling a plug flow reactor (PFR) in the presence of an inert component. For this system, an isoperimetric optimal control problem is formulated to maximize the mean product yield...
There have been a number of works in the literature dealing with the stabilization of processes with input delays for a reaction-diffusion partial differential equation, mainly by using finite-time control strategies, where their settling time depends on the initial conditions. However, in many practical applications, it is difficult or even impossible to obtain system initial values in...
We present a novel machine learning approach to develop energy-preserving reduced-order models (ROMs) by exploiting the multi-symplectic structure of partial differential equations (PDEs). Traditional energy-preserving ROMs often rely on the symplectic Galerkin projection, which requires fully discrete operators - often unavailable in black-box PDE solvers. Our method circumvents this...
The numerical integration of Hamiltonian systems is a central topic in computational physics and theoretical chemistry. Intrinsic structure-preserving symplectic networks (SympNets) [1] are a widely used tool when it comes to learning symplectic integrators for Hamiltonian systems from data. More often then not Hamiltonian systems of interest depend on a set of parameters. Currently, SympNets...
Model-order Reduction (MOR) is a technique used to reduce the complexity of high-dimensional dynamical systems while preserving their essential properties. It has been been successfully used in many scientific and engineering fields, such as Fluid Dynamics and Control Theory by methods like Proper Orthogonal Decomposition (POD), Autoencoder, Dynamic Mode decomposition (DMD) and so on. However,...
Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE when solving the optimal control problem.
Specifically, we explore the use of the shifted...
For linear control systems, we study algebraic stability domains which are defined by polynomial inequalities in the complex plane. Based on Kharitonov's characterization via generalized Lyapunov matrix inequalities, we define a suitable pair of Gramians and a balancing transformation. Then we analyze the preservation of these stability domains by balanced truncation. For domains bounded by...
We propose a least-squares formulation for parabolic PDEs in the natural $L^2(0,T;V^*)\times H$ norm which avoids regularity assumptions on the data of the problem, which e.g. appear in the recently proposed First Order System Least Squares (FOSLS) method. For abstract parabolic equations the resulting bilinear form then is symmetric, continuous, and coercive. This among other things paves the...
Model order reduction (MOR) is a method designed to simplify high-dimensional systems by representing their states using a low-dimensional parametrization. It enhances computational efficiency and reduces memory usage but can lead to reduced simulation accuracy and difficulties in preserving system properties, such as sparsity, positivity, and incompressibility. Consequently, choosing an...
