In many applications such as parameter-dependent partial differential equations and control theory, matrix equations of the form
$\sum \limits_{k=1}^K A_k X B_k^T=C$
are of interest. For $k$, $N$, $m \in \mathbb{N}$, the matrices involved are $A_k\in \mathbb{R}^{N\times N}$, $B_k \in \mathbb{R}^{m \times m}$, $C \in \mathbb{R}^{N \times m}$ and the unknown is $X \in \mathbb{R}^{N \times...
Nonlinear model predictive control schemes solving the step-wise local optimization problems via, e.g. ipopt lead to the necessity to solve large KKT systems in each step of the optimizer. The KKT system itself is a saddle point matrix and, due to the outer MPC approach, ideally has Kronecker structure.
Ipopt uses the matrix inertia of the KKT system to accelerate computations. Classic...
Our goal is the feedback stabilization of a two-dimensional
two-phase Stefan problem, which can model solidification and melting of pure materials.
The solid and liquid phases are separated by an interface.
We want to control and stabilize the interface position.
After linearization and discretization, the stabilization problem results in a
non-autonomous differential Riccati equation...
We will discuss open problems in our approach to structure-preserving model order reduction using clustering and projection. The first is developing more general a priori error bounds. The second is guaranteeing synchronization-preservation for nonlinear multi-agent systems. Finally, we will discuss potential relation to hyper-reduction.
The machine learning model for binary classification for tensor input data was proposed in my previous work. The main key point was to compute the kernel matrix for each pair of tensor input data, more efficiently. Along with it, we have explained TT-CP expansion and other theoretical aspects of this model which controls the stability and reliability aspects of it.
In the...
The Generalized Matrix Sign Function (GMSF) of a Matrix pair $(A,B)$ is typically computed using Newton's method. The naive implementation of the iteration step $A_{k+1} = \frac{1}{2}(A + B A_k^{-1} B)$ takes at least $4\frac{2}{3}m^3$ flops, which makes it a computational tough task. With the help of a preprocessing step, the complexity can be reduced down to $2m^3$ flops. Typical ways to...
Dampers are used for instance in structure engineering to stabilize constructions, e.g. buildings, bridges or dams. The dampers are externally included to avoid strong movements caused by external forces with frequencies close to the eigenfrequencies of the structure. We want to optimize these dampers in order to stabilize the constructions against external forces.
Our aim is to optimize...
The main contribution of this work is to provide new insights for capturing the behavior of hard nonlinearities beyond the classical weakly nonlinear Volterra input-output map (VIOMAP). The VIOMAP approach is a special polynomial map unable to capture hard non-linearities due to the ill-conditioned high polynomial terms. On the other hand, the describing functions (DF) framework allows...
The Reduced Basis Method has been quite successful for reducing fully-discretized system of parametric partial differential equations. However, not much is known in terms of applying it to a semi-discretized system of ordinary differential equations, solved by standard black-box solvers, like $\texttt{ode45,ode15s}$, etc. We intend to investigate the question of how $\textit{a posteriori}$ ...
The Loewner framework is one of the most successful data-driven model order reduction techniques.
Given $k$ right interpolation data and $h$ left interpolation data, the standard layout of this approach is composed of two stages.
First, the $kh\times kh$ Loewner matrix $\mathbb{L}$ and shifted Loewner matrix $\mathbb{L}_s$ are constructed. Then, an SVD of $\mathbb{L}_s-\gamma \mathbb{L}$,...
Publications on model reduction research typically culminate in the numerical results section, which (supposedly) illustrates or justifies the previous theoretical findings. These numerical experiments generally show results supporting the proposed model reduction algorithm or modification thereof. However, an experimental design or an experiment itself involving model reduction is easily...
In this talk, we present a data-driven approach to identify second-order systems of the form
\begin{equation}
\begin{array}{rcl}
\mathbf{M}\ddot{\mathbf{x}}(t) + \mathbf{D}\dot{\mathbf{x}}(t)+ \mathbf{K}\mathbf{x}(t) &=& \mathbf{B}\mathbf{u}(t),\qquad \mathbf{x}(0) = 0,~\dot{\mathbf{x}}(0) = 0, \
\mathbf{y}(t) &=& \mathbf{C}\mathbf{x}(t).
\end{array}
\end{equation}
These systems...
Newman's bound in rational interpolation and its relation with the Loewner framework will be explored.
Iterative linear solvers and strategies for accelerating their convergence have been extensively studied by researchers for many decades. Recently, two classes of iterative solvers for tensor-represented systems have proven their effectiveness in solving high dimensional problems. These are low rank Krylov-based solvers and alternating-least-squares-based solvers. Nonetheless, rounding the...
We present some approaches for obtaining the reduced-order model of incompressible Navier-Stokes equations from the existing literature. The focus of our work is ultimately going to be towards the development of non-intrusive model reduction techniques which would be useful when one has no information about the discretized system. The talk will be given keeping this in mind and potential...
Let $f:\Omega\times D\to R$ be some parameterized function with the parameter domain $D$. We develop a sparse empirical quadrature to compute $I(\mu) = \int_\Omega f(x,\mu)dx$. To this end, we compute $I(\mu)$ for a set of training parameters and compute the weights of the quadrature using sparse recovery. At least computationally, we observe that the number of non-zero weights in such a...
In recent work, we have reduced nonlinear second-order power grid models using balanced truncation techniques for quadratic systems. In order to do that, we 1) transform the original system to a first-order system and 2) use lifting to quadratize the first-order system. However, although the original quadratized system reaches steady state, the reduced quadratic system may only reach a partial...
As structured interpolation of transfer functions has been proven to be effective in the setting of structured linear and bilinear systems, we are aiming for a similar structure-preserving model order reduction approach for quadratic-bilinear systems, e.g., for quadratic-bilinear mechanical systems \begin{align*} \begin{aligned} 0 & = M\ddot{q}(t) + D\dot{q} + Kq(t)\ & \quad{}+{}...
Let $A,S,X_0\in \mathbb R^{n\times n}$ and $ B \in \mathbb R^{n\times b}$ be. Moreover assume that $S$, $X_0$ are symmetric positive semidefinite.
We consider
\begin{align}
\dot{X}(t) &= A^T X (t) + X(t) A - X(t) S X(t) + C^T C, \\
X(0)&=X_0.
\end{align}
It is well known that the solution $X$ can be obtained from the Hamiltonian system
\begin{align}
\begin{bmatrix}...
