This talk will include two parts: (1) a scalable fast iterative solver for the frequency-domain Maxwell equations at high frequencies and (2) a practice of characterization and optimal control of a real quantum computer. The frequency-domain Maxwell equations at high frequencies has indefinite nature, and as a result, it is challenging for iterative linear solvers to invert it efficiently. It also has high-resolution requirement, and the memory needed by standard multifrontal direct solvers may become prohibitive for practical high-frequency 3D problems. In the first part of the talk, we present an efficient scalable frequency-domain Maxwell solver built from scalable time-domain solvers called EM-WaveHoltz. Due to its high potential, quantum computing recently draw a lot of attention from the scientific community. In the second part of the talk, we briefly introduce some main difference between a quantum computer and a classical computer. We also use deterministic and Bayesian methods to characterize a quantum computer based on experimental data. Control pulses are designed based on the characterization results, and their performance is demonstrated through experimental validations.

Solving high-dimensional PDEs is a long-standing challenge in scientific computing. Luckily, studies discovered that the neural networks can be used to approximate a certain class of functions without the curse of dimensionality. It is therefore natural to expect that the solutions to high dimensional PDEs can be accurately approximated with the neural networks under controllable numerical costs. Along this line, three questions have been widely considered: 1) How can a PDE problem be formulated into an optimization problem so that it fits into the frame of deep learning? 2) How accurate will a neural network approximation be? 3) How could the training be conducted in a systematic way so that we could be close to a global minimum? In this talk, I will briefly describe a few initial attempts I have made to answer the questions above.

Motivated by $\mathcal{H}_2$-optimal model order reduction (MOR) for non-parametric linear time-invariant (LTI) systems and $\mathcal{H}_2 \otimes \mathcal{L}_2$-optimal parametric MOR for parametric LTI systems, we investigate $\mathcal{L}_2$-optimal parametric MOR for parametric stationary problems arising from, e.g., discretizations of parametric stationary partial differential equations. We first develop gradients of the squared $\mathcal{L}_2$ error with respect to the reduced system operators, which then leads to a gradient-based optimization method for MOR of parametric stationary problems. We also illustrate that the optimization algorithm can be performed purely in a data-driven manner using only the samples of the quantities of interest without access to full-order operators. Furthermore, we develop interpolatory conditions for optimal MOR of a class of parametric stationary problems. Finally, we discuss MOR methods based on (Petrov-)Galerkin projection and whether $\mathcal{L}_2$-optimal reduced-order models are necessarily of such type. We illustrate the theory via various numerical examples and compare our framework to standard projection-based approaches.

Dynamical modeling of a process is essential to study its dynamical behavior and perform engineering studies such as control and optimization. With the ease of accessibility of data, learning models directly from the data have recently drawn much attention. It is also desirable to construct simple models describing complex nonlinear dynamics for efficient simulations and engineering studies. The simplest model--one can think of--is the linear model, but they are often not expressive enough to model complex dynamics. In this work, we propose \emph{McCormick-envelope} inspired modeling of nonlinear dynamics and discuss a common framework to model nonlinear dynamic processes. The preeminent idea, coming from the envelope, is smooth nonlinear systems can be written as quadratic systems in appropriate lifted coordinate systems without any approximation. We utilize deep learning capabilities and discuss suitable neural network architectures to find such a coordinate system using data. We also discuss an extension to high-dimensional data, which exhibits a slow decay of singular values. We showcase the approach using data coming from applications in engineering and biology.

In many applied sciences applications, the underlying dynamics of the process under study may be inaccessible to direct modeling, or it may be only partially known. However, with the increasing prevalence of available data from practical experiments, it is of high relevance to include such measurements in the modeling process. Data corresponding to the underlying dynamical systems are available in various formats. For example, among others, in the form of the frequency response. In such cases, one could construct a simplified empirical model of lower dimension that fits the measured data, and hence accurately approximates the original system. This reduced-order system may then be used as a surrogate to predict behavior or derive control strategies. The main motivation of the methods discussed here is that measured system response data can be used in a beneficial way without the need of accessing any prescribed realization of the original model. The balanced truncation (BT) method (introduced by Moore '81) and Loewner-matrix (LM) methodologies (introduced by Mayo/Antoulas '07) are common model reduction approaches. However, only the latter approach (LM) is purely data-driven, while the first is not. We show how to implement a data-driven counterpart of the classical BT approach by using only frequency response data, without explicitly using the model (system's matrices). This recent method (G./Gugercin/Beattie '22) is based on implicitly imposing quadrature approximations on the infinite Gramians, and on constructing the reduced-order model by accessing the transfer function values only.