Michael S. Ackermann

Ph.D candidate at Virginia Tech, Blacksburg, USA

Research Projects

Research Interests

Reduced order modeling of transport-dominated problems

Computational models are essential in modern engineering practice. Standard finite-element discretizations of governing partial differential equations can result in very large systems of coupled ordinary differential equations which may be computationally expensive to evaluate, especially in many-query applications. When the underlying physics are dispersive, standard linear model reduction and reduced order modeling techniques have been highly successful at lessening this computational burden. When the physics are dominated by transport phenomena such as moving shock waves, linear methods lose their effectiveness. Thus, specialized nonlinear reduced order modeling methods are required to handle transport-dominated problems. In particular, special care must be taken with the numerical implementation of these methods as the nonlinearities in the reduced modeling process often add significant computational burden to the online phase of the ROM. In this project, we develop a novel reduced modeling framework for transport-dominated problems which is both exceptionally accurate at predicting shock locations and results in a significant speed up over the full order model.

Data-driven modeling of structured dynamical systems

When a computational model is either completely unavailable or accessible only as a black-box, the only option to form a lower cost surrogate model is to learn a model from data. When some information about the underlying physics is available, this information should be exploited to enable reinterpretation of the model and ensure the data-driven surrogate is as faithful as possible. This leads to the structured data-driven reduced order modeling problem, where one optimizes parameters of a model form that enforces the appropriate structure. In this project, we develop an adaptive structured rational approximation algorithm for modeling second-order structured dynamical systems from measurements of its frequency response.

Optimal frequency-based reduced models from time-domain data

Reduced models of dynamical systems constructed from frequency (Laplace)-domain data come with powerful theoretical guarantees for their performance. However, at times frequency domain data is unavailable and must be inferred from time-domain data. While there exist methods to infer a system’s frequency response from time-domain simulations, using standard techniques to learn the system’s response to an exponentially growing input can be impractical due to restrictions on the power of the input. When constructing optimal reduced models from data, one requires samples of the system’s transfer function at points corresponding to exponentially growing inputs. In this project, we develop a computationally robust method to learn values and derivatives of a system’s transfer function at any point in the complex plane from only a single input-output trajectory in the time-domain. Further analysis enables an effective algorithm for learning optimal reduced models from a single input-output trajectory.

Adaptive rational approximation

Rational approximations enjoy exponential convergence rates to many classes of functions, including non-analytic functions and functions with branch cuts. While a very accurate rational approximation of a given degree may exist, computing this rational approximation is a different problem. In this project, we develop a novel method for rational approximation based on interpolation and nonlinear least-squares fitting of a provided data set. The method is adaptive in the sense that the degree of the rational approximation grows until the data are fit to a prescribed tolerance. Due to an adaptive initialization strategy for the nonlinear least-squares problems we solve at each step, the method is guaranteed to monotonically decrease the approximation error.