## 2020 |

Galioto, Nicholas; Gorodetsky, Alex Arkady Bayesian System ID: optimal management of parameter, model, and measurement uncertainty Journal Article Forthcoming Forthcoming. Abstract | Links | BibTeX | Tags: Bayesian Inference, Dynamical Systems, Kalman Filtering, Markov Chain Monte Carlo, System ID @article{Galioto2020, title = {Bayesian System ID: optimal management of parameter, model, and measurement uncertainty}, author = {Nicholas Galioto and Alex Arkady Gorodetsky }, url = {http://www.alexgorodetsky.com/wp-content/uploads/2020/03/galioto_gorodetsky_bayesid.pdf}, year = {2020}, date = {2020-03-05}, abstract = { We evaluate the robustness of a probabilistic formulation of system identification (ID) to sparse, noisy, and indirect data. Specifically, we compare estimators of future system behavior derived from the Bayesian posterior of a learning problem to several commonly used least squares-based optimization objectives used in system ID. Our comparisons indicate that the log posterior has improved geometric properties compared with the objective function surfaces of traditional methods that include differentially constrained least squares and least squares reconstructions of discrete time steppers like dynamic mode decomposition (DMD). These properties allow it to be both more sensitive to new data and less affected by multiple minima -- overall yielding a more robust approach. Our theoretical results indicate that least squares and regularized least squares methods like dynamic mode decomposition and sparse identification of nonlinear dynamics (SINDy) can be derived from the probabilistic formulation by assuming noiseless measurements. We also analyze the computational complexity of a Gaussian filter-based approximate marginal Markov Chain Monte Carlo scheme that we use to obtain the Bayesian posterior for both linear and nonlinear problems. We then empirically demonstrate that obtaining the marginal posterior of the parameter dynamics and making predictions by extracting optimal estimators (e.g., mean, median, mode) yields orders of magnitude improvement over the aforementioned approaches. We attribute this performance to the fact that the Bayesian approach captures parameter, model, and measurement uncertainties, whereas the other methods typically neglect at least one type of uncertainty.}, keywords = {Bayesian Inference, Dynamical Systems, Kalman Filtering, Markov Chain Monte Carlo, System ID}, pubstate = {forthcoming}, tppubtype = {article} } We evaluate the robustness of a probabilistic formulation of system identification (ID) to sparse, noisy, and indirect data. Specifically, we compare estimators of future system behavior derived from the Bayesian posterior of a learning problem to several commonly used least squares-based optimization objectives used in system ID. Our comparisons indicate that the log posterior has improved geometric properties compared with the objective function surfaces of traditional methods that include differentially constrained least squares and least squares reconstructions of discrete time steppers like dynamic mode decomposition (DMD). These properties allow it to be both more sensitive to new data and less affected by multiple minima -- overall yielding a more robust approach. Our theoretical results indicate that least squares and regularized least squares methods like dynamic mode decomposition and sparse identification of nonlinear dynamics (SINDy) can be derived from the probabilistic formulation by assuming noiseless measurements. We also analyze the computational complexity of a Gaussian filter-based approximate marginal Markov Chain Monte Carlo scheme that we use to obtain the Bayesian posterior for both linear and nonlinear problems. We then empirically demonstrate that obtaining the marginal posterior of the parameter dynamics and making predictions by extracting optimal estimators (e.g., mean, median, mode) yields orders of magnitude improvement over the aforementioned approaches. We attribute this performance to the fact that the Bayesian approach captures parameter, model, and measurement uncertainties, whereas the other methods typically neglect at least one type of uncertainty. |

## 2017 |

Gorodetsky, Alex; Karaman, Sertac; Marzouk, Youssef Low-rank tensor integration for Gaussian filtering of continuous time nonlinear systems Inproceedings 2017 IEEE 56th Annual Conference on Decision and Control (CDC), pp. 2789-2794, Melbourne, VIC, Australia, 2017, ISBN: 978-1-5090-2873-3. Abstract | Links | BibTeX | Tags: Dynamical Systems, Kalman Filtering, Tensor decompositions @inproceedings{Gorodetsky2017, title = {Low-rank tensor integration for Gaussian filtering of continuous time nonlinear systems}, author = {Alex Gorodetsky and Sertac Karaman and Youssef Marzouk}, url = {https://alexgorodetsky.com/wp-content/uploads/2018/02/main.pdf}, doi = {10.1109/CDC.2017.8264064}, isbn = {978-1-5090-2873-3}, year = {2017}, date = {2017-12-12}, booktitle = {2017 IEEE 56th Annual Conference on Decision and Control (CDC)}, pages = {2789-2794}, address = {Melbourne, VIC, Australia}, abstract = {Integration-based Gaussian filters such as un-scented, cubature, and Gauss-Hermite filters are effective ways to assimilate data and models within nonlinear systems. Traditionally, these filters have only been applicable for systems with a handful of states due to stability and scalability issues. In this paper, we present a new integration method for scaling quadrature-based filters to higher dimensions. Our approach begins by decomposing the dynamics and observation models into separated, low-rank tensor formats. Once in low-rank tensor format, adaptive integration techniques may be used to efficiently propagate the mean and covariance of the distribution of the system state with computational complexity that is polynomial in dimension and rank. Simulation results are shown on nonlinear chaotic systems with 20 state variables.}, keywords = {Dynamical Systems, Kalman Filtering, Tensor decompositions}, pubstate = {published}, tppubtype = {inproceedings} } Integration-based Gaussian filters such as un-scented, cubature, and Gauss-Hermite filters are effective ways to assimilate data and models within nonlinear systems. Traditionally, these filters have only been applicable for systems with a handful of states due to stability and scalability issues. In this paper, we present a new integration method for scaling quadrature-based filters to higher dimensions. Our approach begins by decomposing the dynamics and observation models into separated, low-rank tensor formats. Once in low-rank tensor format, adaptive integration techniques may be used to efficiently propagate the mean and covariance of the distribution of the system state with computational complexity that is polynomial in dimension and rank. Simulation results are shown on nonlinear chaotic systems with 20 state variables. |

Gorodetsky, Alex Massachusetts Institute of Technology, 2017. Abstract | Links | BibTeX | Tags: Dynamic Programming, Kalman Filtering, Stochastic Optimal Control, Surrogate models, Tensor decompositions @phdthesis{Gorodetsky2017b, title = {Continuous low-rank tensor decompositions, with applications to stochastic optimal control and data assimilation}, author = {Alex Gorodetsky}, url = {http://hdl.handle.net/1721.1/108918}, year = {2017}, date = {2017-02-02}, address = {Cambridge, MA}, school = {Massachusetts Institute of Technology}, abstract = {Optimal decision making under uncertainty is critical for control and optimization of complex systems. However, many techniques for solving problems such as stochastic optimal control and data assimilation encounter the curse of dimensionality when too many state variables are involved. In this thesis, we propose a framework for computing with high-dimensional functions that mitigates this exponential growth in complexity for problems with separable structure. Our framework tightly integrates two emerging areas: tensor decompositions and continuous computation. Tensor decompositions are able to effectively compress and operate with low-rank multidimensional arrays. Continuous computation is a paradigm for computing with functions instead of arrays, and it is best realized by Chebfun, a MATLAB package for computing with functions of up to three dimensions. Continuous computation provides a natural framework for building numerical algorithms that effectively, naturally, and automatically adapt to problem structure. The first part of this thesis describes a compressed continuous computation framework centered around a continuous analogue to the (discrete) tensor-train decomposition called the function-train decomposition. Computation with the function-train requires continuous matrix factorizations and continuous numerical linear algebra. Continuous analogues are presented for performing cross approximation; rounding; multilinear algebra operations such as addition, multiplication, integration, and differentiation; and continuous, rank-revealing, alternating least squares. Advantages of the function-train over the tensor-train include the ability to adaptively approximate functions and the ability to compute with functions that are parameterized differently. For example, while elementwise multiplication between tensors of different sizes is undefined, functions in FT format can be readily multiplied together. Next, we develop compressed versions of value iteration, policy iteration, and multilevel algorithms for solving dynamic programming problems arising in stochastic optimal control. These techniques enable computing global solutions to a broader set of problems, for example those with non-affine control inputs, than previously possible. Examples are presented for motion planning with robotic systems that have up to seven states. Finally, we use the FT to extend integration-based Gaussian filtering to larger state spaces than previously considered. Examples are presented for dynamical systems with up to twenty states.}, keywords = {Dynamic Programming, Kalman Filtering, Stochastic Optimal Control, Surrogate models, Tensor decompositions}, pubstate = {published}, tppubtype = {phdthesis} } Optimal decision making under uncertainty is critical for control and optimization of complex systems. However, many techniques for solving problems such as stochastic optimal control and data assimilation encounter the curse of dimensionality when too many state variables are involved. In this thesis, we propose a framework for computing with high-dimensional functions that mitigates this exponential growth in complexity for problems with separable structure. Our framework tightly integrates two emerging areas: tensor decompositions and continuous computation. Tensor decompositions are able to effectively compress and operate with low-rank multidimensional arrays. Continuous computation is a paradigm for computing with functions instead of arrays, and it is best realized by Chebfun, a MATLAB package for computing with functions of up to three dimensions. Continuous computation provides a natural framework for building numerical algorithms that effectively, naturally, and automatically adapt to problem structure. The first part of this thesis describes a compressed continuous computation framework centered around a continuous analogue to the (discrete) tensor-train decomposition called the function-train decomposition. Computation with the function-train requires continuous matrix factorizations and continuous numerical linear algebra. Continuous analogues are presented for performing cross approximation; rounding; multilinear algebra operations such as addition, multiplication, integration, and differentiation; and continuous, rank-revealing, alternating least squares. Advantages of the function-train over the tensor-train include the ability to adaptively approximate functions and the ability to compute with functions that are parameterized differently. For example, while elementwise multiplication between tensors of different sizes is undefined, functions in FT format can be readily multiplied together. Next, we develop compressed versions of value iteration, policy iteration, and multilevel algorithms for solving dynamic programming problems arising in stochastic optimal control. These techniques enable computing global solutions to a broader set of problems, for example those with non-affine control inputs, than previously possible. Examples are presented for motion planning with robotic systems that have up to seven states. Finally, we use the FT to extend integration-based Gaussian filtering to larger state spaces than previously considered. Examples are presented for dynamical systems with up to twenty states. |