Probabilistic Linear Solvers: A Unifying View

Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the matrix inverse. These approaches have typically focused on replicating the behavior of the conjugate gradient method as a prototypical iterative method. In this work surprisingly general conditions for equivalence of these disparate methods are presented. We also describe connections between probabilistic linear solvers and projection methods for linear systems, providing a probabilistic interpretation of a far more general class of iterative methods. In particular, this provides such an interpretation of the generalised minimum residual method. A probabilistic view of preconditioning is also introduced. These developments unify the literature on probabilistic linear solvers, and provide foundational connections to the literature on iterative solvers for linear systems

Probabilistic Methods for Model Validation

This dissertation develops a probabilistic method for validation and verification (V&V) of uncertain nonlinear systems. Existing systems-control literature on model and controller V&V either deal with linear systems with norm-bounded uncertainties,or consider nonlinear systems in set-based and moment based framework. These existing methods deal with model invalidation or falsification, rather than assessing the quality of a model with respect to measured data. In this dissertation, an axiomatic framework for model validation is proposed in probabilistically relaxed sense, that instead of simply invalidating a model, seeks to quantify the "degree of validation". To develop this framework, novel algorithms for uncertainty propagation have been proposed for both deterministic and stochastic nonlinear systems in continuous time. For the deterministic flow, we compute the time-varying joint probability density functions over the state space, by solving the Liouville equation via method-of-characteristics. For the stochastic flow, we propose an approximation algorithm that combines the method-of-characteristics solution of Liouville equation with the Karhunen-Lo eve expansion of process noise, thus enabling an indirect solution of Fokker-Planck equation, governing the evolution of joint probability density functions. The efficacy of these algorithms are demonstrated for risk assessment in Mars entry-descent-landing, and for nonlinear estimation. Next, the V&V problem is formulated in terms of Monge-Kantorovich optimal transport, naturally giving rise to a metric, called Wasserstein metric, on the space of probability densities. It is shown that the resulting computation leads to solving a linear program at each time of measurement availability, and computational complexity results for the same are derived. Probabilistic guarantees in average and worst case sense, are given for the validation oracle resulting from the proposed method. The framework is demonstrated for nonlinear robustness veri cation of F-16 flight controllers, subject to probabilistic uncertainties. Frequency domain interpretations for the proposed framework are derived for linear systems, and its connections with existing nonlinear model validation methods are pointed out. In particular, we show that the asymptotic Wasserstein gap between two single-output linear time invariant systems excited by Gaussian white noise, is the difference between their average gains, up to a scaling by the strength of the input noise. A geometric interpretation of this result allows us to propose an intrinsic normalization of the Wasserstein gap, which in turn allows us to compare it with classical systems-theoretic metrics like v-gap. Next, it is shown that the optimal transport map can be used to automatically refine the model. This model refinement formulation leads to solving a non-smooth convex optimization problem. Examples are given to demonstrate how proximal operator splitting based computation enables numerically solving the same. This method is applied for nite-time feedback control of probability density functions, and for data driven modeling of dynamical systems

Probabilistic Numerics and Uncertainty in Computations

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data has led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimisers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.Comment: Author Generated Postprint. 17 pages, 4 Figures, 1 Tabl

A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zero-th order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P

Probabilistic Interpretation of Linear Solvers

This manuscript proposes a probabilistic framework for algorithms that iteratively solve unconstrained linear problems $Bx = b$ with positive definite $B$ for $x$. The goal is to replace the point estimates returned by existing methods with a Gaussian posterior belief over the elements of the inverse of $B$, which can be used to estimate errors. Recent probabilistic interpretations of the secant family of quasi-Newton optimization algorithms are extended. Combined with properties of the conjugate gradient algorithm, this leads to uncertainty-calibrated methods with very limited cost overhead over conjugate gradients, a self-contained novel interpretation of the quasi-Newton and conjugate gradient algorithms, and a foundation for new nonlinear optimization methods.Comment: final version, in press at SIAM J Optimizatio

Abstraction-Based Data-Driven Control

Our world is living a paradigm shift in technology policy, often referred to as the Cyber-Physical Revolution or Industry 4.0. Nowadays, Cyber-Physical Systems are ubiquitous in modern control engineering, including automobiles, aircraft, building control systems, chemical plants, transportation systems, and so on. The interactions of the physical processes with the machines that control them are becoming increasingly complex, and in a growing number of situations either the model of the system is unavailable, or it is too difficult to describe accurately. Therefore, embedded computers need to "learn" the optimal way to control the systems by the mere observation of data. What seems the best approach to control these complex systems is often by discretizing the different variables, thus transforming the model into a combinatorial problem on a finite-state automaton, which is called an abstraction of the real system. Until now, this approach, often referred to as "abstraction-based control" or "symbolic control", has not been proved useful beyond small academic examples. In this project I aim to show the potential of this approach by implementing a novel data-driven approach based on a probabilistic interpretation of the discretization error. I have developed a toolbox (github.com/davidedl-ucl/master-thesis) implementing this kind of control with the aim of integrating it in the Dionysos software github.com/dionysos-dev). With this software, I succeeded in efficiently solving problems for non-linear control systems such as a path planning for an autonomous vehicle and a cart-pole balancing problem. The long-term objective of this project is to improve the methods implemented in my current software by employing a variable discretization of the state space and to consider complex specifications such as LTL formulas.Our world is living a paradigm shift in technology policy, often referred to as the Cyber-Physical Revolution or Industry 4.0. Nowadays, Cyber-Physical Systems are ubiquitous in modern control engineering, including automobiles, aircraft, building control systems, chemical plants, transportation systems, and so on. The interactions of the physical processes with the machines that control them are becoming increasingly complex, and in a growing number of situations either the model of the system is unavailable, or it is too difficult to describe accurately. Therefore, embedded computers need to "learn" the optimal way to control the systems by the mere observation of data. What seems the best approach to control these complex systems is often by discretizing the different variables, thus transforming the model into a combinatorial problem on a finite-state automaton, which is called an abstraction of the real system. Until now, this approach, often referred to as "abstraction-based control" or "symbolic control", has not been proved useful beyond small academic examples. In this project I aim to show the potential of this approach by implementing a novel data-driven approach based on a probabilistic interpretation of the discretization error. I have developed a toolbox (github.com/davidedl-ucl/master-thesis) implementing this kind of control with the aim of integrating it in the Dionysos software github.com/dionysos-dev). With this software, I succeeded in efficiently solving problems for non-linear control systems such as a path planning for an autonomous vehicle and a cart-pole balancing problem. The long-term objective of this project is to improve the methods implemented in my current software by employing a variable discretization of the state space and to consider complex specifications such as LTL formulas