arXiv subjects

Melvin Leok

Publications and source records attributed to Melvin Leok.

At least 37 records · Page 2Linked to original sources

arXiv2022

Safe Control Synthesis with Uncertain Dynamics and Constraints

This paper considers safe control synthesis for dynamical systems with either probabilistic or worst-case uncertainty in both the dynamics model and the safety constraints. We formulate novel probabilistic and robust (worst-case) control Lyapunov function (CLF) and control barrier function (CBF) constraints that take into account the effect of uncertainty in either case. We show that either the probabilistic or the robust (worst-case) formulation leads to a second-order cone program (SOCP), which enables efficient safe and stable control synthesis. We evaluate our approach in PyBullet simulations of an autonomous robot navigating in unknown environments and compare the performance with a baseline CLF-CBF quadratic programming approach.

cs.RO↗

arXiv2022

Variational Structures in Cochain Projection Based Variational Discretizations of Lagrangian PDEs

Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite element methods. Multisymplectic variational integrators are a class of geometric numerical integrators for Lagrangian and Hamiltonian field theories, and they yield methods that preserve the multisymplectic structure and momentum-conservation properties of the continuous system. In this paper, we investigate the synthesis of these two approaches, by constructing discretization of the variational principle for Lagrangian field theories utilizing structure-preserving finite element projections. In our investigation, compatible discretization by cochain projections plays a pivotal role in the preservation of the variational structure at the discrete level, allowing the discrete variational structure to essentially be the restriction of the continuum variational structure to a finite-dimensional subspace. The preservation of the variational structure at the discrete level will allow us to construct a discrete Cartan form, which encodes the variational structure of the discrete theory, and subsequently, we utilize the discrete Cartan form to naturally state discrete analogues of Noether's theorem and multisymplecticity, which generalize those introduced in the discrete Lagrangian variational framework by Marsden et al. [29]. We will study both covariant spacetime discretization and canonical spatial semi-discretization, and subsequently relate the two in the case of spacetime tensor product finite element spaces.

math.NA↗

arXiv2022

Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups

This paper develops the theory of discrete Dirac reduction of discrete Lagrange-Dirac systems with an abelian symmetry group acting on the configuration space. We begin with the linear theory and, then, we extend it to the nonlinear setting using retraction compatible charts. We consider the reduction of both the discrete Dirac structure and the discrete Lagrange-Pontryagin principle, and show that they both lead to the same discrete Lagrange-Poincaré-Dirac equations. The coordinatization of the discrete reduced spaces relies on the notion of discrete connections on principal bundles. At last, we demonstrate the method obtained by applying it to a charged particle in a magnetic field, and to the double spherical pendulum.

math-ph↗

arXiv2022

Accelerated Optimization on Riemannian Manifolds via Discrete Constrained Variational Integrators

A variational formulation for accelerated optimization on normed vector spaces was recently introduced in Wibisono et al., and later generalized to the Riemannian manifold setting in Duruisseaux and Leok. This variational framework was exploited on normed vector spaces in Duruisseaux et al. using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization, and it was observed that geometric discretizations which respect the time-rescaling invariance and symplecticity of the Lagrangian and Hamiltonian flows were substantially less prone to stability issues, and were therefore more robust, reliable, and computationally efficient. As such, it is natural to develop time-adaptive Hamiltonian variational integrators for accelerated optimization on Riemannian manifolds. In this paper, we consider the case of Riemannian manifolds embedded in a Euclidean space that can be characterized as the level set of a submersion. We will explore how holonomic constraints can be incorporated in discrete variational integrators to constrain the numerical discretization of the Riemannian Hamiltonian system to the Riemannian manifold, and we will test the performance of the resulting algorithms by solving eigenvalue and Procrustes problems formulated as optimization problems on the unit sphere and Stiefel manifold.

math.NA↗

arXiv2022

High-order symplectic Lie group methods on $SO(n)$ using the polar decomposition

A variational integrator of arbitrarily high-order on the special orthogonal group $SO(n)$ is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second-order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie--Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on $SO(3)$ which demonstrate that they are comparable to variational Runge--Kutta--Munthe-Kaas methods in terms of computational efficiency. However, the methods we have proposed preserve the Lie group structure much more accurately and and exhibit better near energy preservation.

math.NA↗

arXiv2022

A Variational Formulation of Accelerated Optimization on Riemannian Manifolds

It was shown recently by Su et al. (2016) that Nesterov's accelerated gradient method for minimizing a smooth convex function $f$ can be thought of as the time discretization of a second-order ODE, and that $f(x(t))$ converges to its optimal value at a rate of $\mathcal{O}(1/t^2)$ along any trajectory $x(t)$ of this ODE. A variational formulation was introduced in Wibisono et al. (2016) which allowed for accelerated convergence at a rate of $\mathcal{O}(1/t^p)$, for arbitrary $p>0$, in normed vector spaces. This framework was exploited in Duruisseaux et al. (2021) to design efficient explicit algorithms for symplectic accelerated optimization. In Alimisis et al. (2020), a second-order ODE was proposed as the continuous-time limit of a Riemannian accelerated algorithm, and it was shown that the objective function $f(x(t))$ converges to its optimal value at a rate of $\mathcal{O}(1/t^2)$ along solutions of this ODE. In this paper, we show that on Riemannian manifolds, the convergence rate of $f(x(t))$ to its optimal value can also be accelerated to an arbitrary convergence rate $\mathcal{O}(1/t^p)$, by considering a family of time-dependent Bregman Lagrangian and Hamiltonian systems on Riemannian manifolds. This generalizes the results of Wibisono et al. (2016) to Riemannian manifolds and also provides a variational framework for accelerated optimization on Riemannian manifolds. An approach based on the time-invariance property of the family of Bregman Lagrangians and Hamiltonians was used to construct very efficient optimization algorithms in Duruisseaux et al. (2021), and we establish a similar time-invariance property in the Riemannian setting. One expects that a geometric numerical integrator that is time-adaptive, symplectic, and Riemannian manifold preserving will yield a class of promising optimization algorithms on manifolds.

math.OC↗

arXiv2022

Accelerated Optimization on Riemannian Manifolds via Projected Variational Integrators

A variational formulation of accelerated optimization on normed spaces was recently introduced by considering a specific family of time-dependent Bregman Lagrangian and Hamiltonian systems whose corresponding trajectories converge to the minimizer of the given convex function at an arbitrary accelerated rate of $\mathcal{O}(1/t^p)$. This framework has been exploited using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization. It was observed that geometric discretizations were substantially less prone to stability issues, and were therefore more robust, reliable, and computationally efficient. More recently, this variational framework has been extended to the Riemannian manifold setting by considering a more general family of time-dependent Bregman Lagrangian and Hamiltonian systems on Riemannian manifolds. It is thus natural to develop time-adaptive Hamiltonian variational integrators for accelerated optimization on Riemannian manifolds. In the past, Hamiltonian variational integrators have been constructed with holonomic constraints, but the resulting algorithms were implicit in nature, which significantly increased their cost per iteration. In this paper, we will test the performance of explicit methods based on Hamiltonian variational integrators combined with projections that constrain the numerical solution to remain on the constraint manifold.

math.OC↗

arXiv2021

Multisymplectic Hamiltonian Variational Integrators

Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics using Hamiltonian variational integrators. In this paper, we will extend these results to the setting of Hamiltonian multisymplectic field theories. We demonstrate that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law, and establish a discrete Noether's theorem for discretizations that are invariant under a Lie group action on the discrete dual jet bundle. In addition, we demonstrate that for spacetime tensor product discretizations, one can recover the multisymplectic integrators of Bridges and Reich, and show that a variational multisymplectic discretization of a Hamiltonian multisymplectic field theory using spacetime tensor product Runge--Kutta discretizations is well-defined if and only if the partitioned Runge--Kutta methods are symplectic in space and time.

math.NA↗

arXiv2021

Adaptive Hamiltonian Variational Integrators and Symplectic Accelerated Optimization

It is well known that symplectic integrators lose their near energy preservation properties when variable step sizes are used. The most common approach to combine adaptive step sizes and symplectic integrators involves the Poincaré transformation of the original Hamiltonian. In this article, we provide a framework for the construction of variational integrators using the Poincaré transformation. Since the transformed Hamiltonian is typically degenerate, the use of Hamiltonian variational integrators based on Type II or Type III generating functions is required instead of the more traditional Lagrangian variational integrators based on Type I generating functions. Error analysis is provided and numerical tests based on the Taylor variational integrator approach of Schmitt, Shingel, Leok (2018) to time-adaptive variational integration of Kepler's 2-Body problem are presented. Finally, we use our adaptive framework together with the variational approach to accelerated optimization presented in Wibisono, Wilson, Jordan (2016) to design efficient variational and non-variational explicit integrators for symplectic accelerated optimization.

math.NA↗

arXiv2021

Variational Symplectic Accelerated Optimization on Lie Groups

There has been significant interest in generalizations of the Nesterov accelerated gradient descent algorithm due to its improved performance guarantee compared to the standard gradient descent algorithm, and its applicability to large scale optimization problems arising in deep learning. A particularly fruitful approach is based on numerical discretizations of differential equations that describe the continuous time limit of the Nesterov algorithm, and a generalization involving time-dependent Bregman Lagrangian and Hamiltonian dynamics that converges at an arbitrarily fast rate to the minimum. We develop a Lie group variational discretization based on an extended path space formulation of the Bregman Lagrangian on Lie groups, and analyze its computational properties with two examples in attitude determination and vision-based localization.

math.OC↗

arXiv2019

Geometric Symmetry Reduction of the Unobservable Subspace for Kalman Filtering

In this article, we consider the implications of unobservable subspaces in the construction of a Kalman filter. In particular, we consider dynamical systems which are invariant with respect to a group action, and which are therefore unobservable in the group direction. We obtain reduced propagation and measurement equations that are invariant with respect to the group action, and we decompose the state space into unobservable and observable parts. Based on the decomposition, we propose a reduced Bayesian inference method, which exhibits superior accuracy for orientation and position estimation, and that is more robust to large measurement noise.

math.OC↗

arXiv2017

An Empirical Chaos Expansion Method for Uncertainty Quantification

Uncertainty quantification seeks to provide a quantitative means to understand complex systems that are impacted by parametric uncertainty. The polynomial chaos method is a computational approach to solve stochastic partial differential equations (SPDE) by projecting the solution onto a space of orthogonal polynomials of the stochastic variables and solving for the deterministic coefficients. Polynomial chaos can be more efficient than Monte Carlo methods when the number of stochastic variables is low, and the integration time is not too large. When performing long-term integration, however, achieving accurate solutions often requires the space of polynomial functions to become unacceptably large. This paper presents an alternative approach, where sets of empirical basis functions are constructed by examining the behavior of the solution for fixed values of the random variables. The empirical basis functions are evolved over time, which means that the total number can be kept small, even when performing long-term integration. We introduce this method of empirical chaos expansion, and apply it to a number of model equations, demonstrating that the computational time scales linearly with the final integration time. That is not the case for polynomial chaos in general, since achieving accuracy for long-term integration usually requires larger polynomial bases, causing a nonlinear scaling with the final integration time. We also present an analytical method that uses the dynamics of the SPDE to evolve the empirical basis functions and demonstrate how it can be applied to extend the validity of empirical basis functions in time without the need to sample additional realizations of the original SPDE.

math.NA↗

arXiv2017

High-Order Retractions on Matrix Manifolds using Projected Polynomials

We derive a family of high-order, structure-preserving approximations of the Riemannian exponential map on several matrix manifolds, including the group of unitary matrices, the Grassmannian manifold, and the Stiefel manifold. Our derivation is inspired by the observation that if $Ω$ is a skew-Hermitian matrix and $t$ is a sufficiently small scalar, then there exists a polynomial of degree $n$ in $tΩ$ (namely, a Bessel polynomial) whose polar decomposition delivers an approximation of $e^{tΩ}$ with error $O(t^{2n+1})$. We prove this fact and then leverage it to derive high-order approximations of the Riemannian exponential map on the Grassmannian and Stiefel manifolds. Along the way, we derive related results concerning the supercloseness of the geometric and arithmetic means of unitary matrices.

math.NA↗

arXiv2017

Lie group variational integrators for rigid body problems using quaternions

Rigid body dynamics on the rotation group have typically been represented in terms of rotation matrices, unit quaternions, or local coordinates, such as Euler angles. Due to the coordinate singularities associated with local coordinate charts, it is common in engineering applications to adopt the unit quaternion representation, and the numerical simulations typically impose the unit length condition using constraints or by normalization at each step. From the perspective of geometric structure-preserving, such approaches are undesirable as they are either computationally less efficient, or interfere with the preservation of other geometric properties of the dynamics. In this paper, we adopt the approach used in constructing Lie group variational integrators for rigid body dynamics on the rotation group to the representation in terms of unit quaternions. In particular, the rigid body dynamics is lifted to unit quaternions, and the Lie group structure of unit quaternions is used to represent tangent vector intrinsically, thereby avoiding the use of a Lagrange multiplier. A Lie group variational integrator in the unit quaternion representation is derived, and numerical results are presented.

math.NA↗

arXiv2017

Lagrangian and Hamiltonian Taylor Variational Integrators

In this paper, we present a variational integrator that is based on an approximation of the Euler--Lagrange boundary-value problem via Taylor's method. This can viewed as a special case of the shooting-based variational integrator. The Taylor variational integrator exploits the structure of the Taylor method, which results in a shooting method that is one order higher compared to other shooting methods based on a one-step method of the same order. In addition, this method can generate quadrature nodal evaluations at the cost of a polynomial evaluation, which may increase its efficiency relative to other shooting-based variational integrators. A symmetric version of the method is proposed, and numerical experiments are conducted to exhibit the efficacy and efficiency of the method.

math.NA↗

arXiv2017

Constructing equivalence-preserving Dirac variational integrators with forces

The dynamical motion of mechanical systems possesses underlying geometric structures, and preserving these structures in numerical integration improves the qualitative accuracy and reduces the long-time error of the simulation. For a single mechanical system, structure preservation can be achieved by adopting the variational integrator construction. This construction has been generalized to more complex systems involving forces or constraints as well as to the setting of Dirac mechanics. Variational integrators have recently been applied to interconnected systems in Parks and Leok (2017), which are an important class of practically useful mechanical systems whose description in terms of Dirac structures and Dirac mechanical systems was elucidated in Jacobs and Yoshimura (2014). Since these interconnected systems are modeled as a collection of subsystems with forces of interconnection, we revisit some of the properties of forced variational integrators. In particular, we derive a class of Dirac variational integrators with forces that exhibit preservation properties that are critical when applying variational integrators to the discretization of interconnected Dirac systems. We close with a discussion of ongoing and future research based on these findings.

math.NA↗

arXiv2017

Geometric Exponential Integrators

In this paper, we consider exponential integrators for semilinear Poisson systems. Two types of exponential integrators are constructed, one preserves the Poisson structure, and the other preserves energy. Numerical experiments for semilinear Possion systems obtained by semi-discretizing Hamiltonian PDEs are presented. These geometric exponential integrators exhibit better long time stability properties as compared to non-geometric integrators, and are computationally more efficient than traditional symplectic integrators and energy-preserving methods based on the discrete gradient method.

math.NA↗

arXiv2016

Properties of Hamiltonian Variational Integrators

Discrete Hamiltonian variational integrators are derived from Type II and Type III generating functions for symplectic maps, and in this paper we establish a variational error analysis result that relates the order of accuracy of the associated numerical methods with the extent to which these generating functions approximate the exact discrete Hamiltonians. We also introduce the notion of an adjoint discrete Hamiltonian, and relate it to the adjoint of the associated symplectic integrator. We show that when constructing discrete Lagrangians and discrete Hamiltonians using the Taylor variational integrator approach, the same underlying one-step method and quadrature rule does not necessarily lead to the same symplectic integrator, and the same observation holds when developing variational integrators based on averaging techniques. Numerical experiments also indicate that the resonance behavior of variational integrators also depend on the type of generating functions used, and we relate this resonance behavior to the ill-posedness of the boundary-value problems used to define the exact discrete Lagrangian and exact discrete Hamiltonian.

math.NA↗