Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

Fantuzzi, Giovanni; Goluskin, David; Huang, Deqing; Chernyshenko, S. I.

doi:10.1137/15m1053347

Cited by 66 publications

(116 citation statements)

References 38 publications

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“…The first step in this approach is to construct near-optimal auxiliary functions using SOS optimisation. We prove that this is always possible with a small modification to the computational methods described in the previous works [13][14][15]. This follows from the argument in [24] using a standard argument in SOS optimisation, and is the dual version of Theorem 2 in [23].…”

Section: Introductionmentioning

confidence: 76%

“…Tobasco et al [24] proved that (4) is a well-posed optimisation problem and that there exist optimal initial conditions a * 0 such that Φ(a * 0 ) = Φ * . In fact, there are clearly infinitely many such optimal initial conditions because Φ[a(t ; a * 0 )] = Φ(a * 0 ) for any fixed time t. The same authors also proved that Φ * can be characterised equivalently as the optimal value of a minimisation problem, originally proposed in [13] and further studied in [14][15][16]31], over continuously differentiable auxiliary functions V : R n → R. Precisely,…”

Section: Infinite-time Averages and Auxiliary Functionsmentioning

confidence: 99%

“…This has already been clearly demonstrated by the results in [16,36], where nearly optimal polynomial auxiliary functions for the Lorenz system and the Kuramoto-Sivashinsky equation were constructed numerically using SOS optimisation. In section 4 we show that if f and Φ are polynomials and Ω is a compact semialgebraic set subject to a mild technical condition, then arbitrarily sharp bounds on Φ * and the corresponding near-optimal auxiliary functions can be constructed numerically using a variation of the computational approach utilised in [13,14,16,36].…”

Section: Infinite-time Averages and Auxiliary Functionsmentioning

confidence: 99%

“…To see this, suppose that a fully optimal auxiliary function V * exists and is available. Then, the extremal trajectory a(t) must satisfy [14,24] λ…”

Section: Approximation Of Extremal Trajectoriesmentioning

confidence: 99%

“…By increasing d, the degree of V , one obtains a sequence of nonincreasing bounds on Φ * . This is the approach followed in [13,14,16,36].…”

Section: Optimising Auxiliary Functions With Sos Optimisationmentioning

confidence: 99%

See 4 more Smart Citations

Finding Extremal Periodic Orbits with Polynomial Optimization, with Application to a Nine-Mode Model of Shear Flow

Lakshmi

Fantuzzi²,

Fernández-Caballero³

et al. 2020

SIAM J. Appl. Dyn. Syst.

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Tobasco et al. [Physics Letters A, 382:382-386, 2018] recently suggested that trajectories of ODE systems which optimise the infinite-time average of a certain observable can be localised using sublevel sets of a function that arise when bounding such averages using socalled auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimisation is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localise the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed efficiently through direct nonlinear optimisation. Such points provide good initial conditions for UPO computations. We then combine these methods with a singleshooting Netwon-Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow. We discover three previously unknown families of UPOs, one of which simultaneously minimises the mean energy dissipation rate and maximises the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125.

show abstract

Section: Introductionmentioning

confidence: 76%

Section: Infinite-time Averages and Auxiliary Functionsmentioning

confidence: 99%

Section: Infinite-time Averages and Auxiliary Functionsmentioning

confidence: 99%

“…To see this, suppose that a fully optimal auxiliary function V * exists and is available. Then, the extremal trajectory a(t) must satisfy [14,24] λ…”

Section: Approximation Of Extremal Trajectoriesmentioning

confidence: 99%

“…By increasing d, the degree of V , one obtains a sequence of nonincreasing bounds on Φ * . This is the approach followed in [13,14,16,36].…”

Section: Optimising Auxiliary Functions With Sos Optimisationmentioning

confidence: 99%

See 3 more Smart Citations

Finding Extremal Periodic Orbits with Polynomial Optimization, with Application to a Nine-Mode Model of Shear Flow

Lakshmi

Fantuzzi²,

Fernández-Caballero³

et al. 2020

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

show abstract

Bootstrapping the gap in quantum spin systems

Nancarrow,

Xin

2023

J. High Energ. Phys.

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In this work we report on a new bootstrap method for quantum mechanical problems that closely mirrors the setup from conformal field theory (CFT). We use the equations of motion to develop an analogue of the conformal block expansion for matrix elements and impose crossing symmetry in order to place bounds on their values. The method can be applied to any quantum mechanical system with a local Hamiltonian, and we test it on an anharmonic oscillator model as well as the (1 + 1)-dimensional transverse field Ising model (TFIM). For the anharmonic oscillator model we show that a small number of crossing equations provides an accurate solution to the spectrum and matrix elements. For the TFIM we show that the Hamiltonian equations of motion, translational invariance and global symmetry selection rules imposes a rigorous bound on the gap and the matrix elements of TFIM in the thermodynamic limit. The bound improves as we consider larger systems of crossing equations, ruling out more finite-volume solutions. Our method provides a way to probe the low energy spectrum of an infinite lattice from the Hamiltonian rigorously and without approximation.

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Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

Goluskin

2017

J Nonlinear Sci

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We describe methods for proving bounds on infinite-time averages in differential dynamical systems. The methods rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proved using Lyapunov functions. Nonnegativity is enforced by requiring the polynomials to be sums of squares, a condition which is then formulated as a semidefinite program (SDP) that can be solved computationally. Although such computations are subject to numerical error, we demonstrate two ways to obtain rigorous results: using interval arithmetic to control the error of an approximate SDP solution, and finding exact analytical solutions to relatively small SDPs. Previous formulations are extended to allow for bounds depending analytically on parametric variables. These methods are illustrated using the Lorenz equations, a system with three state variables (x, y, z) and three parameters (β, σ, r). Bounds are reported for infinite-time averages of all eighteen moments x l y m z n up to quartic degree that are symmetric under (x, y) → (−x, −y). These bounds apply to all solutions regardless of stability, including chaotic trajectories, periodic orbits, and equilibrium points. The analytical approach yields two novel bounds that are sharp: the mean of z 3 can be no larger than its value of (r − 1) 3 at the nonzero equilibria, and the mean of xy 3 must be nonnegative. The interval arithmetic approach is applied at the standard chaotic parameters to bound eleven average moments that all appear to be maximized on the shortest periodic orbit. Our best upper bound on each such average exceeds its value on the maximizing orbit by less than 1%. Many bounds reported here are much tighter than would be possible without computer assistance. *

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Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

Cited by 66 publications

References 38 publications

Finding Extremal Periodic Orbits with Polynomial Optimization, with Application to a Nine-Mode Model of Shear Flow

Finding Extremal Periodic Orbits with Polynomial Optimization, with Application to a Nine-Mode Model of Shear Flow

Bootstrapping the gap in quantum spin systems

Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

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