# Dr Giovanni Fantuzzi

Faculty of EngineeringDepartment of Aeronautics

Imperial College Research Fellow

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### Location

420BCity and Guilds BuildingSouth Kensington Campus

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## Publications

Publication Type
Year
to

28 results found

Lakshmi MV, Fantuzzi G, Chernyshenko SI, Lasagna Det al., 2021, Finding unstable periodic orbits: A hybrid approach with polynomial optimization

We present a novel method to compute unstable periodic orbits (UPOs) that optimize the infinite-time average of a given quantity for polynomial ODE systems. The UPO search procedure relies on polynomial optimization to construct nonnegative polynomials whose sublevel sets approximately localize parts of the optimal UPO, and that can be used to implement a simple yet effective control strategy to reduce the UPO's instability. Precisely, we construct a family of controlled ODE systems, parameterized by a scalar k, such that the original ODE system is recovered for k=0 and such that the optimal orbit is less unstable, or even stabilized, for k>0. Periodic orbits for the controlled system can often be more easily converged with traditional methods, and numerical continuation in k allows one to recover optimal UPOs for the original system. The effectiveness of this approach is illustrated on three low-dimensional ODE systems with chaotic dynamics.

Working paper

Arslan A, Fantuzzi G, Craske J, Wynn Aet al., 2021, Bounds on internally heated convection with fixed boundary heat flux, Journal of Fluid Mechanics, Vol: 992, Pages: R1-R1, ISSN: 0022-1120

We prove a new rigorous bound for the mean convective heat transport ⟨wT⟩, where w and T are the non-dimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity ⟨wT⟩ is equal to half the ratio of convective to conductive vertical heat transport, and also to 12 plus the mean temperature difference between the top and bottom boundaries. An analytical application of the background method based on the construction of a quadratic auxiliary function yields ⟨wT⟩≤12(12+13√)−1.6552R−(1/3) uniformly in the Prandtl number, where R is the non-dimensional control parameter measuring the strength of the internal heating. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the −1/3 exponent are optimal within our bounding framework. This new result halves the best existing (uniform in R) bound (Goluskin, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, table 1.2), and its dependence on R is consistent with previous conjectures and heuristic scaling arguments. Contrary to physical intuition, however, it does not rule out a mean heat transport larger than 12 at high R, which corresponds to the top boundary being hotter than the bottom one on average.

Journal article

Fantuzzi G, Arslan A, Wynn A, 2021, The background method: Theory and computations

Working paper

Arslan A, Fantuzzi G, Craske J, Wynn Aet al., 2021, Bounds on heat transport for convection driven by internal heating, Journal of Fluid Mechanics, Vol: 919, Pages: 1-34, ISSN: 0022-1120

The mean vertical heat transport ⟨wT⟩ in convection between isothermal plates driven by uniform internal heating is investigated by means of rigorous bounds. These are obtained as a function of the Rayleigh number R by constructing feasible solutions to a convex variational problem, derived using a formulation of the classical background method in terms of quadratic auxiliary functions. When the fluid's temperature relative to the boundaries is allowed to be positive or negative, numerical solution of the variational problem shows that best previous bound ⟨wT⟩≤1/2 can only be improved up to finite R. Indeed, we demonstrate analytically that ⟨wT⟩≤2−21/5R1/5 and therefore prove that ⟨wT⟩<1/2 for R<65536. However, if the minimum principle for temperature is invoked, which asserts that internal temperature is at least as large as the temperature of the isothermal boundaries, then numerically optimised bounds are strictly smaller than 1/2 until at least R=3.4×105. While the computational results suggest that the best bound on ⟨wT⟩ approaches 1/2 asymptotically from below as R→∞, we prove that typical analytical constructions cannot be used to prove this conjecture.

Journal article

Zheng Y, Fantuzzi G, Papachristodoulou A, 2021, Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization

Chordal and factor-width decomposition methods for semidefinite programmingand polynomial optimization have recently enabled the analysis and control oflarge-scale linear systems and medium-scale nonlinear systems. Chordaldecomposition exploits the sparsity of semidefinite matrices in a semidefiniteprogram (SDP), in order to formulate an equivalent SDP with smallersemidefinite constraints that can be solved more efficiently. Factor-widthdecompositions, instead, relax or strengthen SDPs with dense semidefinitematrices into more tractable problems, trading feasibility or optimality forlower computational complexity. This article reviews recent advances inlarge-scale semidefinite and polynomial optimization enabled by these two typesof decomposition, highlighting connections and differences between them. Wealso demonstrate that chordal and factor-width decompositions allow forsignificant computational savings on a range of classical problems from controltheory, and on more recent problems from machine learning. Finally, we outlinepossible directions for future research that have the potential to facilitatethe efficient optimization-based study of increasingly complex large-scaledynamical systems.

Working paper

Zheng Y, Fantuzzi G, 2020, Sum-of-squares chordal decomposition of polynomial matrix inequalities

We prove decomposition theorems for sparse positive (semi)definite polynomialmatrices that can be viewed as sparsity-exploiting versions of theHilbert--Artin, Reznick, Putinar, and Putinar--Vasilescu Positivstellens\"atze.First, we establish that a polynomial matrix $P(x)$ with chordal sparsity ispositive semidefinite for all $x\in \mathbb{R}^n$ if and only if there exists asum-of-squares (SOS) polynomial $\sigma(x)$ such that $\sigma P$ is a sum ofsparse SOS matrices. Second, we show that setting $\sigma(x)=(x_1^2 + \cdots +x_n^2)^\nu$ for some integer $\nu$ suffices if $P$ is homogeneous and positivedefinite globally. Third, we prove that if $P$ is positive definite on acompact semialgebraic set $\mathcal{K}=\{x:g_1(x)\geq 0,\ldots,g_m(x)\geq 0\}$satisfying the Archimedean condition, then $P(x) = S_0(x) + g_1(x)S_1(x) +\cdots + g_m(x)S_m(x)$ for matrices $S_i(x)$ that are sums of sparse SOSmatrices. Finally, if $\mathcal{K}$ is not compact or does not satisfy theArchimedean condition, we obtain a similar decomposition for $(x_1^2 + \ldots +x_n^2)^\nu P(x)$ with some integer $\nu\geq 0$ when $P$ and $g_1,\ldots,g_m$are homogeneous of even degree. Using these results, we find sparse SOSrepresentation theorems for polynomials that are quadratic and correlativelysparse in a subset of variables, and we construct new convergent hierarchies ofsparsity-exploiting SOS reformulations for convex optimization problems withlarge and sparse polynomial matrix inequalities. Numerical examples demonstratethat these hierarchies can have a significantly lower computational complexitythan traditional ones.

Working paper

Zheng Y, Fantuzzi G, Papachristodoulou A, Goulart P, Wynn Aet al., 2020, Chordal decomposition in operator-splitting methods for sparse semidefinite programs, MATHEMATICAL PROGRAMMING, Vol: 180, Pages: 489-532, ISSN: 0025-5610

Journal article

Fantuzzi G, Nobili C, Wynn A, 2020, New bounds on the vertical heat transport for Benard-Marangoni convection at infinite Prandtl number, JOURNAL OF FLUID MECHANICS, Vol: 885, ISSN: 0022-1120

Journal article

Pachev B, Whitehead JP, Fantuzzi G, Grooms Iet al., 2020, Rigorous bounds on the heat transport of rotating convection with Ekman pumping, JOURNAL OF MATHEMATICAL PHYSICS, Vol: 61, ISSN: 0022-2488

Journal article

Lakshmi M, Fantuzzi G, Fernandez-Caballero JD, Hwang Y, Chernyshenko Set al., 2020, Finding Extremal Periodic Orbits with Polynomial Optimization, with Application to a Nine-Mode Model of Shear Flow, SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, Vol: 19, Pages: 763-787, ISSN: 1536-0040

Journal article

Fantuzzi G, Goluskin D, 2020, Bounding Extreme Events in Nonlinear Dynamics Using Convex Optimization, SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, Vol: 19, Pages: 1823-1864, ISSN: 1536-0040

Journal article

Zheng Y, Fantuzzi G, Papachristodoulou A, 2019, Fast ADMM for sum-of-squares programs using partial orthogonality, IEEE Transactions on Automatic Control, Vol: 64, Pages: 3869-3876, ISSN: 0018-9286

IEEE When sum-of-squares (SOS) programs are recast as semidefinite programs (SDPs) using the standard monomial basis, the constraint matrices in the SDP possess a structural property that we call partial orthogonality. In this paper, we leverage partial orthogonality to develop a fast first-order method, based on the alternating direction method of multipliers (ADMM), for the solution of the homogeneous self-dual embedding of SDPs describing SOS programs. Precisely, we show how a &#x201C;diagonal plus low rank&#x201D; structure implied by partial orthogonality can be exploited to project efficiently the iterates of a recent ADMM algorithm for generic conic programs onto the set defined by the affine constraints of the SDP. The resulting algorithm, implemented as a new package in the solver CDCS, is tested on a range of large-scale SOS programs arising from constrained polynomial optimization problems and from Lyapunov stability analysis of polynomial dynamical systems. These numerical experiments demonstrate the effectiveness of our approach compared to common state-of-the-art solvers.

Journal article

Fantuzzi G, 2019, Duality of convex relaxations for constrained variational problems

We prove weak duality between two recent convex relaxation methods forbounding the optimal value of a constrained variational problem in which theobjective is an integral functional. The first approach, proposed by Valmorbidaet al. (IEEE Trans. Automat. Control 61(6):1649--1654, 2016), replaces thevariational problem with a convex program over sufficiently smooth functions,subject to pointwise non-negativity constraints. The second approach, discussedby Korda et al. (arXiv:1804.07565v1 [math.OC]), relaxes the variational probleminto a convex program over scaled probability measures. We also prove that theduality between these infinite-dimensional convex programs is strong, meaningthat their optimal values coincide, when the range and gradients of admissiblefunctions in the variational problem are constrained to bounded sets. Forvariational problems with polynomial data, the optimal values of each convexrelaxation can be approximated by solving weakly dual hierarchies offinite-dimensional semidefinite programs (SDPs). These are strongly dual understandard constraint qualification conditions irrespective of whether strongduality holds at the infinite-dimensional level. Thus, the two relaxationapproaches are equivalent for the purposes of computations.

Working paper

Goluskin D, Fantuzzi G, 2019, Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming, NONLINEARITY, Vol: 32, Pages: 1705-1730, ISSN: 0951-7715

Journal article

Zheng Y, Fantuzzi G, Papachristodoulou A, 2019, Sparse sum-of-squares (SOS) optimization: A bridge between DSOS/SDSOS and SOS optimization for sparse polynomials, American Control Conference (ACC), Publisher: IEEE, Pages: 5513-5518, ISSN: 0743-1619

Conference paper

Fantuzzi G, Pershin A, Wynn A, 2018, Bounds on heat transfer for Benard-Marangoni convection at infinite Prandtl number, JOURNAL OF FLUID MECHANICS, Vol: 837, Pages: 562-596, ISSN: 0022-1120

Journal article

Fantuzzi G, 2018, Bounds for Rayleigh-Benard convection between free-slip boundaries with an imposed heat flux, JOURNAL OF FLUID MECHANICS, Vol: 837, ISSN: 0022-1120

Journal article

Zheng Y, Fantuzzi G, Papachristodoulou A, 2018, Decomposition methods for large-scale semidefinite programs with chordal aggregate sparsity and partial orthogonality, Large-Scale and Distributed Optimization, Editors: Giselsson, Rantzer, Publisher: Springer International Publishing AG, Pages: 33-55, ISBN: 978-3-319-97477-4

Many semidefinite programs (SDPs) arising in practical applications have useful structural properties that can be exploited at the algorithmic level. In this chapter, we review two decomposition frameworks for large-scale SDPs characterized by either chordal aggregate sparsity or partial orthogonality. Chordal aggregate sparsity allows one to decompose the positive semidefinite matrix variable in the SDP, while partial orthogonality enables the decomposition of the affine constraints. The decomposition frameworks are particularly suitable for the application of first-order algorithms. We describe how the decomposition strategies enable one to speed up the iterations of a first-order algorithm, based on the alternating direction method of multipliers, for the solution of the homogeneous self-dual embedding of a primal-dual pair of SDPs. Precisely, we give an overview of two structure-exploiting algorithms for semidefinite programming, which have been implemented in the open-source MATLAB solver CDCS. Numerical experiments on a range of large-scale SDPs demonstrate that the decomposition methods described in this chapter promise significant computational gains.

Book chapter

Zheng Y, Fantuzzi G, Papachristodoulou A, 2018, Decomposition and Completion of Sum-of-Squares Matrices, 57th IEEE Conference on Decision and Control (CDC), Publisher: IEEE, Pages: 4026-4031, ISSN: 0743-1546

Conference paper

Fantuzzi G, Wynn A, Goulart PJ, Papachristodoulou Aet al., 2017, Optimization With Affine Homogeneous Quadratic Integral Inequality Constraints, IEEE TRANSACTIONS ON AUTOMATIC CONTROL, Vol: 62, Pages: 6221-6236, ISSN: 0018-9286

Journal article

Zheng Y, Fantuzzi G, Papachristodoulou A, 2017, Exploiting Sparsity in the Coefficient Matching Conditions in Sum-of-Squares Programming Using ADMM, IEEE CONTROL SYSTEMS LETTERS, Vol: 1, Pages: 80-85, ISSN: 2475-1456

Journal article

Fantuzzi G, Wynn A, 2017, Exact energy stability of Bénard–Marangoni convection at infinite Prandtl number, Journal of Fluid Mechanics, Vol: 822, ISSN: 1469-7645

Using the energy method we investigate the stability of pure conduction in Pearson’s model for Bénard–Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh–Bénard convection set-up, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.

Journal article

Zheng Y, Fantuzzi G, Papachristodoulou A, Goulart P, Wynn Aet al., 2017, Fast ADMM for Semidefinite Programs with Chordal Sparsity, American Control Conference (ACC), Publisher: IEEE, Pages: 3335-3340, ISSN: 0743-1619

Conference paper

Zheng Y, Fantuzzi G, Papachristodoulou A, Goulart P, Wynn Aet al., 2017, Fast ADMM for homogeneous self-dual embedding of sparse SDPs, 20th World Congress of the International-Federation-of-Automatic-Control (IFAC), Publisher: ELSEVIER, Pages: 8411-8416, ISSN: 2405-8963

Conference paper

Fantuzzi G, Wynn A, 2016, Optimal bounds with semidefinite programming: An application to stress driven shear flows, Physical Review E, Vol: 93, ISSN: 1550-2376

We introduce an innovative numerical technique based on convex optimization to solve a range ofinfinite dimensional variational problems arising from the application of the background method tofluid flows. In contrast to most existing schemes, we do not consider the Euler-Lagrange equationsfor the minimizer. Instead, we use series expansions to formulate a finite dimensional semidefiniteprogram (SDP) whose solution converges to that of the original variational problem. Our formulationaccounts for the influence of all modes in the expansion and the feasible set of the SDP is strictlycontained within the feasible set of the original problem. Moreover, SDPs can be easily formulatedwhen the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditionalmethods. We apply this technique to compute rigorous and near-optimal upper bounds on thedissipation coefficient for flows driven by a surface stress. We improve previous analytical boundsby more than 10 times, and show that the bounds become independent of the domain aspect ratioin the limit of vanishing viscosity. We also confirm that the dissipation properties of stress drivenflows are similar to those of flows subject to a body force localized in a narrow layer near the surface.Finally, we show that SDP relaxations are an efficient method to investigate the energy stability oflaminar flows driven by a surface stress.

Journal article

Fantuzzi G, Wynn A, 2016, Semidefinite relaxation of a class of quadratic integral inequalities, 55th IEEE Conference on Decision and Control (CDC), Publisher: IEEE, Pages: 6192-6197, ISSN: 0743-1546

Conference paper

Fantuzzi G, Goluskin D, Huang D, Chernyshenko SIet al., 2016, Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization, SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, Vol: 15, Pages: 1962-1988, ISSN: 1536-0040

Journal article

Fantuzzi G, Wynn A, 2015, Construction of an optimal background profile for the Kuramoto-Sivashinsky equation using semidefinite programming, PHYSICS LETTERS A, Vol: 379, Pages: 23-32, ISSN: 0375-9601

Journal article

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