## Publications

32 results found

Arslan A, Fantuzzi G, Craske J, et al., 2022, Rigorous scaling laws for internally heated convection at infinite Prandtl number

Kumar A, Arslan A, Fantuzzi G,
et al., 2022, Analytical bounds on the heat transport in internally heated convection, *Journal of Fluid Mechanics*, ISSN: 0022-1120

We obtain an analytical bound on the mean vertical convective heat flux$\langle w T \rangle$ between two parallel boundaries driven by uniforminternal heating. We consider two configurations, one with both boundaries heldat the same constant temperature, and the other one with a top boundary held atconstant temperature and a perfectly insulating bottom boundary. For the firstconfiguration, Arslan et al. (J. Fluid Mech. 919:A15, 2021) recently providednumerical evidence that Rayleigh-number-dependent corrections to the only knownrigorous bound $\langle w T \rangle \leq 1/2$ may be provable if the classicalbackground method is augmented with a minimum principle stating that thefluid's temperature is no smaller than that of the top boundary. Here, weconfirm this fact rigorously for both configurations by proving bounds on$\langle wT \rangle$ that approach $1/2$ exponentially from below as theRayleigh number is increased. The key to obtaining these bounds are innerboundary layers in the background fields with a particular inverse-powerscaling, which can be controlled in the spectral constraint using Hardy andRellich inequalities. These allow for qualitative improvements in the analysisnot available to standard constructions.

Fantuzzi G, 2022, Verification of some functional inequalities via polynomial optimization

Lakshmi MV, Fantuzzi G, Chernyshenko SI,
et al., 2021, Finding unstable periodic orbits: A hybrid approach with polynomial optimization, *Physica D: Nonlinear Phenomena*, Vol: 427, ISSN: 0167-2789

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.

Fantuzzi G, Arslan A, Wynn A, 2021, The background method: Theory and computations, *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 380, ISSN: 1364-503X

The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader "auxiliary function" framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh-Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined.

Zheng Y, Fantuzzi G, 2021, Sum-of-squares chordal decomposition of polynomial matrix inequalities, *Mathematical Programming*, ISSN: 0025-5610

We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert--Artin, Reznick, Putinar, and Putinar--Vasilescu Positivstellensätze. First, we establish that a polynomial matrix $P(x)$ with chordal sparsity is positive semidefinite for all $x\in \mathbb{R}^n$ if and only if there exists a sum-of-squares (SOS) polynomial $\sigma(x)$ such that $\sigma P$ is a sum of sparse 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 positive definite globally. Third, we prove that if $P$ is positive definite on a compact 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 SOS matrices. Finally, if $\mathcal{K}$ is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for $(x_1^2 + \cdots + x_n^2)^\nu P(x)$ 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 SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones.

Chernyavsky A, Bramburger JJ, Fantuzzi G, et al., 2021, Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization, Publisher: ArXiv

We present a method for finding lower bounds on the global infima of integralvariational problems, wherein $\int_\Omega f(x,u(x),\nabla u(x)){\rm d}x$ isminimized over functions $u\colon\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$satisfying given equality or inequality constraints. Each constraint may beimposed over $\Omega$ or its boundary, either pointwise or in an integralsense. These global minimizations are generally non-convex and intractable. Weformulate a particular convex maximization, here called the pointwise dualrelaxation (PDR), whose supremum is a lower bound on the infimum of theoriginal problem. The PDR can be derived by dualizing and relaxing the originalproblem; its constraints are pointwise equalities or inequalities overfinite-dimensional sets, rather than over infinite-dimensional function spaces.When the original minimization can be specified by polynomial functions of$(x,u,\nabla u)$, the PDR can be further relaxed by replacing pointwiseinequalities with polynomial sum-of-squares (SOS) conditions. The resulting SOSprogram is computationally tractable when the dimensions $m,n$ and number ofconstraints are not too large. The framework presented here generalizes anapproach of Valmorbida, Ahmadi, and Papachristodoulou (IEEE Trans. Automat.Contr., 61:1649--1654, 2016). We prove that the optimal lower bound given bythe PDR is sharp for several classes of problems, whose special cases includeleading eigenvalues of Sturm-Liouville problems and optimal constants ofPoincar\'e inequalities. For these same classes, we prove that SOS relaxationsof the PDR converge to the sharp lower bound as polynomial degrees areincreased. Convergence of SOS computations in practice is illustrated forseveral examples.

Arslan A, Fantuzzi G, Craske J,
et 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.

Arslan A, Fantuzzi G, Craske J,
et 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.

Zheng Y, Fantuzzi G, Papachristodoulou A, 2021, Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization, *Annual Reviews in Control*, Vol: 52, Pages: 243-279, ISSN: 1367-5788

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

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- Citations: 18

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

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- Citations: 2

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

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- Citations: 1

Lakshmi M, Fantuzzi G, Fernandez-Caballero JD,
et 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

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- Citations: 6

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

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- Citations: 7

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 “diagonal plus low rank” 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.

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.

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

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- Citations: 21

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

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- Citations: 5

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

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- Citations: 11

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

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- Citations: 8

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

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- Citations: 3

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.

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

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- Citations: 5

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

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- Citations: 12

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.

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

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- Citations: 24

Zheng Y, Fantuzzi G, Papachristodoulou A, et 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

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- Citations: 7

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.

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

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- Citations: 2

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