Optimal bounds with semidefinite programming: an application to stress driven shear flows

We introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing schemes, we do not consider the Euler--Lagrange equations for the minimizer. Instead, we use series expansions to formulate a finite dimensional semidefinite program (SDP) whose solution converges to that of the original variational problem. Our formulation accounts for the influence of all modes in the expansion, and the feasible set of the SDP corresponds to a subset of the feasible set of the original problem. Moreover, SDPs can be easily formulated when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional methods. We apply this technique to compute rigorous and near-optimal upper bounds on the dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds by more than 10 times, and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress driven flows 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 of laminar flows driven by a surface stress.Comment: 17 pages; typos removed; extended discussion of linear matrix inequalities in Section III; revised argument in Section IVC, results unchanged; extended discussion of computational setup and limitations in Sectios IVE-IVF. Submitted to Phys. Rev.

Bounds for Rayleigh-Bénard convection between free-slip boundaries with an imposed heat flux

We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh-Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number $Nu$ is bounded by $Nu \leq 0.5999 R^{1/3}$ uniformly in the Prandtl number, where $R$ is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, $Ra$, we obtain $Nu \leq 0.4646 Ra^{1/2}$. The scaling with Rayleigh number is the same as that of bounds obtained with no-slip isothermal, free-slip isothermal, and no-slip fixed flux boundaries, and numerical optimisation of the bound suggests that it cannot be improved within our bounding framework. Contrary to the two-dimensional case, therefore, the $Ra$-dependence of rigorous upper bounds on the heat transfer obtained with the background method for three-dimensional Rayleigh-Bénard convection is insensitive to both the thermal and the velocity boundary conditions

Semidefinite relaxation of a class of quadratic integral inequalities

We propose a novel technique to solve optimization problems subject to a class of integral inequalities whose integrand is quadratic and homogeneous with respect to the dependent variables, and affine in the parameters. We assume that the dependent variables are subject to homogeneous boundary conditions. Specifically, we derive rigorous relaxations of such integral inequalities in terms of semidefinite constraints, so a strictly feasible and near-optimal point for the original problem can be computed using semidefinite programming. Simple examples arising from the stability analysis of partial differential equations illustrate the potential of our method compared to existing techniques

Fast ADMM for sum-of-squares programs using partial orthogonality

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

Exploiting Sparsity in the Coefficient Matching Conditions in Sum-of-Squares Programming Using ADMM

This letter introduces an efficient first-order method based on the alternating direction method of multipliers (ADMM) to solve semidefinite programs arising from sum-of-squares (SOS) programming. We exploit the sparsity of the coefficient matching conditions when SOS programs are formulated in the usual monomial basis to reduce the computational cost of the ADMM algorithm. Each iteration of our algorithm requires one projection onto the positive semidefinite cone and the solution of multiple quadratic programs with closed-form solutions free of any matrix inversion. Our techniques are implemented in the open-source MATLAB solver SOSADMM. Numerical experiments on SOS problems arising from unconstrained polynomial minimization and from Lyapunov stability analysis for polynomial systems show speed-ups compared to the interior-point solver SeDuMi, and the first-order solver CDCS

The background method: Theory and computations

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

Rigorous scaling laws for internally heated convection at infinite Prandtl number

New bounds are proven on the mean vertical convective heat transport, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯, for uniform internally heated (IH) convection in the limit of infinite Prandtl number. For fluid in a horizontally-periodic layer between isothermal boundaries, we show that ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−2, where R is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and c=216. Then, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯=0 corresponds to vertical heat transport by conduction alone, while ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯>0 represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−4, with c≈0.0107. This result implies that the Nusselt number Nu, defined as the ratio of the total-to-conductive heat transport, satisfies Nu≲R4. Both bounds are obtained by combining the background method with a minimum principle for the fluid's temperature and with Hardy--Rellich inequalities to exploit the link between the vertical velocity and temperature. In both cases, power-law dependence on R improves the previously best-known bounds, which, although valid at both infinite and finite Prandtl numbers, approach the uniform bound exponentially with R

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

Accuracy of eight-polar bioelectrical impedance analysis for the assessment of total and appendicular body composition in peritoneal dialysis patients

Objective: To establish the accuracy of bioelectrical impedance analysis (BIA) for the assessment of total and appendicular body composition in peritoneal dialysis (PD) patients. Design: Cross-sectional study. Setting: University Nephrology Clinic. Subjects: In all, 20 PD patients and 77 healthy controls matched for gender, age and body mass index. Methods: Whole-body fat-free mass (FFM) and appendicular lean tissue mass (LTM) were measured by dual-energy X-ray absorptiometry. Resistance ( R) of arms, trunk and legs was measured by eight-polar BIA at frequencies of 5, 50, 250 and 500 kHz. Whole-body resistance was calculated as the sum of R of arms, trunk and legs. The resistance index ( RI) was calculated as the ratio between squared height and whole-body or segmental R. Results: RI at 500 kHz was the best predictor of FFM, LTMarm and LTMleg in both PD patients and controls. Equations developed on controls overestimated FFM and LTMarm and underestimated LTMleg when applied to PD patients. Specific equations were thus developed for PD patients. Using these equations, the percent root mean-squared errors of the estimate for PD patients vs controls were 5 vs 6% for FFM, 8 vs 8% for LTMarm and 7 vs 8% for LTMleg. Conclusion: Eight-polar BIA offers accurate estimates of total and appendicular body composition in PD patients, provided that population-specific equations are used

Analytical bounds on the heat transport in internally heated convection

We obtain an analytical bound on the mean vertical convective heat flux $\langle w T \rangle$ between two parallel boundaries driven by uniform internal heating. We consider two configurations, one with both boundaries held at the same constant temperature, and the other one with a top boundary held at constant temperature and a perfectly insulating bottom boundary. For the first configuration, Arslan et al. (J. Fluid Mech. 919:A15, 2021) recently provided numerical evidence that Rayleigh-number-dependent corrections to the only known rigorous bound $\langle w T \rangle \leq 1/2$ may be provable if the classical background method is augmented with a minimum principle stating that the fluid's temperature is no smaller than that of the top boundary. Here, we confirm this fact rigorously for both configurations by proving bounds on $\langle wT \rangle$ that approach $1/2$ exponentially from below as the Rayleigh number is increased. The key to obtaining these bounds are inner boundary layers in the background fields with a particular inverse-power scaling, which can be controlled in the spectral constraint using Hardy and Rellich inequalities. These allow for qualitative improvements in the analysis not available to standard constructions