1,048 research outputs found
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.
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
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
Single and Double Photoionization and Photodissociation of Toluene by Soft X-rays in Circumstellar Environment
The formation of polycyclic aromatic hydrocarbons (PAHs) and their methyl
derivatives occurs mainly in the dust shells of asymptotic giant branch (AGB)
stars. The bands at 3.3 and 3.4 m, observed in infrared emission spectra
of several objects, are attributed C-H vibrational modes in aromatic and
aliphatic structures, respectively. In general, the feature at 3.3 m is
more intense than the 3.4 m. Photoionization and photodissociation
processes of toluene, the precursor of methylated PAHs, were studied using
synchrotron radiation at soft X-ray energies around the carbon K edge with
time-of-flight mass spectrometry. Partial ion yields of a large number of ionic
fragments were extracted from single and 2D-spectra, where electron-ion
coincidences have revealed the doubly charged parent-molecule and several
doubly charged fragments containing seven carbon atoms with considerable
abundance. \textit{Ab initio} calculations based on density functional theory
were performed to elucidate the chemical structure of these stable dicationic
species. The survival of the dications subjected to hard inner shell ionization
suggests that they could be observed in the interstellar medium, especially in
regions where PAHs are detected. The ionization and destruction of toluene
induced by X-rays were examined in the T Dra conditions, a carbon-rich AGB
star. In this context, a minimum photodissociation radius and the half-life of
toluene subjected to the incidence of the soft X-ray flux emitted from a
companion white dwarf star were determined.Comment: 11 pages, 4 figures, accept for publication in Ap
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
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
Analytical bounds on the heat transport in internally heated convection
We obtain an analytical bound on the mean vertical convective heat flux 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 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 that approach 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
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