Quantitative Stability Results Research Articles

A Talenti-type comparison theorem for the [formula omitted]-Laplacian on [formula omitted] spaces and some applications

In this paper, we prove a Talenti-type comparison theorem for the p-Laplacian with Dirichlet boundary conditions on open subsets of a normalized RCD(K,N) space with K>0 and N∈(1,∞). The obtained Talenti-type comparison theorem is sharp, rigid and stable with respect to measured Gromov–Hausdorff topology. As an application of such Talenti-type comparison, we establish a sharp and rigid reverse Hölder inequality for first eigenfunctions of the p-Laplacian and a related quantitative stability result.

Exploring a modification of [formula omitted] convergence

In the work by M. C. Lee, A. Naber, and R. Neumayer (Lee et al., 2023) a beautiful ɛ-regularity theorem is proved under small negative scalar curvature and entropy bounds. In that paper, the dp distance for Riemannian manifolds is introduced and the quantitative stability results are given in terms of this notion of distance, with important examples showing why other existing notions of convergence are not adequate in their setting. Due to the presence of an entropy bound, the possibility of long, thin splines forming along a sequence of Riemannian manifolds whose scalar curvature is becoming almost positive is ruled out. It is important to rule out such examples since the dp distance is not well behaved in the presences of splines that persist in the limit. Since there are many geometric stability conjectures (Gromov, 2023; Sormani, 2023) where we want to allow for the presence of splines that persist in the limit, it is crucial to be able to modify the dp distance to retain its positive qualities and prevent it from being sensitive to splines. In this paper we explore one such modification of the dp distance and give a theorem which allows one to estimate the modified dp distance, which we expect to be useful in practice.

A quantitative stability result for the sphere packing problem in dimensions 8 and 24

Abstract We prove explicit stability estimates for the sphere packing problem in dimensions 8 and 24, showing that, in the lattice case, if a lattice is ∼ ε {\sim\varepsilon} close to satisfying the optimal density, then it is, in a suitable sense, close to the E 8 {E_{8}} and Leech lattices, respectively. In the periodic setting, we prove that, under the same assumptions, we may take a large “frame” through which our packing locally looks like E 8 {E_{8}} or Λ 24 {\Lambda_{24}} . Our methods make explicit use of the magic functions constructed in [M. S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 2017, 3, 991–1015] in dimension 8 and in [H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 2017, 3, 1017–1033] in dimension 24, together with results of independent interest on the abstract stability of the lattices E 8 {E_{8}} and Λ 24 {\Lambda_{24}} .

A quantitative stability result for the Prékopa–Leindler inequality for arbitrary measurable functions

We prove that if a triplet of functions satisfies almost-equality in the Prékopa–Leindler inequality, then these functions are close to a common log-concave function, up to multiplication and rescaling. Our result holds for general measurable functions in all dimensions, and provides a quantitative stability estimate with computable constants.

Controlled polyhedral sweeping processes: Existence, stability, and optimality conditions

This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations.

A reverse Hölder inequality for first eigenfunctions of the Dirichlet Laplacian on 𝑅𝐶𝐷(𝐾,𝑁) spaces

In the framework of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by a positive constant in a synthetic sense, we establish a sharp and rigid reverse-Hölder inequality for first eigenfunctions of the Dirichlet Laplacian. This generalises to the positively curved and non-smooth setting the classical “Chiti Comparison Theorem”. We also prove a related quantitative stability result which seems to be new even for smooth Riemannian manifolds.

Uncertainty relations for multiple operators without covariances

In this paper, we prove the sum and product uncertainty relations conjectured by V Dodonov for multiple observables. The uncertainty relations for linear combinations of position and momentum recently obtained by Kechrimparis and Weigert are recovered. Furthermore, the entropic uncertainty relations conjectured by the latter authors are proved for specific cases. At last, we revisit the uncertainty relation for triple canonical operators and obtain a tighter bound on real Hilbert space. A quantitative stability result is given as well.

Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle

<p style='text-indent:20px;'>We prove quantitative statistical stability results for a large class of small <inline-formula><tex-math id="M1">\begin{document}$ C^{0} $\end{document}</tex-math></inline-formula> perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.</p>

Quantitative stability and numerical analysis of Markovian quadratic BSDEs with reflection

<p style='text-indent:20px;'>We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations (BSDEs) with bounded terminal data. By virtue of bounded mean oscillation martingale and change of measure techniques, we obtain stability estimates for the variation of the solutions with different underlying forward processes. In addition, we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.</p>

Quantitative Stability and Empirical Approximation of Risk-Averse Models Induced by Two-Stage Stochastic Programs with Full Random Recourse

In this paper, we consider the quantitative stability analysis and empirical approximation of risk-averse models induced by two-stage stochastic programs with full random recourse. We first establish the quantitative stability under the mean-coherent risk measure framework and the expected utility framework, respectively, under suitable probability metrics. Based on the obtained quantitative stability results, we then investigate the empirical approximation to these models, and estimate the rates of convergence for the optimal value and optimal solution set with the aid of Ky Fan distance.

A sharp stability result for the Gauss mean value formula

We prove a quantitative stability result for the Gauss mean value formula. We also show by an example that the estimate proved here is sharp.

Quantitative stability for the Heisenberg–Pauli–Weyl inequality

We prove a quantitative stability result for the Heisenberg–Pauli–Weyl inequality. This leads to a next, and next-to-next order correction terms in the inequality.

On distributionally robust optimization problems with k-th order stochastic dominance constraints induced by full random quadratic recourse

In this paper, we consider the optimization problems with k-th order stochastic dominance constraint on the objective function of the two-stage stochastic programs with full random quadratic recourse. By establishing the Lipschitz continuity of the feasible set mapping under some pseudo-metric, we show the Lipschitz continuity of the optimal value function and the upper semicontinuity of the optimal solution mapping of the problem. Furthermore, by the Hölder continuity of parameterized ambiguity set under the pseudo-metric, we demonstrate the quantitative stability results of the feasible set mapping, the optimal value function and the optimal solution mapping of the corresponding distributionally robust problem.

Stability of a class of risk-averse multistage stochastic programs and their distributionally robust counterparts

In this paper, we consider the quantitative stability of a class of risk-averse multistage stochastic programs, whose objective functions are defined by multi-period \begin{document}$ p $\end{document} th order lower partial moments (LPM) with given targets, and their distributionally robust counterparts. We first derive the upper bounds of feasible solutions as preliminaries. Then, by employing calm modifications, the quantitative stability results are obtained under a special measurable perturbation of stochastic process, which extend the present results under risk-neutral cases to risk-averse ones. Moreover, we recast the risk-averse model by probability measures of stochastic process, and obtain new quantitative stability estimations on the basis of proper probability metrics under the general perturbation of stochastic process. Finally, motivated by the availability of only partial information about probability measures, we further consider the distributionally robust counterpart of our recasting model, and establish the discrepancy of optimal values with respect to the perturbation of ambiguity sets.

The micro-hybrid method to assess transient stability quantitatively in pooled off-grid microgrids

In interconnected off-grid microgrids an in-depth knowledge of dynamic limits can be obtained by performing quantitative transient stability assessment. Quantitative stability results using the new hybrid method can be used for optimal generation dispatching, adjusting protection system settings, optimal clustering of microgrids etc. However, a discrepancy in the profiles of kinetic and potential energy has been noticed. Therefore, in this research work an improved new hybrid method, the so-called micro-hybrid method, was developed by adjusting the transient energy function, which is valid for transmission systems. Further, the new approach was validated and applied in a stand-alone multi-machine microgrid comprising detailed dynamic models of diesel generators with fast-reacting controllers. Finally, the transient stability of three clustered microgrids was assessed quantitatively. From the results it can be concluded that the profiles of the energies obtained by using the micro-hybrid method are exact mirror images. The critical fault clearing times can be corrected based on the profiles of the system stability degree and stability reserve degree of individual generators for different clearing times. Lastly, pooling of the investigated microgrids has a negative impact on the critical clearing times as well as on the profiles of system stability degree with respect to various clearing times.

Quantitative Stability Analysis of Two-Stage Stochastic Linear Programs with Full Random Recourse

In this paper, we apply the parametric linear programing technique and pseudo metrics to study the quantitative stability of the two-stage stochastic linear programing problem with full random recourse. Under the simultaneous perturbation of the cost vector, coefficient matrix, and right-hand side vector, we first establish the locally Lipschitz continuity of the optimal value function and the boundedness of optimal solutions of parametric linear programs. On the basis of these results, we deduce the locally Lipschitz continuity and the upper bound estimation of the objective function of the two-stage stochastic linear programing problem with full random recourse. Then by adopting different pseudo metrics, we obtain the quantitative stability results of two-stage stochastic linear programs with full random recourse which improve the current results under the partial randomness in the second stage problem. Finally, we apply these stability results to the empirical approximation of the two-stage stochastic programing model, and the rate of convergence is presented.

Quantitative stability of full random two-stage problems with quadratic recourse

ABSTRACTIn this paper, we discuss quantitative stability of two-stage stochastic programs with quadratic recourse where all parameters in the second-stage problem are random. By establishing the Lipschitz continuity of the feasible set mapping of the restricted Wolfe dual of the second-stage quadratic programming in terms of the Hausdorff distance, we prove the local Lipschitz continuity of the integrand of the objective function of the two-stage stochastic programming problem and then establish quantitative stability results of the optimal values and the optimal solution sets when the underlying probability distribution varies under the Fortet–Mourier metric. Finally, the obtained results are applied to study the asymptotic behaviour of the empirical approximation of the model.

Quantitative Stability of Two-Stage Linear Second-Order Conic Stochastic Programs with Full Random Recourse

In this paper, we consider quantitative stability for full random two-stage linear stochastic program with second-order conic constraints when the underlying probability distribution is subjected to perturbation. We first investigate locally Lipschitz continuity of feasible set mappings of the primal and dual problems in the sense of Hausdorff distance which derives the Lipschitz continuity of the objective function, and then establish the quantitative stability results of the optimal value function and the optimal solution mapping for the perturbation problem. Finally, the obtained results are applied to the convergence analysis of optimal values and solution sets for empirical approximations of the stochastic problems.

Quantitative stability for the Brunn–Minkowski inequality

We prove a quantitative stability result for the Brunn–Minkowski inequality: if |A|=|B|=1, t∈[τ,1−τ] with τ>0, and |tA+(1−t)B|1/n≤1+δ for some small δ, then, up to a translation, both A and B are quantitatively close (in terms of δ) to a convex set K.

Quantitative stability of the Brunn-Minkowski inequality for sets of equal volume

The authors prove a quantitative stability result for the Brunn-Minkowski inequality on sets of equal volume: If |A| = |B| > 0 and |A + B|1/n = (2+δ)|A|1/n for some small δ, then, up to a translation, both A and B are close (in terms of δ) to a convex set K. Although this result was already proved by the authors in a previous paper, the present paper provides a more elementary proof that the authors believe has its own interest. Also, the result here provides a stronger estimate for the stability exponent than the previous result of the authors.