Investigation of determinism-related issues in the Sobol′ low-discrepancy sequence for producing sound global sensitivity analysis indices

A computationally efficient and robust sampling scheme can support a sensitivity analysis of models to discover their behaviour through Quasi Monte Carlo approximation. This is especially useful for complex models, as often occur in environmental domains when model runtime can be prohibitive. The Sobol' sequence is one of the most used quasi-random low-discrepancy sequences as it can explore the parameter space significantly more evenly than pseudo-random sequences. The built-in determinism of the Sobol' sequence assists in achieving this attractive property. However, the Sobol' sequence tends to deteriorate in the sense that the estimated errors are distributed inconsistently across model parameters as the dimensions of a model increase. By testing multiple Sobol' sequence implementations, it is clear that the deterministic nature of the Sobol' sequence occasionally introduces relatively large errors in sensitivity indices produced by well-known global sensitivity analysis methods, and that the errors do not diminish by averaging through multiple replications. Problematic sensitivity indices may mistakenly guide modellers to make type I and II errors in trying to identify sensitive parameters, and this will potentially impact model reduction attempts based on these sensitivity measurements. This work investigates the cause of the Sobol' sequence's determinism-related issues. References I. A. Antonov and V. M. Saleev. An economic method of computing LPτ-sequences. USSR Comput. Math. Math. Phys. 19.1 (1979), pp. 252–256. doi: 10.1016/0041-5553(79)90085-5 P. Bratley and B. L. Fox. Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Soft. 14.1 (1988), pp. 88–100. doi: 10.1145/42288.214372 J. Feinberg and H. P. Langtangen. Chaospy: An open source tool for designing methods of uncertainty quantification. J. Comput. Sci. 11 (2015), pp. 46–57. doi: 10.1016/j.jocs.2015.08.008 on p. C90). S. Joe and F. Y. Kuo. Constructing Sobol sequences with better two-dimensional projections. SIAM J. Sci. Comput. 30.5 (2008), pp. 2635–2654. doi: 10.1137/070709359 S. Joe and F. Y. Kuo. Remark on algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Soft. 29.1 (2003), pp. 49–57. doi: 10.1145/641876.641879 W. J. Morokoff and R. E. Caflisch. Quasi-random sequences and their discrepancies. SIAM J. Sci. Comput. 15.6 (1994), pp. 1251–1279. doi: 10.1137/0915077 X. Sun, B. Croke, S. Roberts, and A. Jakeman. Comparing methods of randomizing Sobol’ sequences for improving uncertainty of metrics in variance-based global sensitivity estimation. Reliab. Eng. Sys. Safety 210 (2021), p. 107499. doi: 10.1016/j.ress.2021.107499 S. Tarantola, W. Becker, and D. Zeitz. A comparison of two sampling methods for global sensitivity analysis. Comput. Phys. Com. 183.5 (2012), pp. 1061–1072. doi: 10.1016/j.cpc.2011.12.015 S. Tezuka. Discrepancy between QMC and RQMC, II. Uniform Dist. Theory 6.1 (2011), pp. 57–64. url: https://pcwww.liv.ac.uk/~karpenk/JournalUDT/vol06/no1/5Tezuka11-1.pdf I. M. Sobol′. On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7.4 (1967), pp. 86–112. doi: 10.1016/0041-5553(67)90144-9 I. M. Sobol′. Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp 1.4 (1993), pp. 407–414.

Optimal L p – L q -Type Decay Rates of Solutions to the Three-Dimensional Nonisentropic Compressible Euler Equations with Relaxation

In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global small H k k ⩾ 3 solution only under the condition of smallness of the H 3 norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal L p – L q 1 ⩽ p ⩽ 2 , 2 ⩽ q ⩽ ∞ -type decay rates of the solution and its higher-order derivatives.

Modified scattering for the nonlinear nonlocal Schrödinger equation in one-dimensional case

We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity $$\begin{aligned} \left\{ \begin{array}{l} i\partial _{t}\left( u-\partial _{x}^{2}u\right) +\partial _{x}^{2}u-a\partial _{x}^{4}u=\lambda \left| u\right| ^{2}u,\text { } t>0,{\ }x\in {\mathbb {R}}\mathbf {,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,{\ }x\in {\mathbb {R}}\mathbf {,} \end{array} \right. \end{aligned}$$ i ∂ t u - ∂ x 2 u + ∂ x 2 u - a ∂ x 4 u = λ u 2 u , t > 0 , x ∈ R , u 0 , x = u 0 x , x ∈ R , where $$a>\frac{1}{5},$$ a > 1 5 , $$\lambda \in {\mathbb {R}}$$ λ ∈ R . We continue to develop the factorization techniques which was started in papers Hayashi and Naumkin (Z Angew Math Phys 59(6):1002–1028, 2008) for Klein–Gordon, Hayashi and Naumkin (J Math Phys 56(9):093502, 2015) for a fourth-order Schrödinger, Hayashi and Kaikina (Math Methods Appl Sci 40(5):1573–1597, 2017) for a third-order Schrödinger to show the modified scattering of solutions to the equation. The crucial points of our approach presented here are based on the $${\mathbf {L}}^{2}$$ L 2 -boundedness of the pseudodifferential operators.

Short wave limit of the Novikov equation and its integrable semi-discretizations

In this paper, we are concerned with one of the generalized short wave equations proposed by Hone et al. (Lett. Math. Phys 108 927 (2018)). We show that the derivative form of this equation can be viewed as a short wave limit of the Novikov (sw-Novikov) equation. Furthermore, this generalized short wave equation and its derivative form are found to be connected to period 3 reduction of two-dimensional CKP(BKP)-Toda hierarchy, same as the short wave limit of the Depasperis-Procesi (sw-DP) equation. We propose a two-component short wave equation which contain the sw-Novikov equation and sw-DP equation as two special cases. As a main result, we construct two types of integrable semi-discretizations via Hirota’s bilinear method and provide multi-soliton solution to the semi-discrete sw-Novikov equation.

On spectral asymptotic of quasi-exactly solvable quartic potential

Motivated by the earlier results of Masoero and De Benedetti (Nonlinearity 23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we discuss below the asymptotic of the solvable part of the spectrum for the quasi-exactly solvable quartic oscillator. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the level crossings of the latter oscillator with the asymptotic shape of zeros of the Yablonskii–Vorob’ev polynomials. Further we present a numerical study of the spectral monodromy for the oscillator in question.

Eigenstate Thermalization Hypothesis for Wigner Matrices

We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalisation Hypothesis by Deutsch (Phys Rev A 43:2046–2049, 1991) for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in Bourgade and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al. (Commun Pure Appl Math 73:1526–1596, 2020).

A PROOF OF MERCA’S CONJECTURES ON SUMS OF ODD DIVISOR FUNCTIONS

Merca [‘Congruence identities involving sums of odd divisors function’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca’s work, we complete the proof of these conjectures.

N-fold Darboux transformation of a six-field integrable lattice system

In this paper, we studied a semidiscrete coupled equation, which is integrable in the sense of admitting Lax representations. Proposed first by Vakhnenko in 2006, local conservation laws and one-fold Darboux transformation were presented with different forms, respectively, in O. O. Vakhnenko, J. Phys. Soc. Jpn. 84, 014003 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015). On the basis of these results, we principally construct [Formula: see text]-fold Darboux transformation by means of researching gauge transformation of its Lax pair, and work out its explicit multisolutions. Given a set of seed solutions and appropriate parameters, we can calculate two-soliton solutions and plot their figures when [Formula: see text].