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Efficient finite element methods for semiclassical nonlinear Schr\"odinger equations with random potentials

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Pith's one-line read This paper proves an $L^2$ error bound for a time-splitting multiscale finite element method for the semiclassical nonlinear Schrödinger equation with random potentials, giving second-order accuracy in space and time and almost…

desk verdict A credible extension of MsFEM to the semiclassical cubic NLSE with random potentials, but the theory covers linear observables while the numerics use quadratic density, and the spatial error analysis leans on an imported localization result. read the letter →

arxiv 2502.07569 v1 pith:66PJX4HW submitted 2025-02-11 math.NA cs.NA

Panchi Li , Zhiwen Zhang This is my paper · ORCID
classification math.NAcs.NA MSC 35Q5565M6081Q0547H40
keywords semiclassicalnonlinearSchrödingerequationrandompotentialsmultiscalefiniteelementmethodtime-splittingmethodsquasi-MonteCarloKarhunen-LoèveexpansionerrorestimateAndersonlocalization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes two Strang time-splitting finite element methods for the semiclassical cubic nonlinear Schrödinger equation with a random potential, and couples them to multiscale finite element spaces to keep the spatial degrees of freedom low. Multiscale basis functions are generated by solving constrained energy-minimization problems, with a rescaling constant that prevents the nonlinear term from introducing a spurious mesh-size dependence. The central result is a root-mean-square error bound for the sampled expectation of any linear functional of the wave function: $C(H^2/\varepsilon^3+\Delta t^2/\varepsilon^4+m^{-\chi}/\varepsilon+C_{\gamma,m}N^{-r})$, which the paper interprets as second-order accuracy in space and time and almost first-order accuracy in the number of quasi-Monte Carlo samples. The paper also shows numerically that the scheme retains its order with discontinuous potentials and that it reproduces Anderson-style localization for the linear equation and delocalization as the nonlinearity strengthens.

What carries the argument

The multiscale basis functions are the central mechanism. They are obtained from the equality-constrained quadratic program $\min a(\phi,\phi)$ subject to $\int_D \phi\,\varphi_q^H\,dx=\lambda(H)\delta_{pq}$, where $a(\phi,\phi)=\frac{\epsilon^2}{2}\|\nabla\phi\|^2+(v\phi,\phi)$ and $\lambda(H)=(1,\varphi_q^H)$. This normalization is what suppresses the mesh-dependent scale in the nonlinear term. A weighted quasi-interpolation operator $I_H$ produces the splitting $V_h=V_{\mathrm{ms}}\oplus W_h$, and an iterative decay argument imported from earlier work gives the exponential localization of the basis. In time, Strang splitting with $L_1$ solved by eigendecomposition (SI) or by a Crank--Nicolson update (SII) avoids nonlinear iterations. In the random dimension, the truncated Karhunen--Loève expansion parameterizes the potential and randomly shifted lattice rules provide the sampling error term $C_{\gamma,m}N^{-r}$.

What would settle it

Compute, for a single random sample satisfying Assumption 2.1, the multiscale basis from (3.14)--(3.15) and measure two things: the decay of $\|\nabla\phi_p\|_{L^2(D\setminus D_\ell)}$ as $\ell$ grows, and the $L^2$ error of the projection of a known smooth function onto $V_{\mathrm{ms}}$ as $H\to0$. If the decay is not exponential with ratio $\beta<1$ or the error does not scale like $H^2/\varepsilon^3$, then the propositions behind Theorem 4.3 fail for that sample.

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Extended reading notes

Core claim

The paper's claim is that a time-splitting multiscale finite element method can solve the semiclassical nonlinear Schrödinger equation with random potentials at the rate stated in Theorem 4.3: if $\psi_{\mathrm{in}}\in H^4$ and the random potential satisfies Assumptions 2.1 and 2.2, the root-mean-square error between $E[G(\psi^\varepsilon(t_n))]$ and its quasi-Monte Carlo approximation $Q_{m,N}[G(\psi^{\varepsilon,n}_{H,m})]$ is bounded by $C(H^2/\varepsilon^3+\Delta t^2/\varepsilon^4+m^{-\chi}/\varepsilon+C_{\gamma,m}N^{-r})$. The spatial component of this bound comes from the multiscale basis construction (3.14)--(3.15): the basis functions minimize the energy with an orthogonality constraint whose scaling $\lambda(H)=(1,\varphi_q^H)$ cancels a mesh-dependent factor that would otherwise break the cubic nonlinearity. The paper further claims that the first splitting variant, which solves the linear part by eigendecomposition, keeps second-order convergence even for discontinuous potentials, while the classical variant requires smoother potentials and time steps of order $\epsilon$.

Load-bearing premise

The spatial half of the headline error bound assumes, from earlier work, that every optimized multiscale basis function stays exponentially localized and that the finite element space splits as $V_h=V_{\mathrm{ms}}\oplus W_h$ for every random potential sample with the assumed boundedness; the paper cites this instead of re-deriving or testing it.

Editorial extensions

If this is right

  • A coarse multiscale mesh with mesh size $H$ gives the same order of spatial accuracy as a fine finite element mesh, so solving on $V_{\mathrm{ms}}$ cuts the number of unknowns while retaining the $\Delta t^2$ temporal order.
  • The MsFEM spatial error depends on $\varepsilon$ only through $H^2/\varepsilon^3$ and needs only $\partial_t\psi^\varepsilon\in L^2$, whereas the classical FEM spatial estimate needs $H^2$ with $\partial_t\psi^\varepsilon\in H^2$; the authors use this to say the MsFEM is better adapted to the semiclassical regime.
  • With the SI splitting, the scheme is second-order in space even for discontinuous potentials, so the method can be used for rough random media.
  • In the random dimension, the quasi-Monte Carlo rate is $N^{-r}$ for any $r=1-\delta$ with $0<\delta<1/2$; combined with the KL truncation error $m^{-\chi}/\varepsilon$, this gives a full uncertainty-quantification estimate.
  • The numerical experiments predict a physical transition: mass density localizes for the linear equation and delocalizes when $\lambda$ is large, tracked by the moment $A(t)$. If these rates hold, the method is a practical tool for long-time wave-in-random-media simulations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The scale-cancelling choice $\lambda(H)=(1,\varphi_q^H)$ is specific to the cubic term; for other nonlinearities such as a quintic term, the same optimization-constraint mechanism would likely need a different normalization, so checking whether the advertised rate survives is a direct test of the mechanism.
  • The POD reduced-basis construction of Appendix A is demonstrated only as a timing improvement in 1D and 2D; an immediate question is whether the same offline/online decomposition preserves the $H^2/\varepsilon^3$ spatial error in 3D, where the constants in the regularity bounds are largest.
  • The analysis assumes the Karhunen--Loève eigenvalues decay polynomially and the potential is almost surely bounded; if the potential is lognormal or unbounded, the $m^{-\chi}$ and $N^{-r}$ terms would need re-derivation, and the numerical evidence here does not cover that case.
  • The finding that nonlinearity destroys Anderson localization is consistent with earlier studies; a natural next step would be to compare the delocalization threshold in $\lambda$ against predictions from lattice or random-matrix models, but the paper does not do that.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes two time-splitting finite element methods (SI and SII) for the semiclassical cubic nonlinear Schrödinger equation with random potentials, and combines them with a multiscale finite element method (MsFEM) whose basis functions are constructed from equality-constrained optimization problems. The main theoretical result is Theorem 4.3, which gives a root-mean-square error bound for the expectation of a linear functional of the solution in terms of H^2/ε^3, Δt^2/ε^4, m^{-χ}/ε, and C_{γ,m}N^{-r}. The paper also presents numerical experiments in 1D and 2D that show second-order spatial and temporal convergence and a near-first-order qMC convergence rate, and it describes a POD-based multiscale reduced basis method in Appendix A.

Significance. If the central error analysis is correct, this would be a useful contribution: it extends MsFEM error estimates to the semiclassical NLSE with random potentials and provides a rigorous rate in the random sampling dimension. The paper has several strengths: it proposes a concrete basis construction via optimization, it gives explicit convergence rates that are stated in a checkable form, and it includes numerical experiments with clean second-order rates and a comparison of MC and qMC. The POD reduced-basis appendix also addresses a practical bottleneck in constructing basis functions for many random samples. However, the significance is currently limited by a mismatch between the theory and the numerics (linear functionals vs. quadratic density) and by gaps in the proof of the load-bearing MsFEM spatial error estimate, as detailed below.

major comments (4)
  1. [§5.3, Theorem 4.3] Theorem 4.3 and Lemma 4.5 are stated only for continuous linear functionals G, yet all random-potential experiments in §5.3 compute the expected density E(|ψ|^2), which is a quadratic functional. The error analysis does not cover this quantity, and an extension to quadratic (or at least Lipschitz) functionals, with control on ∥ψ∥_{H^2}, would be needed before the numerical rates for E(|ψ|^2) can be cited as evidence for the theorem.
  2. [§3.1, Eq. (3.10)] The SI scheme implements the nonlinear substep as pointwise multiplication of nodal coefficients by exp(-iλΔt/(2ε)|U^n|^2). This is not the Galerkin discretization of the nonlinear flow in the FEM space and introduces an interpolation error that is not estimated. The analysis in Lemma 4.2 and Theorem 4.1 concerns the continuous splitting operator, while the fully discrete scheme in (3.10) is what is actually used; the notation L in (4.9) conflates the continuous and discrete operators. Consequently, the claimed O(H^2/ε^3) spatial error is not proven for the implemented scheme.
  3. [§4.2.1, Lemma 4.3] The proof first quotes a bound O(H^2/ε^2) from [52] for the linear Schrödinger step and then concludes (4.18) with O(H^2/ε^3); the additional factor 1/ε is not derived. The lemma is stated for the full NLSE, but the cubic nonlinear term is only treated through a stability factor, and the MsFEM error introduced by the nonlinear substep is not estimated. As a result, the central spatial error term used in Theorem 4.3 is not verified by the argument given.
  4. [§4.2, Prop. 3.2 and Theorem 4.3] The exponential decay estimate (3.22) is imported from the deterministic analysis [52] and is asserted for the present rescaled constraints and for each random potential sample, but the paper does not prove that the decay constant β is uniform in ω. Assumption 2.1 is not stated as a hypothesis of Theorem 4.3, and the numerical test in §5.3.1 appears to violate it: with σ=1, β=0, m=5, ε=1/8, and H≈0.063, one has ∥v∥∞≈8.66 while ε^2/H^2≈3.96. A uniform-in-ω version of Proposition 3.2, or at least a numerical verification of Assumption 2.1 for the experiments, is needed to support the stochastic MsFEM claim.
minor comments (5)
  1. [§4.1, proof of Theorem 4.2] The expression '∥Lnψin − Lnψin∥' should read '∥L^n ψ0_h − L^n ψin∥' to match the error split in (4.9).
  2. [§2, Eq. (2.7)] In the covariance kernel, the notation '|xi − yj|^2' appears to have an index error: the second index j should presumably be i in the sum over dimensions.
  3. [§5.3.2, Fig. 7] The measured convergence slopes for the MC and qMC curves are not reported; stating the numerical slopes would substantiate the claim of 'almost first-order' convergence in the random space.
  4. [§3.2, Prop. 3.1] The proof of Proposition 3.1 is only a sketch; a fuller derivation or a precise quotation of [52, Lemma 3.2] including the required resolution condition would improve verifiability.
  5. [§4.2, Lemma 4.5, Eq. (4.21)] The formula for ϱ(α) appears to have a parenthesis imbalance; please check and correct the expression.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence rates follow from proven regularity estimates and independent prior theorems, not from fitted data.

full rationale

The derivation chain of Theorem 4.3 is not equivalent to its inputs by construction. The temporal splitting error (Theorem 4.1) is proven from Strang-splitting stability and H^4 regularity; the FEM spatial error (Theorem 4.2) follows from a Ritz-projection estimate; the MsFEM spatial error (Lemma 4.3) uses Proposition 3.2 imported from [52], which is a published deterministic theorem with assumptions that do not include the current random NLSE target. The paper's change of constraint scale λ(H)=(1,φ_H^q) is, as the paper states, a rescaling that does not change the basis space, so the transfer of Proposition 3.2 is a scale-invariant statement and not a definitional reduction. The random-space part is self-contained: Lemma 4.4 is proved in Appendix C, and the qMC bound (Lemmas 4.5–4.6) rests on standard lattice-rule theory plus that regularity lemma. Numerical experiments validate the predicted second-order and near-first-order rates against fine-mesh reference solutions; no parameter is fitted and then renamed as a prediction. Concerns about whether every random sample satisfies the resolution condition of [52] are assumption-checking issues, not circularity. Therefore no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central estimate is built on imported prior results (the optimization-based MsFEM of [14], the LOD analysis of [52], and standard KL/qMC machinery) plus smoothness and random-field decay assumptions. The new normalization λ(H) = (1, φ^H_p) is a computed normalization, not a fitted free parameter, so no parameter is fitted to data in the paper.

assumptions (6)
  • domain assumption Assumption 2.1: ∥v(x,ω)∥∞ ≲ ε²/H².
    Binds the potential amplitude to the coarse mesh size; used implicitly in the exponential-decay arguments behind Proposition 3.2 and in Lemma 4.3. It restricts the class of random potentials to those compatible with a coarse mesh.
  • domain assumption Assumption 2.2: KL eigenvalue decay λj ≤ Cj^(−Θ), H² growth bound ∥vj∥H² ≤ Cλj^(−η), and truncation error ∥v − vm∥∞ ≤ Cm^(−χ).
    Needed for the random-field truncation error m^(−χ)/ε in Lemma 2.2 and Lemma 4.6, and for the qMC weight sums in Lemma 4.5.
  • domain assumption ψin ∈ H4 and ψε ∈ L∞([0,T]; H4) ∩ L1([0,T]; H2), with ∂tψε ∈ H2 for the FEM estimate and ∂tψε ∈ L2 for the MsFEM estimate.
    Used in Theorems 4.1-4.3 and Lemmas 2.1 and 4.1; semiclassical solutions may not meet this in the deep semiclassical limit without additional structure.
  • standard math Proposition 3.1 from [52]: the solution of (3.14)-(3.15) gives Vh = Vms ⊕ Wh with a-orthogonality a(φ, w) = 0.
    Imported as background from the authors' prior work; it is load-bearing for the MsFEM error analysis but not re-derived here.
  • standard math Proposition 3.2 from [52]: exponential decay ∥∇φp∥L2(D\Dℓ) ≤ βℓ∥∇φp∥.
    Also imported from [52]; it justifies using global basis functions and the coarse-space spatial convergence, and it is not verified experimentally in this paper.
  • standard math The product estimate ∥∇²(fgh)∥ ≤ C∥f∥H²∥g∥H²∥h∥H² from [49].
    Used in Appendix C to propagate H² bounds on mixed random derivatives in Lemma 4.4.

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Pith. "Pith review of Efficient finite element methods for semiclassical nonlinear Schr\"odinger equations with random potentials." pith.science (2026). https://pith.science/paper/66PJX4HW

@misc{pith2026250207569,
  author       = {Pith},
  title        = {Pith review of: Efficient finite element methods for semiclassical nonlinear Schr\"odinger equations with random potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66PJX4HW}},
  note         = {Machine review of arXiv:2502.07569}
}
abstract

In this paper, we propose two time-splitting finite element methods to solve the semiclassical nonlinear Schr\"odinger equation (NLSE) with random potentials. We then introduce the multiscale finite element method (MsFEM) to reduce the degrees of freedom in the physical space. We construct multiscale basis functions by solving optimization problems and rigorously analyze two time-splitting MsFEMs for the semiclassical NLSE with random potentials. We provide the $L^2$ error estimate of the proposed methods and show that they achieve second-order accuracy in both spatial and temporal spaces and an almost first-order convergence rate in the random space. Additionally, we present a multiscale reduced basis method to reduce the computational cost of constructing basis functions for solving random NLSEs. Finally, we carry out several 1D and 2D numerical examples to validate the convergence of our methods and investigate wave propagation behaviors in the NLSE with random potentials.

Figures

Figures reproduced from arXiv: 2502.07569 by the authors.

Figure 1
Figure 1. Numerical solution computed by the two TS-FEMs with different ∆t. 5.2. Numerical experiments of TS-MsFEMs. In this study, we consider two forms of the multi￾scale solution: ψ ϵ H on the coarse mesh and ψ ϵ H,h on the fine mesh. We begin by employing the harmonic potential and varying the values of H. We then record the error between the numerical solution and the reference solution in [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 2
Figure 2. Numerical convergence rate of SI and SII for the discontinuous potential. In the plots, the L 2 error and H1 error on the coarse mesh are depicted. Furthermore, we consider the small semiclassical constant ϵ = 1 128 and the discontinuous potential as in [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. The convergence rates of FEM and MsFEM for the NLSE with the discon￾tinuous potential and semiclassical constant ϵ = 1 128 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Reference solution (FEM) and the spatial error distribution computed by SI, in which the MsFEM is used with H = 8h and H = 4h [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Reference solution (FEM) and the spatial error distribution computed by SII, in which the MsFEM is used with H = 8h and H = 4h. 5.3. Numerical simulations of NLSE with random potentials. For the 1D case, we consider the random potential (5.2) v(x, ω) = σ Xm j=1 sin(jx)…
Figure 6
Figure 6. Figure 6: Numerical results computed by FEM and MsFEM with different time￾splitting methods for the NLSE with λ = 0 and λ = 1.0. 5.3.2. Convergence of MC sampling and qMC sampling. The MC method and qMC method exhibit different convergence rates. To eliminate the perturbation of…
Figure 7
Figure 7. Figure 7: Numerical convergence rates of the MC and qMC methods [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: The evolution of A(t) and density of expectation at T = 20, as the nonlinear coefficient λ varies. Results computed by the SI and MsFEM [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: The evolution of A(t) for 2D linear case and nonlinear case with λ = 20. Results are computed by SI and MsFEM [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: The localization and delocalization of mass distribution of the 2D linear Schr¨odinger equation and NLSE with random potentials, respectively [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Numerical comparison of FEM, MsFEM and the MsFEM with POD re￾duction methods. Furthermore, we vary the qMC samples and record the corresponding time costs in [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Numerical comparison of MsFEM method and the MsFEM with the POD reduction method for the 1D NLSE with λ = 20 [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

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Reference graph

Works this paper leans on

55 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [52]

    Wu and Z

    Z. Wu and Z. Zhang, Convergence analysis of the localized orthogonal decomposition method for the semiclassical Schr¨ odingerequations with multiscale potentials, J. Sci. Comput. 93 (2022), no. 3, 73

    work page 2022

  2. [1]

    G. D. Akrivis, V. A. Dougalis, and O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schr¨ odingerequation, Numer. Math. 59 (1991), no. 1, 31–53

    work page 1991

  3. [2]

    Altmann, P

    R. Altmann, P. Henning, and D. Peterseim, Numerical homogenization beyond scale separation, Acta Numer. 30 (2021), 1–86

    work page 2021

  4. [3]

    Antoine, W

    X. Antoine, W. Bao, and C. Besse, Computational methods for the dynamics of the nonlinear Schr¨ odinger/Gross-Pitaevskiiequations, Comput. Phys. Comm. 184 (2013), no. 12, 2621–2633

    work page 2013

  5. [4]

    Auzinger, T

    W. Auzinger, T. Kassebacher, O. Koch, and M. Thalhammer, Convergence of a Strang splitting finite element discretization for the Schr¨ odinger-Poissonequation, ESAIM: M2AN 51 (2017), no. 4, 1245–1278

    work page 2017

  6. [5]

    Bao and Y

    W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models 6 (2013), no. 1, 1–135

    work page 2013

  7. [6]

    W. Bao, D. Jaksch, and P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys. 187 (2003), no. 1, 318–342

    work page 2003

  8. [7]

    W. Bao, S. Jin, and P. A. Markowich, On time-splitting spectral approximations for the Schr¨ odingerequation in the semiclassical regime, J. Comput. Phys. 175 (2002), no. 2, 487–524

    work page 2002

Show all 55 references
  1. [8]

    W. Bao, S. Jin, and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schr¨ odingerequations in the semiclassical regimes, SIAM J. Sci. Comput. 25 (2003), no. 1, 27–64

    2003

  2. [9]

    Besse, A relaxation scheme for the nonlinear schr¨ odingerequation, SIAM J

    C. Besse, A relaxation scheme for the nonlinear schr¨ odingerequation, SIAM J. Numer. Anal. 42 (2004), no. 3, 934–952

    2004

  3. [10]

    Besse, B

    C. Besse, B. Bid´ egaray, and S. Descombes,Order estimates in time of splitting methods for the nonlinear Schr¨ odinger equation, SIAM J. Numer. Anal. 40 (2003), no. 1, 26–40

    2003

  4. [11]

    Besse, S

    C. Besse, S. Descombes, G. Dujardin, and I. Lacroix-Violet, Energy-preserving methods for nonlinear Schr¨ odinger equations, IMA J. Numer. Anal. 41 (2020), no. 1, 618–653

    2020

  5. [12]

    J. Chen, S. Li, and Z. Zhang, Efficient multiscale methods for the semiclassical Schr¨ odingerequation with time-dependent potentials, Comput. Methods Appl. Mech. Engrg. 369 (2020), 113232

    2020

  6. [13]

    J. Chen, D. Ma, and Z. Zhang, A multiscale finite element method for the Schr¨ odingerequation with multiscale potentials, SIAM J. Sci. Comput. 41 (2019), no. 5, B1115–B1136

    2019

  7. [14]

    , A multiscale reduced basis method for the Schr¨ odingerequation with multiscale and random potentials, Multiscale Model. Simul. 18 (2020), no. 4, 1409–1434

    2020

  8. [15]

    E. T. Chung, Y. Efendiev, and G. Li, An adaptive GMsFEM for high-contrast flow problems, J. Comput. Phys. 273 (2014), 54–76

    2014

  9. [16]

    Descombes, Convergence of a splitting method of high order for reaction-diffusion systems, Math

    S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comput. 70 (2001), no. 236, 1481–1501

    2001

  10. [17]

    J. Dick, F. Y. Kuo, and I. H. Sloan, High-dimensional integration: The quasi-Monte Carlo way, Acta Numer. 22 (2013), 133–288

    2013

  11. [18]

    D¨ oding, P

    C. D¨ oding, P. Henning, and J. W¨ arneg ˚ ard,An efficient two level approach for simulating Bose-Einstein condensates, arXiv preprint arXiv:2212.07392 (2022)

    2022 arXiv

  12. [19]

    Donsa, H

    S. Donsa, H. Hofst¨ atter, O. Koch, J. Burgd¨ orfer, and I. Bˇ rezinov´ a, Long-time expansion of a Bose-Einstein condensate: Observability of Anderson localization, Phys. Rev. A 96 (2017), 043630

    2017

  13. [20]

    W. E and B. Engquist, The heterognous multiscale methods, Commun. Math. Sci. 1 (2003), no. 1, 87–132

    2003

  14. [21]

    Efendiev, J

    Y. Efendiev, J. Galvis, and T. Y. Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys. 251 (2013), 116–135

    2013

  15. [22]

    Efendiev and T

    Y. Efendiev and T. Y. Hou, Multiscale finite element methods: theory and applications, vol. 4, Springer Science & Business Media, 2009

    2009

  16. [23]

    Fishman, Y

    S. Fishman, Y. Krivolapov, and A. Soffer, The nonlinear Schr¨ odingerequation with a random potential: results and puzzles, Nonlinearity 25 (2012), no. 4, R53

    2012

  17. [24]

    Garc ´ ıa-Mata and D

    I. Garc ´ ıa-Mata and D. L. Shepelyansky,Delocalization induced by nonlinearity in systems with disorder, Phys. Rev. E 79 (2009), 026205

    2009

  18. [25]

    I. G. Graham, F. Y. Kuo, J. A. Nichols, R. Scheichl, Ch. Schwab, and I. H. Sloan, Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients, Numer. Math. 131 (2015), no. 2, 329–368

    2015

  19. [26]

    Henning, A

    P. Henning, A. M ˚ alqvist, and D. Peterseim,Two-level discretization techniques for ground state computations of Bose-Einstein condensates, SIAM J. Numer. Anal. 52 (2014), no. 4, 1525–1550

    2014

  20. [27]

    Henning and A

    P. Henning and A. M ˚ alqvist,Localized orthogonal decomposition techniques for boundary value problems, SIAM J. Sci. Comput. 36 (2014), no. 4, A1609–A1634

    2014

  21. [28]

    Henning and A

    P. Henning and A. Persson, On optimal convergence rates for discrete minimizers of the Gross-Pitaevskii energy in localized orthogonal decomposition spaces, Multiscale Model. Simul. 21 (2023), no. 3, 993–1011. 28 PANCHI LI AND ZHIWEN ZHANG ∗

    2023

  22. [29]

    Henning and J

    P. Henning and J. W¨ arneg ˚ ard,Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation, Kinet. Relat. Models 12 (2019), no. 6, 1247–1271

    2019

  23. [30]

    T. Y. Hou, D. Ma, and Z. Zhang, A model reduction method for multiscale elliptic PDEs with random coefficients using an optimization approach, Multiscale Model. Simul. 17 (2019), no. 2, 826–853

    2019

  24. [31]

    T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134 (1997), no. 1, 169–189

    1997

  25. [32]

    T. Y. Hou and P. Zhang, Sparse operator compression of higher-order elliptic operators with rough coefficients, Res. Math. Sci. 4 (2017), no. 1, 24

    2017

  26. [33]

    Iomin, Subdiffusion in the nonlinear Schr¨ odingerequation with disorder, Phys

    A. Iomin, Subdiffusion in the nonlinear Schr¨ odingerequation with disorder, Phys. Rev. E 81 (2010), 017601

    2010

  27. [34]

    Karhunen, ¨Uber lineare methoden in der wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae: Ser

    K. Karhunen, ¨Uber lineare methoden in der wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae: Ser. A 1, Kirjapaino oy. sana, 1947

    1947

  28. [35]

    Kn¨ oller, A

    M. Kn¨ oller, A. Ostermann, and K. Schratz, A Fourier integrator for the cubic nonlinear Schr¨ odingerequation with rough initial data, SIAM J. Numer. Anal. 57 (2019), no. 4, 1967–1986

    2019

  29. [36]

    Li and Z

    S. Li and Z. Zhang, Computing eigenvalues and eigenfunctions of Schr¨ odingerequations using a model reduction approach, Commun. Comput. Phys. 24 (2018), 1073–1100

    2018

  30. [37]

    Loeve, Probability theory ii, F.W.Gehring P.r.Halmos and C.c.Moore, Springer, 1978

    M. Loeve, Probability theory ii, F.W.Gehring P.r.Halmos and C.c.Moore, Springer, 1978

    1978

  31. [38]

    J. C. L´ opez-Marcos and J. M. Sanz-Serna, A definition of stability for nonlinear problems, Numerical Treatment of Differential Equations 104 (1988), 216–226

    1988

  32. [39]

    M ˚ alqvist and D

    A. M ˚ alqvist and D. Peterseim,Computation of eigenvalues by numerical upscaling, Numer. Math. 130 (2015), no. 2, 337–361

    2015

  33. [40]

    A. V. Milovanov and A. Iomin, Destruction of Anderson localization in quantum nonlinear Schr¨ odingerlattices, Phys. Rev. E 95 (2017), 042142

    2017

  34. [41]

    Ostermann, Y

    A. Ostermann, Y. Wu, and F. Yao, A second-order low-regularity integrator for the nonlinear Schr¨ odingerequation, Adv. Contin. Discrete Models 2022 (2022), no. 1, 1–14

    2022

  35. [42]

    Owhadi, Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games, SIAM Rev

    H. Owhadi, Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games, SIAM Rev. 59 (2017), no. 1, 99–149

    2017

  36. [43]

    Owhadi and L

    H. Owhadi and L. Zhang, Localized bases for finite-dimensional homogenization approximations with nonseparated scales and high contrast, Multiscale Model. Simul. 9 (2011), no. 4, 1373–1398

    2011

  37. [44]

    Peterseim, Eliminating the pollution effect in Helmholtz problems by local subspace correction, Math

    D. Peterseim, Eliminating the pollution effect in Helmholtz problems by local subspace correction, Math. Comput. 86 (2017), no. 305, 1005–1036

    2017

  38. [45]

    A. S. Pikovsky and D. L. Shepelyansky, Destruction of Anderson localization by a weak nonlinearity, Phys. Rev. Lett. 100 (2008), 094101

    2008

  39. [46]

    J. M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schr¨ odingerequation, Math. Comput. 43 (1984), no. 167, 21–27

    1984

  40. [47]

    Schwab and R

    C. Schwab and R. A. Todor, Karhunen-Lo´ eveapproximation of random fields by generalized fast multipole methods, J. Comput. Phys. 217 (2006), no. 1, 100–122, Uncertainty Quantification in Simulation Science

    2006

  41. [48]

    D. L. Shepelyansky, Delocalization of quantum chaos by weak nonlinearity, Phys. Rev. Lett. 70 (1993), 1787–1790

    1993

  42. [49]

    M. E. Taylor, Partial differential equations III: Nonlinear equations, Applied Mathematical Sciences, Springer New York, 2010

    2010

  43. [50]

    Thalhammer, High-order exponential operator splitting methods for time-dependent Schr¨ odingerequations, SIAM J

    M. Thalhammer, High-order exponential operator splitting methods for time-dependent Schr¨ odingerequations, SIAM J. Numer. Anal. 46 (2008), no. 4, 2022–2038

    2008

  44. [51]

    Wang, A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schr¨ odingerequation, J

    J. Wang, A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schr¨ odingerequation, J. Sci. Comput. 60 (2014), no. 2, 390–407

    2014

  45. [53]

    Z. Wu, Z. Zhang, and X. Zhao, Error estimate of a quasi-Monte Carlo time-splitting pseudospectral method for nonlinear Schr¨ odingerequation with random potentials, SIAM/ASA J. Uncertainty Quantif. 12 (2024), no. 1, 1–29

    2024

  46. [54]

    Zhao, Numerical integrators for continuous disordered nonlinear Schr¨ odingerequation, J

    X. Zhao, Numerical integrators for continuous disordered nonlinear Schr¨ odingerequation, J. Sci. Comput. 89 (2021), no. 2, 40

    2021

  47. [55]

    G. E. Zouraris, On the convergence of a linear two-step finite element method for the nonlinear Schr¨ odingerequation, ESAIM: M2AN 35 (2001), no. 3, 389–405. Department of Mathematics, The University of Hong Kong, Hong Kong, P.R. China. Email address : lipch@hku.hk Department ...

    2001

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