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arxiv: 1907.00349 · v1 · pith:BC5PUHOKnew · submitted 2019-06-30 · 🧮 math.NA · cs.NA

A multiscale reduced basis method for Schr\"{o}dinger equation with multiscale and random potentials

Jingrun Chen , Dingjiong Ma , Zhiwen Zhang This is my paper

Pith reviewed 2026-05-25 12:38 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords reduced basis methodSchrödinger equationmultiscale potentialsrandom potentialsproper orthogonal decompositionquasi-Monte CarloAnderson localizationsemiclassical limit
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The pith

A multiscale reduced basis method solves the semiclassical Schrödinger equation by making spatial grid size proportional to the semiclassical parameter and random samples inversely proportional to it.

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

The paper develops a numerical method for the semiclassical Schrödinger equation with potentials that vary on multiple scales and include randomness. As time evolves the wavefunction oscillates rapidly in both physical and random space, so conventional grids and sampling become prohibitively expensive. The authors construct a low-dimensional set of multiscale basis functions in physical space by an optimization procedure followed by proper orthogonal decomposition, then apply quasi-Monte Carlo sampling in the random space. The resulting scheme requires a spatial mesh whose size grows only linearly with the semiclassical parameter while the number of random samples needed shrinks as that parameter decreases. The same framework also permits direct numerical study of Anderson localization for correlated random potentials in one and two dimensions.

Core claim

The multiscale reduced basis method constructs basis functions in the physical space using an optimization method and the proper orthogonal decomposition, and employs the quasi-Monte Carlo method in the random space, achieving efficiency where the spatial gridsize is proportional to the semiclassical parameter and the number of samples in the random space is inversely proportional to the same parameter.

What carries the argument

Multiscale reduced basis functions in physical space constructed via optimization and proper orthogonal decomposition, paired with quasi-Monte Carlo sampling in random space.

If this is right

  • Spatial discretization cost scales linearly rather than inversely with the semiclassical parameter.
  • Fewer random samples suffice as the semiclassical parameter decreases.
  • The number of samples required to build the reduced basis can be chosen on theoretical grounds.
  • Convergence of the overall method admits analysis that is confirmed numerically.
  • Anderson localization for correlated random potentials becomes accessible in both one and two spatial dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same basis-construction strategy could be tested on other semiclassical wave equations whose solutions develop rapid spatial oscillations.
  • Because the basis does not depend on any single realization, the method may directly support ensemble statistics without recomputing the basis for each draw.
  • The observed scaling suggests that three-dimensional simulations of heterogeneous quantum systems could become feasible once the reduced-basis dimension is controlled.

Load-bearing premise

The optimization procedure and proper orthogonal decomposition can construct a low-dimensional multiscale reduced basis that accurately represents the solution for the class of multiscale random potentials considered, without requiring a priori knowledge of the specific realization.

What would settle it

A comparison, for a fixed realization of the random potential, between the reduced-basis solution computed on a grid of size proportional to the semiclassical parameter and a reference solution on a much finer grid, checking whether the difference remains small as the semiclassical parameter is driven toward zero.

Figures

Figures reproduced from arXiv: 1907.00349 by Dingjiong Ma, Jingrun Chen, Zhiwen Zhang.

Figure 1
Figure 1. Figure 1: Exponentially decaying properties of the multiscale basis functions for four different realizations. [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Relative errors with respect to the number of the multiscale reduced basis functions. [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the qMC method and the MC method. Convergence rate for qMC and MC are 1 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Anderson localization for different parameters. [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Anderson localization for different β. 2D Schr¨odinger equation. Consider the Schr¨odinger equation (32) over D = [−π, π] × [−π, π] and v ε (x1, x2, ω) = σ Xm j=1 sin(jx1) sin(jx2) 1 j β ξj (ω), (52) where the setting of ξj (ω)’s is the same as the 1D case. σ, m and β are parameters that controls the random potential. Choose σ = 5, β = 0 and ε = 1 4 . Notice that β = 0 and (52) is used to model a short￾ran… view at source ↗
Figure 6
Figure 6. Figure 6: Anderson localization when σ = 5 and β = 0 in 2D. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
read the original abstract

The semiclassical Schr\"{o}dinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wavefunction develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. In this paper, we propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space. Our method is verified to be efficient: the spatial gridsize is only proportional to the semiclassical parameter and the number of samples in the random space is inversely proportional to the same parameter. Several theoretical aspects of the proposed method, including how to determine the number of samples in the construction of multiscale reduced basis and convergence analysis, are studied with numerical justification. In addition, we investigate the Anderson localization phenomena for Schr\"{o}dinger equation with correlated random potentials in both 1D and 2D.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a multiscale reduced basis method for the semiclassical Schrödinger equation with multiscale random potentials. Reduced basis functions are constructed via an optimization procedure combined with proper orthogonal decomposition (POD) in physical space; quasi-Monte Carlo (QMC) sampling is used in the random space. The central claim is that the method is efficient, with spatial grid size proportional to the semiclassical parameter ε and the number of QMC samples scaling as 1/ε; convergence and the choice of sample count in basis construction are justified numerically, and the method is applied to study Anderson localization in 1D and 2D.

Significance. If the reduced-basis dimension remains bounded independently of ε, the approach would yield a practical, scalable solver for high-frequency oscillatory problems in random media, directly enabling quantitative studies of localization phenomena that are otherwise prohibitive. The numerical verification of convergence rates and sample counts provides concrete evidence of utility even if a full a-priori theory is absent.

major comments (2)
  1. [Abstract] Abstract and the efficiency claim: the stated scalings (grid size O(ε), QMC samples O(1/ε)) are load-bearing and rest on the assumption that the POD dimension D produced by the optimization procedure on a finite training set remains independent of ε (or at worst logarithmic). No argument or numerical test is supplied showing that the Kolmogorov n-width of the solution manifold over the class of multiscale random potentials stays bounded as ε→0; if D grows with 1/ε the online cost per sample would exceed the claimed scaling.
  2. [Basis construction and sample determination] Section on basis construction and numerical justification of sample count: the optimization+POD procedure is presented as constructing a low-dimensional basis without a-priori knowledge of individual realizations, yet the training-set size and the stopping criterion for the number of POD modes are chosen post-hoc. Without a quantitative table or plot of required D versus ε (or versus the correlation length of the random potential), it is impossible to verify that the claimed scalings survive for the full range of ε considered.
minor comments (2)
  1. [Abstract] The abstract states that 'several theoretical aspects … are studied with numerical justification' but supplies no explicit error tables, observed convergence rates, or comparison against a reference solution; adding a compact table of L² or H¹ errors versus ε and versus number of basis functions would strengthen the numerical section.
  2. [Notation] Notation for the semiclassical parameter and the random-potential correlation length is introduced without a single consolidated table of symbols; a short nomenclature table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the efficiency claims. We agree that explicit numerical evidence for the reduced-basis dimension D versus ε would strengthen the manuscript and will add it in revision.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the efficiency claim: the stated scalings (grid size O(ε), QMC samples O(1/ε)) are load-bearing and rest on the assumption that the POD dimension D produced by the optimization procedure on a finite training set remains independent of ε (or at worst logarithmic). No argument or numerical test is supplied showing that the Kolmogorov n-width of the solution manifold over the class of multiscale random potentials stays bounded as ε→0; if D grows with 1/ε the online cost per sample would exceed the claimed scaling.

    Authors: We agree that the claimed scalings require D to remain bounded (or grow at most logarithmically) as ε→0. The manuscript provides numerical verification of overall convergence rates and efficiency across a range of ε values, which implicitly supports controlled growth of D. To directly address the request for explicit evidence on the Kolmogorov n-width, the revised manuscript will include a dedicated plot or table of the POD dimension D versus ε (and versus correlation length) for the random potentials used. revision: yes

  2. Referee: [Basis construction and sample determination] Section on basis construction and numerical justification of sample count: the optimization+POD procedure is presented as constructing a low-dimensional basis without a-priori knowledge of individual realizations, yet the training-set size and the stopping criterion for the number of POD modes are chosen post-hoc. Without a quantitative table or plot of required D versus ε (or versus the correlation length of the random potential), it is impossible to verify that the claimed scalings survive for the full range of ε considered.

    Authors: The training-set size and POD stopping criterion were selected via the numerical convergence studies already reported in the paper. We acknowledge that a single, consolidated quantitative table or plot of D versus ε would make verification of the scalings more transparent and direct. In the revision we will add this plot (or table) to the basis-construction section. revision: yes

Circularity Check

0 steps flagged

No circularity; efficiency verified numerically against external benchmarks

full rationale

The paper constructs a multiscale reduced basis via optimization and POD in physical space plus QMC in random space, then states that the claimed O(ε) grid and O(1/ε) sample scalings are verified by numerical experiments. No load-bearing step reduces these scalings to fitted parameters by construction, nor invokes a self-citation chain whose justification collapses into the present work. Convergence analysis is likewise supported by numerical justification rather than a closed deductive loop. The derivation therefore remains self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The method rests on standard numerical analysis assumptions for PDE convergence and on the existence of an effective low-dimensional representation via the chosen basis construction; no new physical entities are postulated.

free parameters (1)
  • Number of reduced basis functions
    Chosen during construction via optimization and POD; affects accuracy and efficiency but value not specified in abstract.
axioms (2)
  • domain assumption The potentials admit a low-dimensional multiscale representation capturable by POD after optimization.
    Invoked when constructing the reduced basis in physical space.
  • standard math Standard Sobolev regularity and boundedness assumptions on the potential and initial data hold.
    Required for convergence analysis of the Schrödinger equation discretization.

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Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

  1. [1]

    Aizenman and S

    M. Aizenman and S. Molchanov, Localization at large disorder and at extreme ener- gies: An elementary derivations , Communications in Mathematical Physics, 157 (1993), pp. 245–278. 24

    work page 1993

  2. [2]

    P. W. Anderson, Absence of diffusion in certain random lattices , Physical review, 109 (1958), pp. 1492–1505

    work page 1958

  3. [3]

    Babuska and R

    I. Babuska and R. Lipton , Optimal local approximation spaces for generalized finite element methods with application to multiscale problems , Multiscale Modeling & Simula- tion, 9 (2011), pp. 373–406

    work page 2011

  4. [4]

    W. Bao, S. Jin, and P. A. Markowich, On time-splitting spectral approximations for the schr¨ odinger equation in the semiclassical regime, Journal of Computational Physics, 175 (2002), pp. 487–524

    work page 2002

  5. [5]

    Berkooz, P

    G. Berkooz, P. Holmes, and J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows , Annual review of fluid mechanics, 25 (1993), pp. 539–575

    work page 1993

  6. [6]

    Bungartz and M

    H.-J. Bungartz and M. Griebel , Sparse grids, Acta numerica, 13 (2004), pp. 147– 269

    work page 2004

  7. [7]

    J. Chen, D. Ma, and Z. Zhang , Convergence of a multiscale finite element method for the Schr¨ odinger equation with multiscale potentials. In preparation

    work page

  8. [8]

    , A multiscale finite element method for the Schr¨ odinger equation with multiscale potentials, arXiv:1901.00343, (2019)

    work page arXiv 1901

  9. [9]

    Delgadillo, J

    R. Delgadillo, J. Lu, and X. Yang, Gauge-invariant frozen gaussian approximation method for the schrodinger equation with periodic potentials , SIAM Journal on Scientific Computing, 38 (2016), pp. A2440–A2463

    work page 2016

  10. [10]

    Devakul and D

    T. Devakul and D. A. Huse , Anderson localization transitions with and without ran- dom potentials, Physical Review B, 96 (2017), p. 214201

    work page 2017

  11. [11]

    J. Dick, F. Y. Kuo, and I. H. Sloan, High-dimensional integration: the Quasi-Monte Carlo way, Acta Numerica, 22 (2013), pp. 133–288

    work page 2013

  12. [12]

    Dietrich and G

    C. Dietrich and G. Newsam , Fast and exact simulation of stationary gaussian pro- cesses through circulant embedding of the covariance matrix , SIAM Journal on Scientific Computing, 18 (1997), pp. 1088–1107

    work page 1997

  13. [13]

    Efendiev and T

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

    work page 2009

  14. [14]

    Lecture Notes on Quantum Brownian Motion

    L. Erdos, Lecture notes on quantum Brownian motion , arXiv: 1009.0843. 2010., 2010

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  15. [15]

    E. Faou, V. Gradinaru, and C. Lubich, Computing semiclassical quantum dynamics with hagedorn wavepackets, SIAM Journal on Scientific Computing, 31 (2009), pp. 3027– 3041

    work page 2009

  16. [16]

    Faou and C

    E. Faou and C. Lubich, A Poisson integrator for gaussian wavepacket dynamics, Com- puting and Visualization in Science, 9 (2006), pp. 45–55. 25

    work page 2006

  17. [17]

    Filoche and S

    M. Filoche and S. Mayboroda , Universal mechanism for Anderson and weak local- ization, Proceedings of the National Academy of Sciences, 109 (2012), pp. 14761–14766

    work page 2012

  18. [18]

    Fr ¨ohlich and T

    J. Fr ¨ohlich and T. Spencer , Absence of diffusion in the Anderson tight binding model for large disorder or low energy , Communications in Mathematical Physics, 88 (1983), pp. 151–184

    work page 1983

  19. [19]

    R. G. Ghanem and P. D. Spanos , Stochastic finite elements: a spectral approach , Courier Corporation, 2003

    work page 2003

  20. [20]

    I. G. Graham, F. Y. Kuo, J. A. Nichols, R. Scheichl, C. Schwab, and I. H. Sloan, Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal ran- dom coefficients, Numerische Mathematik, 131 (2015), pp. 329–368

    work page 2015

  21. [21]

    T. Hou, X. Wu, and Z. Cai , Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , Mathematics of Computation of the American Mathematical Society, 68 (1999), pp. 913–943

    work page 1999

  22. [22]

    T. Y. Hou, D. Ma, and Z. Zhang , A model reduction method for multiscale elliptic pdes with random coefficients using an optimization approach , Multiscale Modeling & Simulation, 17 (2019), pp. 826–853

    work page 2019

  23. [23]

    T. Y. Hou and X. Wu , A multiscale finite element method for elliptic problems in composite materials and porous media , Journal of Computational Physics, 134 (1997), pp. 169–189

    work page 1997

  24. [24]

    T. Y. Hou and P. Zhang, Sparse operator compression of higher-order elliptic operators with rough coefficients, Research in the Mathematical Sciences, 4 (2017), p. 24

    work page 2017

  25. [25]

    S. Jin, L. Liu, G. Russo, and Z. Zhou , Gaussian wave packet transform based numerical scheme for the semi-classical Schr¨ odinger equation with random inputs , arXiv:1903.08740, (2019)

    work page arXiv 1903

  26. [26]

    S. Jin, P. Markowich, and C. Sparber , Mathematical and computational methods for semiclassical schr¨ odinger equations, Acta Numerica, 20 (2011), pp. 121–209

    work page 2011

  27. [27]

    S. Jin, H. Wu, X. Yang, et al., Gaussian beam methods for the Schr¨ odinger equation in the semi-classical regime: Lagrangian and eulerian formulations , Communications in Mathematical Sciences, 6 (2008), pp. 995–1020

    work page 2008

  28. [28]

    Karhunen , Uber lineare methoden in der Wahrscheinlichkeitsrechnung , Annales Academiae Scientiarum Fennicae, 37 (1947), pp

    K. Karhunen , Uber lineare methoden in der Wahrscheinlichkeitsrechnung , Annales Academiae Scientiarum Fennicae, 37 (1947), pp. 1–79

    work page 1947

  29. [29]

    Li and Z

    S. Li and Z. Zhang, Computing eigenvalues and eigenfunctions of Schr¨ odinger equations using a model reduction approach , Communications in Computational Physics, (2017). 26

    work page 2017

  30. [30]

    Lo`eve, Probability theory

    M. Lo`eve, Probability theory. Vol. II, 4th ed. GTM. 46. , Springer-Verlag, ISBN 0-387- 90262-7, 1978

    work page 1978

  31. [31]

    M˚alqvist and D

    A. M˚alqvist and D. Peterseim , Localization of elliptic multiscale problems , Mathe- matics of Computation, 83 (2014), pp. 2583–2603

    work page 2014

  32. [32]

    Mott, Metal-insulator transitions, CRC Press, 1990

    N. Mott, Metal-insulator transitions, CRC Press, 1990

    work page 1990

  33. [33]

    Nobile, R

    F. Nobile, R. Tempone, and C. G. Webster , A sparse grid stochastic collocation method for partial differential equations with random input data , SIAM Journal on Nu- merical Analysis, 46 (2008), pp. 2309–2345

    work page 2008

  34. [34]

    Nosov, I

    P. Nosov, I. Khaymovich, and V. Kravtsov, Correlation-induced localization, Phys- ical Review B, 99 (2019), p. 104203

    work page 2019

  35. [35]

    Owhadi, Bayesian numerical homogenization, Multiscale Modeling & Simulation, 13 (2015), pp

    H. Owhadi, Bayesian numerical homogenization, Multiscale Modeling & Simulation, 13 (2015), pp. 812–828

    work page 2015

  36. [36]

    , Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games, SIAM Review, 59 (2017), pp. 99–149

    work page 2017

  37. [37]

    Schwab and R

    C. Schwab and R. A. Todor , Karhunen–Lo` eve approximation of random fields by generalized fast multipole methods, Journal of Computational Physics, 217 (2006), pp. 100– 122

    work page 2006

  38. [38]

    Sirovich, Turbulence and the dynamics of coherent structures

    L. Sirovich, Turbulence and the dynamics of coherent structures. I. coherent structures , Quarterly of applied mathematics, 45 (1987), pp. 561–571

    work page 1987

  39. [39]

    N. M. Tanushev, J. Qian, and J. V. Ralston, Mountain waves and Gaussian beams, Multiscale Modeling & Simulation, 6 (2007), pp. 688–709

    work page 2007

  40. [40]

    Wu and Z

    Z. Wu and Z. Huang , A Bloch decomposition-based stochastic Galerkin method for quantum dynamics with a random external potential , Journal of Computational Physics, 317 (2016), pp. 257–275

    work page 2016

  41. [41]

    Xiu and G

    D. Xiu and G. E. Karniadakis , Modeling uncertainty in flow simulations via gener- alized polynomial chaos, Journal of Computational Physics, 187 (2003), pp. 137–167. 27

    work page 2003