pith. sign in

arxiv: 2604.20282 · v1 · submitted 2026-04-22 · 🧮 math.NA · cs.NA

Cayley-transform analysis and numerical validation of the convergent Born series for the Helmholtz equation

Morten Jakobsen This is my paper

Pith reviewed 2026-05-10 00:13 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords convergent born serieshelmholtz equationcayley transformlippmann-schwinger equationself-adjoint operatornumerical rangeabsorbing layersconvergence criterion
0
0 comments X

The pith

The convergent Born series for the Helmholtz equation admits a unitary Cayley-transform representation from the resolvent of a self-adjoint background operator, delivering domain-independent convergence bounds.

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

The paper establishes an operator-theoretic framework for the convergent Born series applied to the Lippmann-Schwinger equation in high-frequency Helmholtz problems. It expresses the preconditioned iteration entirely through the resolvent of a self-adjoint background operator, yielding a unitary Cayley-transform form of the iteration operator. This form supplies basis-independent numerical-range bounds and a general convergence criterion that holds on arbitrary bounded domains and for complex wave numbers. The same structure supports smoothly tapered complex-wavenumber absorbing layers that preserve self-adjointness while improving contractivity, and it extends without modification to other wave and diffusion equations whose fundamental solutions lack closed form. Numerical benchmarks against PML finite-difference simulations confirm stable accuracy over wide ranges of contrast and frequency.

Core claim

The preconditioned Lippmann-Schwinger iteration is expressed entirely in terms of the resolvent of a self-adjoint background operator. This yields a unitary Cayley-transform representation of the CBS iteration operator, from which basis-independent bounds on its numerical range are derived together with a convergence criterion valid on arbitrary bounded domains and for complex-valued wave numbers.

What carries the argument

The unitary Cayley-transform representation of the CBS iteration operator, obtained from the resolvent of a self-adjoint background operator.

If this is right

  • The convergence criterion applies without change to arbitrary bounded domains.
  • The framework works for complex-valued wave numbers.
  • Smoothly tapered complex-wavenumber absorbing layers can be incorporated while preserving self-adjointness and increasing contractivity without altering the differential operator.
  • The method extends directly to frequency-domain wave and diffusion equations whose Green's functions are not available in closed form.
  • Numerical solutions remain accurate and stable across broad ranges of contrasts and frequencies.

Where Pith is reading between the lines

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

  • The approach could be tested on domains with highly irregular boundaries where Fourier-based analysis is unavailable.
  • The self-adjoint-preserving absorbing-layer construction might transfer to other iterative solvers for scattering problems.
  • Comparison with PML-based codes on the same meshes would quantify any efficiency gains at high frequencies.
  • The same resolvent representation could be examined for time-domain or nonlinear extensions of the Helmholtz problem.

Load-bearing premise

The background operator must remain self-adjoint once the smoothly tapered complex-wavenumber absorbing layers are added.

What would settle it

A simulation on a non-rectangular domain with complex wavenumber in which the CBS series diverges even though the derived numerical-range bounds and convergence criterion are satisfied.

Figures

Figures reproduced from arXiv: 2604.20282 by Morten Jakobsen.

Figure 1
Figure 1. Figure 1: Convergence history of the CBS iteration in the hom [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Real (left column) and imaginary (right column) pa [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative amplitude (top) and absolute phase (bott [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Heterogeneous velocity model used to compare CBS a [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence history of the CBS (blue curve) and the [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Frequency domain wavefields computed using the FDF [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Amplitude difference between the wavefields comput [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
read the original abstract

We develop an operator-theoretic framework for the Convergent Born Series (CBS) method applied to the Lippmann--Schwinger equation for high-frequency Helmholtz problems. In contrast to the Fourier-based analysis of Osnabrugge et al., our approach expresses the preconditioned Lippmann--Schwinger iteration entirely in terms of the resolvent of a self-adjoint background operator. This leads to a unitary Cayley-transform representation of the CBS iteration operator, from which we derive basis-independent bounds on its numerical range and a general convergence criterion valid on arbitrary bounded domains and for complex-valued wave numbers. Because the analysis does not rely on an explicit Green's function in the Fourier domain, the Cayley-transform framework extends naturally to a broader class of frequency-domain wave and diffusion equations whose fundamental solutions are not available in closed form. We further incorporate smoothly tapered complex-wavenumber absorbing layers that preserve the self-adjoint structure of the reference operator and enhance the contractivity of the iteration without modifying the differential operator. In addition to this theoretical generalization, we present a detailed numerical validation in which CBS solutions are benchmarked against PML-based finite-difference wavefield simulations. These experiments demonstrate that the operator-theoretic CBS formulation delivers accurate and stable results across a broad range of contrasts and frequencies, thereby significantly extending the applicability and theoretical foundation of the CBS method beyond previously analyzed settings.

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 develops an operator-theoretic framework for the Convergent Born Series (CBS) applied to the Lippmann-Schwinger equation for the Helmholtz problem. It expresses the preconditioned iteration via the resolvent of a self-adjoint background operator, obtains a unitary Cayley-transform representation of the iteration operator, and derives basis-independent numerical-range bounds together with a general convergence criterion valid on arbitrary bounded domains and for complex wave numbers. The framework is extended by incorporating smoothly tapered complex-wavenumber absorbing layers claimed to preserve self-adjointness, and is supported by numerical benchmarking against PML-based finite-difference simulations.

Significance. If the self-adjointness of the background operator is rigorously preserved under the absorbing layers, the work provides a meaningful generalization of CBS analysis beyond the Fourier-domain setting of prior work, enabling application to problems without explicit Green's functions and to a broader class of frequency-domain equations. The numerical validation supplies concrete evidence of stability across contrasts and frequencies, supporting the practical utility of the theoretical extension.

major comments (2)
  1. [Section 4] The absorbing-layers construction (Section 4): the manuscript states that smoothly tapered complex-wavenumber absorbing layers preserve the self-adjoint structure of the reference operator, allowing the Cayley transform to remain unitary. However, for the standard background operator A = −Δ + k(x)^2 the sesquilinear form satisfies ⟨Au, v⟩ − ⟨u, Av⟩ = ∫ [k(x)^2 − conj(k(x)^2)] u conj(v) dx, which vanishes only when Im(k(x)^2) ≡ 0. An explicit weighted inner product, conjugation, or other modification must be supplied to restore self-adjointness; without it the unitarity, numerical-range bounds, and convergence criterion do not apply in the claimed generality.
  2. [Theorem 3.4] Convergence criterion (Theorem 3.4 or equivalent): the basis-independent bound on the numerical range is derived from the unitary Cayley representation; once self-adjointness is clarified, the proof should explicitly verify that the bound remains valid when the absorbing layers are present and the wavenumber is complex.
minor comments (2)
  1. [Section 2] Notation for the resolvent and the iteration operator should be introduced with a single consistent symbol set in Section 2 to avoid later ambiguity.
  2. [Numerical results] Figure captions for the numerical benchmarks should state the exact contrast values, frequency range, and grid resolution used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the careful reading and constructive comments on our manuscript. The points raised concerning the self-adjointness of the background operator with absorbing layers are important and will be addressed through targeted revisions and clarifications.

read point-by-point responses

  1. Referee: [Section 4] The absorbing-layers construction (Section 4): the manuscript states that smoothly tapered complex-wavenumber absorbing layers preserve the self-adjoint structure of the reference operator, allowing the Cayley transform to remain unitary. However, for the standard background operator A = −Δ + k(x)^2 the sesquilinear form satisfies ⟨Au, v⟩ − ⟨u, Av⟩ = ∫ [k(x)^2 − conj(k(x)^2)] u conj(v) dx, which vanishes only when Im(k(x)^2) ≡ 0. An explicit weighted inner product, conjugation, or other modification must be supplied to restore self-adjointness; without it the unitarity, numerical-range bounds, and convergence criterion do not apply in the claimed generality.

    Authors: We thank the referee for this precise observation. The manuscript asserts that the smoothly tapered complex-wavenumber layers preserve self-adjointness without modifying the differential operator, yet we did not explicitly identify the inner product under which this holds. In the revision we will introduce a real, positive weight function w(x) and work in the weighted inner product ⟨u,v⟩_w = ∫ w(x) u conj(v) dx. With this choice the background operator A becomes self-adjoint, the Cayley transform remains unitary, and the numerical-range analysis carries over unchanged. The weight will be chosen to be identically one outside the absorbing region so that the physical solution is unaffected. We will also verify that the numerical experiments remain identical under this equivalent formulation. revision: yes

  2. Referee: [Theorem 3.4] Convergence criterion (Theorem 3.4 or equivalent): the basis-independent bound on the numerical range is derived from the unitary Cayley representation; once self-adjointness is clarified, the proof should explicitly verify that the bound remains valid when the absorbing layers are present and the wavenumber is complex.

    Authors: Once the weighted inner product is introduced to restore self-adjointness, the unitary Cayley-transform representation of the iteration operator holds in the corresponding Hilbert space. The proof of the numerical-range bound in Theorem 3.4 relies only on this unitarity and is therefore independent of the specific form of the absorbing layers or the values of the (possibly complex) wavenumbers. In the revised manuscript we will add a short remark immediately following Theorem 3.4 that explicitly confirms the bound and the ensuing convergence criterion remain valid under the weighted-inner-product formulation with absorbing layers. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on standard operator theory applied to a constructed self-adjoint reference operator.

full rationale

The paper constructs a preconditioned Lippmann-Schwinger iteration expressed via the resolvent of a background operator declared self-adjoint, then applies the Cayley transform to obtain numerical-range bounds and a convergence criterion. The absorbing layers are introduced by explicit construction that the authors state preserves self-adjointness; this is an input assumption, not a result derived from the bounds themselves. No equation is shown to equal its own fitted parameter or prior self-citation by definition, and the framework is presented as extending (rather than presupposing) the Fourier analysis of Osnabrugge et al. The central claims therefore remain independent of the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the self-adjoint character of the background operator and the preservation of that structure by the tapered absorbing layers; these are domain assumptions rather than derived results.

axioms (2)
  • domain assumption The background operator is self-adjoint
    Invoked to guarantee that the Cayley transform is unitary and that numerical-range bounds can be derived.
  • domain assumption Smoothly tapered complex-wavenumber absorbing layers preserve the self-adjoint structure of the reference operator
    Used to enhance contractivity of the iteration while keeping the differential operator unmodified.

pith-pipeline@v0.9.0 · 5539 in / 1425 out tokens · 49482 ms · 2026-05-10T00:13:46.287775+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 30 canonical work pages

  1. [1]

    A. B. Weglein, F. A. Araújo, P. M. Carvalho, R. H. Stolt, K. H. Matson, H. Zhang, F. Liu, Inverse scattering series and seismic exploration, Invers e Problems 19 (6) (2003) R27–R83. doi:10.1088/0266-5611/19/6/R01

    work page doi:10.1088/0266-5611/19/6/r01 2003

  2. [2]

    Jakobsen, R.-S

    M. Jakobsen, R.-S. Wu, Renormalized scattering series f or frequency-domain waveform modelling of strong velocity contrasts, Geophysical Journal Interna tional 206 (2) (2016) 880–899

    2016

  3. [3]

    Sheng (Ed.), Introduction to Wave Scattering, Locali zation and Mesoscopic Phenomena, Springer, Berlin, Heidelberg, 1995

    P. Sheng (Ed.), Introduction to Wave Scattering, Locali zation and Mesoscopic Phenomena, Springer, Berlin, Heidelberg, 1995. doi:10.1007/978-3-662-03225-4

    work page doi:10.1007/978-3-662-03225-4 1995

  4. [4]

    Jakobsen, K

    M. Jakobsen, K. Xiang, K. W. A. van Dongen, Seismic and med ical ultrasound imaging of velocity and density variations by nonlinear vectorial inverse scattering, The Journal of the Acoustical Society of America 153 (5) (2023) 3151–3164. doi:10.1121/10.0019563

    work page doi:10.1121/10.0019563 2023

  5. [5]

    K. J. Marfurt, Accuracy of finite-difference and finite-el ement modeling of the scalar and elastic wave equations, Geophysics 49 (5) (1984) 533–549

    1984

  6. [6]

    Babuška, S

    I. Babuška, S. A. Sauter, Is the pollution effect of the fem avoidable for the helmholtz equation with large wave number?, SIAM Journal on Numerical Analysis 34 (6 ) (2000) 2392–2423

    2000

  7. [7]

    Komatitsch, J.-P

    D. Komatitsch, J.-P. Vilotte, The spectral-element met hod: An efficient tool to simulate the seismic response of 2d and 3d geological structures, Geophysical Jo urnal International 132 (1) (1998) 251– 266

    1998

  8. [8]

    Komatitsch, J

    D. Komatitsch, J. Tromp, Introduction to the spectral el ement method for 3-d seismic wave propa- gation, Geophysical Journal International 139 (3) (1999) 8 06–822

    1999

  9. [9]

    X. Feng, H. Wu, Discontinuous galerkin methods for the he lmholtz equation with large wave number, SIAM Journal on Numerical Analysis 47 (4) (2009) 2872–2896

    2009

  10. [10]

    Jakobsen, J

    M. Jakobsen, J. S. Stokke, K. Kumar, F. A. Radu, Frequenc y domain biot–allard equations for isotropic and anisotrop ic Research Square PreprintPreprint (2025). doi:10.21203/rs.3.rs-8293824/v1. URL https://doi.org/10.21203/rs.3.rs-8293824/v1

    work page doi:10.21203/rs.3.rs-8293824/v1 2025

  11. [11]

    Ihlenburg, I

    F. Ihlenburg, I. Babuška, Dispersion analysis and erro r estimation of Galerkin finite element methods for the Helmholtz equation, Computer Methods in Applied Mec hanics and Engineering 128 (3–4) (1995) 363–380. doi:10.1016/0045-7825(95)00736-4

    work page doi:10.1016/0045-7825(95)00736-4 1995

  12. [12]

    Cessenat, B

    M. Cessenat, B. Després, Application of an ultra weak va riational formulation of elliptic PDEs to the two-dimensional Helmholtz problem, SIAM Journal on Numeri cal Analysis 35 (1) (1998) 255–299. doi:10.1137/S0036142996299717

    work page doi:10.1137/s0036142996299717 1998

  13. [13]

    J. B. Ajo-Franklin, Frequency-domain modeling techni ques for the scalar wave equation: An intro- duction (2005)

    2005

  14. [14]

    Abubakar, T

    A. Abubakar, T. M. Habashy, Three-dimensional visco-a coustic modeling using a renormal- ized integral equation iterative solver, Journal of Comput ational Physics 249 (2013) 1–12. doi:10.1016/j.jcp.2013.04.008

    work page doi:10.1016/j.jcp.2013.04.008 2013

  15. [15]

    Huang, S

    X. Huang, S. Greenhalgh, A finite-difference iterative s olver of the helmholtz equation for frequency- domain seismic wave modeling and full-waveform inversion, Geophysics 86 (2) (2021) T107–T116. doi:10.1190/geo2020-0411.1

    work page doi:10.1190/geo2020-0411.1 2021

  16. [16]

    A. H. Sheikh, D. Lahaye, L. G. Ramos, R. Nabben, C. Vuik, A ccelerating the shifted laplace preconditioner for the helmholtz equation by multilevel de flation, Journal of Computational Physics 322 (2016) 473–490. doi:10.1016/j.jcp.2016.06.025

    work page doi:10.1016/j.jcp.2016.06.025 2016

  17. [17]

    M. J. Gander, H. Zhang, A class of iterative solvers for t he helmholtz equation: Factorizations, sweeping preconditioners, source transfer, single layer p otentials, polarized traces, and optimized schwarz methods, SIAM Review 61 (1) (2019) 3–76. doi:10.1137/16M109781X. 21

    work page doi:10.1137/16m109781x 2019

  18. [18]

    Chaumont-Frelet, E

    T. Chaumont-Frelet, E. A. Spence, The geometric error i s less than the pollution error when solving the high-frequency helmholtz equation with high-order fem on curved domains, IMA Journal of Numerical Analysis (2025). doi:10.1093/imanum/draf035

    work page doi:10.1093/imanum/draf035 2025

  19. [19]

    E. R. Pike, P. C. Sabatier, Scattering, two-volume set: Scattering and inverse scattering in pure and applied science, Elsevier, 2001

    2001

  20. [20]

    Colton, R

    D. Colton, R. Kress, Integral Equation Methods in Scatt ering Theory, Wiley-Interscience, New York, 1983

    1983

  21. [21]

    Osnabrugge, R

    G. Osnabrugge, R. Horstmeyer, I. M. Vellekoop, Converg ent born series for solving the inhomoge- neous helmholtz equation in arbitrarily large media, J. Opt . Soc. Am. A 33 (2016) 2317–2322

    2016

  22. [22]

    Krüger, T

    B. Krüger, T. Brenner, A. Kienle, Solution of the inhomo geneous maxwell’s equations using a born series, Optics Express 25 (21) (2017) 25165–25182. doi:10.1364/OE.25.025165

    work page doi:10.1364/oe.25.025165 2017

  23. [23]

    Malovichko, N

    M. Malovichko, N. Khokhlov, N. Yavich, M. Zhdanov, Acou stic 3d modeling by the method of integral equations, Computers & Geosciences 111 (2018) 223 –234

    2018

  24. [24]

    Huang, M

    X. Huang, M. Jakobsen, R.-S. Wu, On the applicability of a renormalized born series for seismic wavefield modelling in strongly scattering media, Journal o f Geophysics and Engineering 17 (2) (2020) 277–299. doi:10.1093/jge/gxz105

    work page doi:10.1093/jge/gxz105 2020

  25. [25]

    R. E. Kleinman, P. Van den Berg, Iterative methods for so lving integral equations, Radio Science 26 (1) (1991) 175–181

    1991

  26. [26]

    Colton, R

    D. Colton, R. Kress, Integral Equation Methods in Scatt ering Theory, SIAM, 2013

    2013

  27. [27]

    Stevenson, Piecewise polynomial C0 approximations for compact operators, Mathematics of Computation 54 (190) (1990) 465–477

    R. Stevenson, Piecewise polynomial C0 approximations for compact operators, Mathematics of Computation 54 (190) (1990) 465–477

    1990

  28. [28]

    C. L. Epstein, L. Greengard, Debye sources and the numer ical solution of the time harmonic maxwell equations, Communications on Pure and Applied Mathematics 71 (6) (2018) 1095–1150

    2018

  29. [29]

    K. S. Eikrem, G. Nævdal, M. Jakobsen, Iterative solutio n of the Lippmann–Schwinger equation in strongly scattering acoustic media by randomized constr uction of preconditioners, Geophysical Journal International 224 (3) (2021) 2121–2130. doi:10.1093/gji/ggaa503

    work page doi:10.1093/gji/ggaa503 2021

  30. [30]

    Ying, Sparsifying preconditioner for the Lippmann– Schwinger equation, Multiscale Modeling & Simulation 13 (2) (2015) 644–660

    L. Ying, Sparsifying preconditioner for the Lippmann– Schwinger equation, Multiscale Modeling & Simulation 13 (2) (2015) 644–660. doi:10.1137/140985147

    work page doi:10.1137/140985147 2015

  31. [31]

    F. Liu, L. Ying, Sparsify and sweep: An efficient precondi tioner for the Lippmann–Schwinger equa- tion, SIAM Journal on Scientific Computing 40 (2) (2018). doi:10.1137/17M1132057

    work page doi:10.1137/17m1132057 2018

  32. [32]

    Greengard, V

    L. Greengard, V. Rokhlin, A fast algorithm for particle simulations, Journal of Computational Physics 73 (2) (1987) 325–348

    1987

  33. [33]

    Jakobsen, X

    M. Jakobsen, X. Huang, R.-S. Wu, Homotopy analysis of th e lippmann–schwinger equation for seismic wavefield modelling in strongly scattering media, G eophysical Journal International 222 (2) (2020) 743–753. doi:10.1093/gji/ggaa159

    work page doi:10.1093/gji/ggaa159 2020

  34. [34]

    R. G. Newton, Scattering Theory of Waves and Particles, 2nd Edition, Dover Publications, New York, 1992

    1992

  35. [35]

    Weinberg, Quasiparticles and the born series, Physi cal Review 131 (1) (1963) 440–460

    S. Weinberg, Quasiparticles and the born series, Physi cal Review 131 (1) (1963) 440–460. doi:10.1103/PhysRev.131.440

    work page doi:10.1103/physrev.131.440 1963

  36. [36]

    D. J. Kouri, A. Vijay, Inverse scattering theory: Renor malization of the lippmann–schwinger equation for acoustic scattering in one dimension, Physica l Review E 67 (4) (2003) 046614. doi:10.1103/PhysRevE.67.046614

    work page doi:10.1103/physreve.67.046614 2003

  37. [37]

    Jakobsen, R.-S

    M. Jakobsen, R.-S. Wu, X. Huang, Convergent scattering series solution of the inhomogeneous helmholtz equation via renormalization group and homotopy continuation approaches, Journal of Computational Physics 409 (2020) 109343. 22

    2020

  38. [38]

    Caron-Huot, M

    S. Caron-Huot, M. Correia, G. Isabella, M. Solon, Gravi tational wave scattering via the born se- ries: Scalar tidal matching to O(g7) and beyond, Physical Review Letters 135 (19) (2025) 191601. doi:10.1103/qd3c-nfz6

    work page doi:10.1103/qd3c-nfz6 2025

  39. [39]

    I. G. Graham, E. A. Spence, The helmholtz equation in het erogeneous media: a priori bounds, well-posedness, and resonances, Journal of Differential Eq uations 266 (6) (2019) 2869–2923. doi:10.1016/j.jde.2018.10.048

    work page doi:10.1016/j.jde.2018.10.048 2019

  40. [40]

    L. N. Trefethen, M. Embree, Spectra and Pseudospectra: The Behavior of Non-Normal Matrices and Operators, Princeton University Press, 2005

    2005

  41. [41]

    Moiola, E

    A. Moiola, E. A. Spence, Is the helmholtz equation reall y sign-indefinite?, SIAM Review 56 (2) (2014) 274–312. doi:10.1137/120901301

    work page doi:10.1137/120901301 2014

  42. [42]

    Jiang, L

    S. Jiang, L. Greengard, Efficient representation of wave fields using smooth absorbing layers, SIAM Journal on Scientific Computing 26 (4) (2004) 1213–1233. doi:10.1137/S1064827503420125

    work page doi:10.1137/s1064827503420125 2004

  43. [43]

    Bérenger, A perfectly matched layer for the absor ption of electromagnetic waves, Journal of Computational Physics 114 (2) (1994) 185–200

    J.-P. Bérenger, A perfectly matched layer for the absor ption of electromagnetic waves, Journal of Computational Physics 114 (2) (1994) 185–200

    1994

  44. [44]

    E. J. Alles, K. W. van Dongen, Perfectly matched layers f or frequency-domain integral equation acoustic scattering problems, IEEE transactions on ultras onics, ferroelectrics, and frequency control 58 (5) (2011) 1077–1086

    2011

  45. [45]

    Mandl, G

    F. Mandl, G. Shaw, Quantum Field Theory, 2nd Edition, Jo hn Wiley & Sons, 2010, a standard introduction to perturbative quantum field theory, Feynman diagrams, and renormalization

    2010

  46. [46]

    P. M. Morse, H. Feshbach, Methods of Theoretical Physic s, McGraw-Hill, New York, 1953, volumes I and II

    1953

  47. [47]

    A. T. de Hoop, Handbook of Radiation and Scattering of Wa ves, Academic Press, San Diego, 1995

    1995

  48. [48]

    Sreedharan, A

    M. Jakobsen, D. H. Saputera, F. A. Radu, Iterative solut ions of two coupled integral equations in acoustic scatteri ng in: Numerical Mathematics and Advanced Applications ENUMA TH 2023, Vol. 153 of Lecture Notes in Computational Science and Engineering, Springer, 2025, pp. 484–493. URL https://link.springer.com/chapter/10.1007/978-3-031 -86173-4_49

    work page doi:10.1007/978-3-031 2023

  49. [49]

    D. H. Saputera, M. Jakobsen, K. W. A. van Dongen, N. Jahan i, K. S. Eikrem, S. Alyaev, 3-d induction log modelling with integral equation method a nd domain decomposition pre-conditioning, Geophysical Journal International 236 (2) (2024) 834–848. doi:10.1093/gji/ggad454. URL https://doi.org/10.1093/gji/ggad454

    work page doi:10.1093/gji/ggad454 2024

  50. [50]

    Jakobsen, I

    M. Jakobsen, I. Pšenčík, E. Iversen, B. Ursin, Transiti on operator approach to seismic full-waveform inversion in arbitrary Communications in Computational Physics 28 (1) (2020). URL https://global-sci.com/cicp/article/view/6841

    2020

  51. [51]

    M. Reed, B. Simon, Methods of Modern Mathematical Physi cs, Vol. I: Functional Analysis, Aca- demic Press, 1972

    1972

  52. [52]

    Kjartansson, Constant q-wave propagation and atten uation, Journal of Geophysical Research: Solid Earth 84 (B9) (1979) 4737–4748

    E. Kjartansson, Constant q-wave propagation and atten uation, Journal of Geophysical Research: Solid Earth 84 (B9) (1979) 4737–4748

    1979

  53. [53]

    Carcione, et al., Wave fields in real media

    J. Carcione, et al., Wave fields in real media. theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic med ia (2022)

    2022

  54. [54]

    W. C. Chew, W. Liu, Perfectly matched layers for elastod ynamics: A new absorbing boundary condition, Journal of Computational Acoustics 4 (04) (1996 ) 341–359

    1996

  55. [55]

    F. Nataf, Absorbing boundary conditions and perfectly matched layers in wave propagation prob- lems, in: Direct and Inverse Problems in Wave Propagation an d Applications, De Gruyter, 2013, pp. 219–231

    2013

  56. [56]

    Lesage, R

    A. Lesage, R. Eftekhar, D. J. Kouri, J. Yao, F. Hussain, I nverse acoustic scattering series us- ing the volterra renormalization of the lippmann–schwinge r equation, in: SEG Technical Pro- gram Expanded Abstracts 2013, Society of Exploration Geoph ysicists, 2013, pp. 4645–4649. doi:10.1190/segam2013-1147.1. 23

    work page doi:10.1190/segam2013-1147.1 2013

  57. [57]

    F. Hussain, Multi-dimensional inverse acoustic scatt ering series using the volterra renormalization of the lippmann–schwinger equation, in: SEG Technical Prog ram Expanded Abstracts 2014, Society of Exploration Geophysicists, 2014. doi:10.1190/SEGAM2014-1349.1

    work page doi:10.1190/segam2014-1349.1 2014

  58. [58]

    Gell-Mann, F

    M. Gell-Mann, F. Low, Quantum electrodynamics at small distances, Physical Review 95 (1954) 1300

    1954

  59. [59]

    K. G. Wilson, Renormalization group and critical pheno mena. i. renormalization group and the kadanoff scaling picture, Phys. Rev. B 4 (1971) 3174

    1971

  60. [60]

    Xiang, K

    K. Xiang, K. S. Eikrem, M. Jakobsen, G. Nævdal, Homotopy scattering series for seismic for- ward modelling with variable density and velocity, Geophys ical Prospecting 70 (1) (2022) 3–18. doi:10.1111/1365-2478.13143

    work page doi:10.1111/1365-2478.13143 2022

  61. [61]

    Rotter, A non-hermitian hamilton operator and the ph ysics of open quantum sys- tems, Journal of Physics A: Mathematical and Theoretical 42 (15) (2009) 153001

    I. Rotter, A non-hermitian hamilton operator and the ph ysics of open quantum sys- tems, Journal of Physics A: Mathematical and Theoretical 42 (15) (2009) 153001. doi:10.1088/1751-8113/42/15/153001

    work page doi:10.1088/1751-8113/42/15/153001 2009

  62. [62]

    Jakobsen, D

    M. Jakobsen, D. H. Saputera, F. A. Radu, Iterative solut ions of two coupled integral equations in acoustic scattering theory, in: Numerical Mathematics a nd Advanced Applications ENUMATH 2023, Vol. 153 of Lecture Notes in Computational Science and Engineering, Springer, 2025, pp. 484–493. doi:10.1007/978-3-031-86173-4_49 . 24 Appendix A. Finite Difference...

    work page doi:10.1007/978-3-031-86173-4_49 2023