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Iowa State University

1. Jacobs, Matthew Aaron. Asymptotic solutions for high frequency Helmholtz equations.

Degree: 2020, Iowa State University

URL: https://lib.dr.iastate.edu/etd/18151

In this thesis, we will investigate and develop asymptotic methods for numerically solving high frequency Helmholtz equations with point-source conditions. Due to the oscillatory nature of the wave, such equations are highly challenging to solve by conventional methods, such as the finite difference and finite element methods, since they often suffer from a large number of degrees of freedom to avoid the `pollution effect' (large dispersion errors). We shall first apply the geometrical optics (GO) approximation to compute the wave locally near the primary source, where instead of computing the oscillatory wave directly, its phase and amplitudes are computed through the eikonal equation and a recurrent system of transport equations, respectively, and are used to reconstruct the wave for any high frequencies. The GO approximation is efficient for providing locally valid approximations of the wave. We propose to further propagate the wave to the whole domain of interest through an appropriate time-dependent Schrödinger equation whose steady-state solution in the domain of interest will provide globally valid approximations of the wave. The wavefunction of the Schrödinger equation can be propagated by a Strang operator splitting based pseudo-spectral method that is unconditionally stable, which allows large time step sizes to reach the steady state efficiently. In the pseudo-spectral method, wherever the matrix exponential is involved, the Krylov subspace method can be used to compute the relevant matrix-vector products. The proposed asymptotic method will be effective since: (1) it is able to obtain globally valid approximations of the wave, (2) it has complexity O(Nlog N) where N is the total number of simulation points for a prescribed accuracy requirement, and (3) the number of simulation points per wavelength can be fixed as the frequency increases. Numerical experiments in both two- and three-dimensional spaces will be performed to demonstrate the method.

Subjects/Keywords: Anisotropic Helmholtz equation; Babich's expansion; Geometrical optics; Pseudo-spectral method; Strang operator splitting; Time-dependent Schr\"{o}dinger equation

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Jacobs, M. A. (2020). Asymptotic solutions for high frequency Helmholtz equations. (Thesis). Iowa State University. Retrieved from https://lib.dr.iastate.edu/etd/18151

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Jacobs, Matthew Aaron. “Asymptotic solutions for high frequency Helmholtz equations.” 2020. Thesis, Iowa State University. Accessed March 03, 2021. https://lib.dr.iastate.edu/etd/18151.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Jacobs, Matthew Aaron. “Asymptotic solutions for high frequency Helmholtz equations.” 2020. Web. 03 Mar 2021.

Vancouver:

Jacobs MA. Asymptotic solutions for high frequency Helmholtz equations. [Internet] [Thesis]. Iowa State University; 2020. [cited 2021 Mar 03]. Available from: https://lib.dr.iastate.edu/etd/18151.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Jacobs MA. Asymptotic solutions for high frequency Helmholtz equations. [Thesis]. Iowa State University; 2020. Available from: https://lib.dr.iastate.edu/etd/18151

Not specified: Masters Thesis or Doctoral Dissertation

2. B. Langella. NORMAL FORM AND KAM METHODS FOR HIGHER DIMENSIONAL LINEAR PDES.

Degree: 2020, Università degli Studi di Milano

URL: http://hdl.handle.net/2434/798372

In this thesis an approach to linear PDEs on higher dimensional spatial domains is proposed. I prove two kinds of results: first I develop an algorithm which enables to obtain reducibility for linear PDEs which depend quasi-periodically on time, and I apply it to a quasilinear transport equation of the form
∂_{t} u= ν•∇u+ ε P (ωt)u on the d-dimensional torus T^{d}, where ε is a small parameter, ν and ω are Diophantine vectors, P (ωt)=V(x,ωt)•∇+W(ωt), V is a smooth function on T^(d+n) and W(ωt) is an unbounded pseudo-differential operator of order strictly less than 1. The strategy is an extension of the methods originally developed in the context of quasilinear one dimensional equations. It consists in first using quantum normal form techniques in order to conjugate the original system to a new one with a smoothing perturbation, and then exploiting the smoothing nature of the new perturbation in order to balance the effects of the small denominators, which in this problem accumulate very fast to 0.
The quantum normal form procedure developed in order to obtain reducibility for the above transport equation is global in phase space. In order to overcome such a limitation, the second problem I tackle in this thesis is that of developing a local quantum normal form procedure, which could be applied to much more general systems. As the simplest relevant model containing all the difficulties of the general case, I consider the operator H=-∆+V(x) with Floquet boundary conditions on the flat torus T^{d}_Γ, where T^{d}_Γ is the manifold obtained as quotient between the d-dimensional space R^{d} and an arbitrary d-dimensional lattice Γ, with the purpose of adapting the quantum normal form procedure to deal with this operator. As a result, I prove for the operator H a Structure Theorem à la Nekhoroshev, and I characterize the asymptotic behavior of all its eigenvalues. The asymptotic expansion is in |λ|^{-δ}, with δ ∊ (0, 1) for most of the eigenvalues λ (stable eigenvalues), while it is a "directional expansion" for the remaining eigenvalues (unstable eigenvalues).
*Advisors/Committee Members: tutor: D. Bambusi, co-tutor: R. Montalto, coordinatore: V. Mastropietro, BAMBUSI, DARIO PAOLO, MASTROPIETRO, VIERI.*

Subjects/Keywords: reducibility; pseudo-differential operators; normal form; Schrö; dinger operator; spectral asymptotics; Nekhoroshev theorem; Settore MAT/07 - Fisica Matematica

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Langella, B. (2020). NORMAL FORM AND KAM METHODS FOR HIGHER DIMENSIONAL LINEAR PDES. (Thesis). Università degli Studi di Milano. Retrieved from http://hdl.handle.net/2434/798372

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Langella, B.. “NORMAL FORM AND KAM METHODS FOR HIGHER DIMENSIONAL LINEAR PDES.” 2020. Thesis, Università degli Studi di Milano. Accessed March 03, 2021. http://hdl.handle.net/2434/798372.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Langella, B.. “NORMAL FORM AND KAM METHODS FOR HIGHER DIMENSIONAL LINEAR PDES.” 2020. Web. 03 Mar 2021.

Vancouver:

Langella B. NORMAL FORM AND KAM METHODS FOR HIGHER DIMENSIONAL LINEAR PDES. [Internet] [Thesis]. Università degli Studi di Milano; 2020. [cited 2021 Mar 03]. Available from: http://hdl.handle.net/2434/798372.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Langella B. NORMAL FORM AND KAM METHODS FOR HIGHER DIMENSIONAL LINEAR PDES. [Thesis]. Università degli Studi di Milano; 2020. Available from: http://hdl.handle.net/2434/798372

Not specified: Masters Thesis or Doctoral Dissertation