Computing diffraction anomalies as nonlinear eigenvalue problems.

Preconditioned iterative methods for a class of nonlinear eigenvalue problems

In the present paper, the differential nonlinear eigenvalue problem describing longitudinal eigenvibrations of an elastic bar with elastically attached load to the bar end is considered. The original differential eigenvalue problem is approximated by a mesh scheme of the finite difference method on a uniform grid. The finite difference scheme is equivalent to the matrix nonlinear eigenvalue problem. To solve this matrix nonlinear eigenvalue problem, the bisection method is proposed and studied. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with elastically attached loads.

Fictitious eigenfrequencies in the BEM for interior acoustic problems

Investigation of finite/infinite unidirectional elastic phononic plates by BEM

We consider the nonlinear eigenvalue problem <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta u plus g left-parenthesis u right-parenthesis equals lamda sine u normal i normal n normal upper Omega comma u greater-than 0 normal i normal n normal upper Omega comma u equals 0 normal o normal n partial-differential normal upper Omega comma"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> λ </mml:mi> <mml:mi>sin</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>u</mml:mi> <mml:mspace width="1em"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mspace width="1em"/> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>u</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="1em"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mspace width="1em"/> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="1em"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mspace width="1em"/> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">-\Delta u + g(u) = \lambda \sin u \quad \mathrm {in} \quad \Omega , \quad u &gt; 0\quad \mathrm {in} \quad \Omega , \quad u = 0 \quad \mathrm {on} \quad \partial \Omega ,</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega subset-of bold upper R Superscript upper N Baseline left-parenthesis upper N greater-than-or-equal-to 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo> ⊂ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:mtext> </mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega \subset {\mathbf {R}}^N\ (N \ge 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an appropriately smooth bounded domain and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> λ </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda &gt; 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a parameter. It is known that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda much-greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi> λ </mml:mi> <mml:mo> ≫ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \gg 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then the corresponding solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript lamda"> <mml:semantics> <mml:msub> <mml:mi>u</mml:mi> <mml:mi> λ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">u_\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is almost flat and almost equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> inside <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We establish an asymptotic expansion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript lamda Baseline left-parenthesis x right-parenthesis left-parenthesis

Positive solutions for nonlinear m-point eigenvalue problems

This paper is to introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We will prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.

Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth problem in a damaged medium

On the basis of computer simulation, we demonstrate the possibility of a new type of “wandering” solitons implementation in nonlinear periodic layered structures. “Wandering” soliton moves across the layers, repeatedly changing its direction of motion due to the reflection from the photonic crystal (PC) boundaries with the ambient medium. The initial soliton is located inside a PC and occupies several of its layers. Its profile can be found as the solution of the corresponding nonlinear eigenvalue problem. “Wandering” solitons are formed as a result of a large perturbation of the wave vector, which leads to the soliton motion across photonic crystal layers. In the process of reflection from the boundary with the ambient medium, the soliton partly penetrates into the ambient medium at a depth equal to the width of several PC layers. A slow return of light energy, which previously left the PC, can take place at this moment.

The computation of dominant eigenvalues of second-order linear control systems with multiple time-delays is tackled by using a contour integral method. The proposed approach depends on a reduced characteristic function and the associated characteristic matrix comprised of measured open-loop receptances. This reduced characteristic function is derived from the original characteristic function of the second-order time delayed systems based on the reasonable assumption that eigenvalues of the closed-loop system are distinct from those of the open-loop system, and has the same eigenvalues as those of the original. Then, the eigenvalues computation is equivalent to solve a nonlinear eigenvalue problem of the associated characteristic matrix by using a contour integral method. The proposed approach also utilizes the spectrum distribution features of the retarded time-delay systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed approach.

In the present paper approximate analytical and numeric solutions to nonlinear eigenvalue problems arising in nonlinear fracture mechanics in analysis of stress — strain fields near a crack tip under mixed mode loading are presented. Asymptotic solutions are obtained via perturbation method technique (small parameter method). The artificial small parameter is the difference between the eigenvalue corresponding to the nonlinear eigenvalue problem and the eigenvalue related to the linear ”undisturbed” problem. It is shown that the perturbation technique gives an effective method of solving nonlinear eigenvalue problems in nonlinear fracture mechanics. Comparison results of numeric and asymptotic results for different value of the mixity parameter and hardening exponent shows good agreement. Thus the perturbation theory technique for study of nonlinear eigenvalue problems is offered and applied for eigenvalue problems arising from fracture mechanics analysis in the case of mixed mode loading.

The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental equation can be integrated once to a first order nonlinear equation, e.g., the Ricatti equation. It is shown that the nonlinear eigenvalue problems of these semi-transcendental equations are equivalent to linear eigenvalue problems. They share the exactly same eigenvalues. The eigensolutions in the two problems are closely related. The nonlinear eigenvalue problem equivalent to the (half) harmonic oscillator in quantum mechanics is solved exactly. This is the first solvable nonlinear eigenvalue problem. The nonlinear eigenvalue problems of some extended equations are also studied.

A novel bi-infinite approach to compute the band structures of 2D photonic superlattices with 1D quasicrystal sequences is devised. Leveraging strategically the bi-infinite characteristic, the approach first transforms the infinite-dimensional eigenvalue problem into a finite-dimensional nonlinear eigenvalue problem (NEVP) on a single cell for efficient numerical solution. Challengingly, the NEVP is built upon the solutions to two systems of cyclic nonlinear matrix equations (NMEs) that have to be solved repeatedly during iteratively solving the NEVP. The solutions are efficiently calculated by a newly developed highly efficient coalescing technique followed by a structure-preserving doubling algorithm. It is showed that the cost of coalescing is proportional to the logarithm of N, the length of the truncated quasicrystal sequence, which is significant as the cost of coalescing becomes more noticeable as N gets bigger for highly accurate simulations. Finally, through mathematical analysis, inclusion intervals for eigenvalue of interest are estimated so as to significantly narrow down the scope of search, and that significantly contributes to the overall efficiency of the approach, as the NEVP is nonlinear in nature and has to be solved iteratively.

The partial Schur factorization can be used to represent several eigenpairs of a matrix in a numerically robust way. Different adaptions of the Arnoldi method are often used to compute partial Schur factorizations. We propose here a technique to compute a partial Schur factorization of a nonlinear eigenvalue problem (NEP). The technique is an extension of our algorithm from [E. Jarlebring, W. Michiels, and K. Meerbergen, Numer. Math., 122 (2012), pp. 169--195], now called the infinite Arnoldi method. The infinite Arnoldi method is a method designed for NEPs, and can be interpreted as Arnoldi's method applied to a linear infinite-dimensional operator, whose reciprocal eigenvalues are the solutions to the NEP. As a first result we show that the invariant pairs of the operator are equivalent to invariant pairs of the NEP. We characterize the structure of the invariant pairs of the operator and show how one can carry out a modification of the infinite Arnoldi method by respecting the structure. This also allows us to naturally add the feature known as locking. We nest this algorithm with an outer iteration, where the infinite Arnoldi method for a particular type of structured functions is appropriately restarted. The restarting exploits the structure and is inspired by the well-known implicitly restarted Arnoldi method for standard eigenvalue problems. The final algorithm is applied to examples from a benchmark collection, showing that both processing time and memory consumption can be considerably reduced with the restarting technique.

We present a new computational approach for a class of large-scale nonlinear eigenvalue problems (NEPs) that are nonlinear in the eigenvalue. The contribution of this paper is twofold. We derive a new iterative algorithm for NEPs, the tensor infinite Arnoldi method (TIAR), which is applicable to a general class of NEPs, and we show how to specialize the algorithm to a specific NEP: the waveguide eigenvalue problem. The waveguide eigenvalue problem arises from a finite-element discretization of a partial differential equation used in the study waves propagating in a periodic medium. The algorithm is successfully applied to accurately solve benchmark problems as well as complicated waveguides. We study the complexity of the specialized algorithm with respect to the number of iterations $m$ and the size of the problem $n$, both from a theoretical perspective and in practice. For the waveguide eigenvalue problem, we establish that the computationally dominating part of the algorithm has complexity $\mathcal{O}(nm^2+\sqrt{n}m^3)$. Hence, the asymptotic complexity of TIAR applied to the waveguide eigenvalue problem, for $n\rightarrow\infty$, is the same as for Arnoldi's method for standard eigenvalue problems.