The bisection method for solving the nonlinear bar eigenvalue problem

Preconditioned iterative methods for a class of nonlinear eigenvalue problems

The nonlinear second-order differential eigenvalue problem describing eigenvibrations of a bar with elastically attached load is investigated. The existence of an increasing sequence of positive simple eigenvalues with a limit point at infinity is established. To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. The initial nonlinear differential eigenvalue problem is approximated by the finite element method on a uniform grid. The accuracy of approximate solutions is studied. The investigations in the present paper can be extended to the cases of more complicated and important problems on eigenvibrations of beams, plates, and shells with elastically attached loads.

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

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.

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.

The aim of this study is to efficiently calculate parametric thermoacoustic maps of typical combustion chambers. Two configurations are considered: an academic configuration based on a Rijke tube, and an industrial combustion chamber, which is the core of a recently developed microturbine for power generation. Such maps can be understood as the collection of loci of thermoacoustic eigenfrequencies obtained under systematic variations of some defined parameters, while considering the Helmholtz equation as the thermoacoustic model of interest. In this study we consider variations on two parameters: the gain n and time-delay τ associated with a generic flame response model. We also show the feasibility of the proposed approach when considering more realistic flame responses. A straight-forward way to calculate such a thermoacoustic map is by solving the Helmholtz equation, and, thus, the corresponding nonlinear eigenvalue problem (NLEVP), one time per parameter combination. With that approach, the nonlinear eigenvalue problem needs to be solved hundreds or thousands of times if an adequate resolution of the thermoacoustic map is sought. Such a strategy may be computationally unaffordable. In order to overcome this difficulty, this work utilizes an adjoint-based, high-order perturbation method. The actual eigenvalue problem is only solved once at a baseline point. After applying the perturbation equations at that point, a polynomial rational function—the Padé approximant—is obtained to estimate the eigenfrequency drift that results for a small or large perturbation in the flame response. It is demonstrated, for both academic and industrial test cases, that the obtained maps are accurate. Additionally, it is shown that these maps reveal a large variety of thermoacoustic features, such as stability boundaries, intrinsic thermoacoustic modes, and exceptional points. The numerical costs for such calculations are negligible even for the industrial combustion chamber investigated.

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.

A quadratically converging algorithm for the computation of the distance to instability of a broad class of nonlinear eigenvalue problems is presented, including the polynomial eigenvalue problem and the delay eigenvalue problem. The algorithm is grounded in a recently presented approach for computing the pseudospectral abscissa. The application of the algorithm only relies on the availability of a method to compute the rightmost eigenvalue of perturbed problems obtained by adding rank one perturbations to the coefficient matrices, for which, in case of large and sparse matrices, efficient iterative algorithms can be used.

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.

Chapter 4 Nonlinear eigenvalue problem with quantization

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