Multiparameter Eigenvalue Problem Research Articles

Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding

AbstractIt is well known that a family of $${n}\times {n}$$ n × n commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the n joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.

Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models

AbstractStandard multiparameter eigenvalue problems (MEPs) are systems of linear ‐parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time‐invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.

Globally Optimal SISO H2-Norm Model Reduction Using Walsh’s Theorem

We present a novel methodology for single-input single-output (SISO) H2-norm model reduction that guarantees global optimality of the obtained solution(s). By exploiting Walsh’s theorem, which is an elegant formulation of the first-order necessary conditions for optimality, we reformulate the model reduction problem as a multiparameter eigenvalue problem (MEVP), the real-valued eigentuples of which characterize the globally optimal solution(s) of the model reduction problem. While aiming for global optimality comes at the cost of a combinatorial growth of the problem complexity for increasing model orders, the novel methodology allows us to tackle larger problems compared to the few other globally optimal approaches in the literature. In particular, the degree of the obtained MEVP is independent of the order of the original higher order and obtained reduced-order model, a property that is favorable from a computational point of view. We perform three numerical experiments to illustrate the effectiveness of the methodology.

Two Double Recursive Block Macaulay Matrix Algorithms to Solve Multiparameter Eigenvalue Problems

We present two double recursive block Macaulay matrix algorithms to solve multiparameter eigenvalue problems (MEPs). In earlier work, we have developed a non-recursive approach that finds the solutions of an MEP via a multidimensional realization problem in the null space of the block Macaulay matrix constructed from the coefficient matrices of that MEP. However, this approach requires an iterative increase of the degree of the block Macaulay matrix: in order to determine whether the null space contains all the (affine) solutions of the MEP, we need to compute a basis matrix of the null space for every degree and check its dimension or rank structure. In this letter, we employ a recursive/sparse technique to compute a basis matrix of the null space of the block Macaulay matrix and a recursive technique to perform the necessary rank checks. We provide two system identification examples to show our improvements in computation time and memory usage.

Pointwise estimates for systems of coupled $ p $-Laplacian elliptic equations

This work examines positive solutions of systems of inequalities$ \pm \Delta_{{\bf{p}}} {\bf{u}} \ge \rho(x) {\bf{f}}({\bf{u}}), \qquad \mbox{ in } \Omega, $where $ {\bf{p}} = (p_1,...,p_k) $, $ p_i > 1 $ and $ \Delta_{{\bf{p}}} $ is the diagonal-matrix $ \text{diag}(\Delta_{p_1},...,\Delta_{p_k}) $, $ \Delta_{p_i} $ is the $ p_i $-Laplace operator, $ \Omega $ is an arbitrary domain (bounded or not) in $ {\mathbb{R}}^N $ ($ N\ge 2 $), $ {\bf{u}} = (u_1,...,u_k)^T $ and $ {\bf{f}} = (f_1,...,f_k)^T $ are vector-valued functions and $ \rho(x) $ is a nonnegative function in $ \Omega $ which is locally bounded. Using a maximum principle-based argument we provide explicit estimates on positive solutions $ u $ at each point $ x\in\Omega $, and as applications we find Liouville type results in unbounded domains such as $ {\mathbb{R}}^N $, exterior domains or generally unbounded domains with the property that $ \sup_{x\in\Omega}dist (x,\partial\Omega) = \infty $, for various nonlinearities $ {\bf{f}} $ and weights $ \rho $. We also give explicit upper bounds on extremal parameters of related nonlinear multi-parameter eigenvalue problems in bounded domains.

On checking dissipativity of parameterized linear and time‐invariant circuits and systems

SummaryA novel dissipativity characterization is proposed for linear time‐invariant circuits and systems whose dynamic behavior depends on one external parameter. The proposed formulation extends to the parameterized case standard Hamiltonian‐based algebraic dissipativity characterizations and is structured as an underdetermined multiparameter eigenvalue problem. Resulting from a polynomial parameterization, the proposed characterization leads to an effective set of algorithms that are able to determine the regions of local dissipativity and local activity in the frequency‐parameter plane, which in turn can be exploited by dissipativity enforcement algorithms to produce uniformly passive models. The reference application of proposed technique is behavioral modeling of complex circuits or systems, whose dynamic behavior can be compressed into a low‐order system through dedicated model order‐reduction processes. Various examples ranging from integrated components to microstrip filters and networks are used to illustrate the proposed characterization.

Two complementary block Macaulay matrix algorithms to solve multiparameter eigenvalue problems

We consider two algorithms that use the block Macaulay matrix to solve (rectangular) multiparameter eigenvalue problems (MEPs). On the one hand, a multidimensional realization problem in the null space of the block Macaulay matrix constructed from the coefficient matrices of an MEP results in a standard eigenvalue problem (SEP), the eigenvalues and eigenvectors of which yield the solutions of that MEP. On the other hand, we propose a complementary algorithm to solve MEPs that considers the data in the column space of the sparse and structured block Macaulay matrix directly, avoiding the computation of a numerical basis matrix of the null space. This paper generalizes, in a certain sense, traditional Macaulay matrix techniques from multivariate polynomial system solving to the block Macaulay matrix in the MEP setting.

On the singular two-parameter eigenvalue problem II

In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems, which is a key relation for many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems. In 2009, Muhič and Plestenjak extended the above relation to a class of singular two-parameter eigenvalue problems with coprime characteristic polynomials and such that all finite eigenvalues are algebraically simple. They introduced a way to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems. Using new tools, in particular the stratification theory, we extend this connection to singular two-parameter eigenvalue problems with possibly multiple eigenvalues and such that characteristic polynomials can have a nontrivial common factor.

Linearizable Eigenvector Nonlinearities

We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear functions of the eigenvector. The exact linearization relies on an equivalent multiparameter eigenvalue problem (MEP) that contains the exact solutions of the NEPv. Due to the characterization of MEPs in terms of a generalized eigenvalue problem this provides a direct way to compute all NEPv solutions for small problems, and it opens up the possibility to develop locally convergent iterative methods for larger problems. Moreover, the linear formulation allows us to easily determine the number of solutions of the NEPv. We propose two numerical schemes that exploit the structure of the linearization: inverse iteration and residual inverse iteration. We show how symmetry in the MEP can be used to improve reliability and reduce computational cost of both methods. Two numerical examples verify the theoretical results, and a third example shows the potential of a hybrid scheme that is based on a combination of the two proposed methods.

Applications of Multiparameter Eigenvalue Problems

It was mainly due to Atkinson works, who introduced Linear Multiparameter Eigenvalue problems (LMEPs), based on determinantal operators on the Tensor Product Space. Later, in the area of Multiparameter eigenvalue problems has received attention from the Mathematicians in the recent years also, who pointed out that there exist a variety of mixed eigenvalue problems with several parameters in different scientific domains. This article aims to bring into a light variety of scientific problems that appear naturally as LMEPs. Of course, with all certainty, the list of collection of applications presented here are far from complete, and there are bound to be many more applications of which we are currently unaware. The paper may provide a review on applications of Multiparameter eigenvalue problems in different scientific domains and future possible applicatios both in theoretical and applied disciplines.

Fiber product homotopy method for multiparameter eigenvalue problems

We develop a new homotopy method for solving multiparameter eigenvalue problems (MEPs) called the fiber product homotopy method. For a k-parameter eigenvalue problem with matrices of sizes \(n_1,\ldots ,n_k = O(n)\), fiber product homotopy method requires deformation of O(1) linear equations, while existing homotopy methods for MEPs require O(n) nonlinear equations. We show that the fiber product homotopy method theoretically finds all eigenpairs of an MEP with probability one. It is especially well-suited for a class of problems we call dimension-deficient singular problems that are generic with respect to intrinsic dimension, as the fiber product homotopy method is provably convergent with probability one for such problems as well, a fact borne out by numerical experiments. More generally, our numerical experiments indicate that the fiber product homotopy method significantly outperforms the standard Delta method in terms of accuracy, with consistent backward errors on the order of \(10^{-16}\) without any use of extended precision. In terms of speed, it significantly outperforms previous homotopy-based methods on all problems and outperforms the Delta method on larger problems, and is also highly parallelizable. We show that the fiber product MEP that we solve in the fiber product homotopy method, although mathematically equivalent to a standard MEP, is typically a much better conditioned problem.

Shapley–Snow Kernels, Multiparameter Eigenvalue Problems, and Stochastic Games

We propose a connection between finite zero-sum stochastic games (henceforth stochastic games) and multiparameter eigenvalue problems. This connection, which relies on the theory developed by Shapley and Snow for matrix games, opens new possibilities in the study of stochastic games. In particular, we derive from this connection a handful of new results for stochastic games.

Globally Optimal H2-Norm Model Reduction: A Numerical Linear Algebra Approach

We show that the H2-norm model reduction problem for single-input/single-output (SISO) linear time-invariant (LTI) systems is essentially an eigenvalue problem (EP), from which the globally optimal solution(s) can be retrieved. The first-order optimality conditions of this model reduction problem constitute a system of multivariate polynomial equations that can be converted to an affine (or inhomogeneous) multiparameter eigenvalue problem (AMEP). We solve this AMEP by using the so-called augmented block Macaulay matrix, which is introduced in this paper and has a special (block) multi-shift invariant null space. The set of all stationary points of the optimization problem, i.e., the (2r)-tuples (r is the order of the reduced model) of affine eigenvalues and eigenvectors of the AMEP, follows from a standard EP related to the structure of that null space. At least one of these (2r)-tuples corresponds to the globally optimal solution of the H2-norm model reduction problem. We present a simple numerical example to illustrate our approach.

Koszul-Type Determinantal Formulas for Families of Mixed Multilinear Systems

Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We construct determinantal formulas for two kind of multilinear systems related to the Multiparameter Eigenvalue Problem (MEP): first, when the polynomials agree in all but one block of variables; second, when the polynomials are bilinear with different supports, related to a bipartite graph. We use the Weyman complex to construct Koszul-type determinantal formulas that generalize Sylvester-type formulas. We can use the matrices associated to these formulas to solve square systems without computing the resultant. The combination of the resultant matrices with the eigenvalue and eigenvector criterion for polynomial systems leads to a new approach for solving MEP.

Multiparameter Eigenvalue Problems and Shift-invariance

We discuss four eigenvalue problems of increasing generality and complexity: rooting a univariate polynomial, solving the polynomial eigenvalue problem, rooting a set of multivariate polynomials and solving multi-parameter eigenvalue problems. In doing so, we provide a unifying framework for solving these eigenvalue problems, where we exploit properties of (block-) (multi-) shift-invariant subspaces and use multi-dimensional realization algorithms.

Accelerating the Sinkhorn–Knopp iteration by Arnoldi-type methods

It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a constrained nonlinear multiparameter eigenvalue problem. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the diagonal transformations. Numerical results illustrate that our novel methods accelerate significantly the convergence of the customary Sinkhorn-Knopp iteration for matrix balancing in the case of clustered dominant eigenvalues.

Globally Optimal Least-Squares ARMA Model Identification is an Eigenvalue Problem

We show that globally optimal least-squares identification of autoregressive moving-average (ARMA) models is an eigenvalue problem (EP). The first order optimality conditions of this identification problem constitute a system of multivariate polynomial equations, in which most variables appear linearly. This system is basically a multiparameter eigenvalue problem (MEP), which we solve by iteratively building a so-called block Macaulay matrix, the null space of which is block multi-shift-invariant. The set of all stationary points of the optimization problem, i.e., the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -tuples of eigenvalues and eigenvectors of the MEP, follows from a standard EP related to the multidimensional realization problem in that null space. At least one of these <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -tuples corresponds to the global minimum of the original least-squares objective function. Contrary to existing heuristic techniques, this approach yields the globally optimal parameters of the ARMA model. We provide a numerical example to illustrate the new identification method.

Nonlinear Multiparameter Eigenvalue Problems in Aeroelasticity

In this work, we devise solution algorithms for nonlinear multiparameter eigenvalue problems arising in the analysis of aeroelastic flutter. Two iterative algorithms are devised, as well as a restriction method for simplifying the system behavior away from the desired flutter points. The algorithms are tested on a sectional model and on the Goland wing. Both are found to have fast and reliable convergence properties, yielding flutter point solutions which are validated by the literature. The definition of the eigenvalues is found to have a significant influence on the convergence properties of the algorithm; preferable choices for eigenvalue definitions are noted. The computational costs of the algorithms are tested and discussed; they are found to be favorable relative to other approaches from the literature. Opportunities for extending these methods are also tested and discussed.

Perturbation of nonlinear operators in the theory of nonlinear multifrequency electromagnetic wave propagation

The paper develops an original approach to study nonlinear multiparameter eigenvalue problems arising in the theory of nonlinear multifrequency electromagnetic wave propagation. The problem under consideration is a multiparameter eigenvalue problem that under some conditions degenerates into n nonlinear one-parameter eigenvalue problems. Further simplification reduces the one-parameter nonlinear problems to linear (one-parameter) eigenvalue problems. Each of the linear problems has a finite number of positive eigenvalues, whereas each of the nonlinear (one-parameter) problems has an infinite number of positive eigenvalues. Using the nonlinear one-parameter problems as ’nonperturbed’ ones, one can prove existence of eigentuples of the multiparameter problem that have no connections with solutions to the linear (one-parameter) problems even if the nonlinear terms have small factors.

Nonlinear multiparameter eigenvalue problems: Linearised and nonlinearised solutions

The paper focuses on a nonlinear multiparameter eigenvalue problem called P. This problem involves n spectral parameters and also depends on n2 numerical parameters, where n⩾2 is an integer. If these numerical parameters vanish, P degenerates into n linear (one-parameter) eigenvalue problems called Pi0 (i=1,n‾). In connection with P one can consider another n nonlinear one-parameter eigenvalue problems called Pi. The problems Pi have eigenvalues with as well as without linear counterparts. The paper suggests to consider Pi as ‘nonunperturbed’ instead of Pi0. Using properties of eigenvalues of Pi, one manages to prove existence of eigentuples of P. Among the eigentuples found in this way, there are eigetuples with as well as without linear counterparts. Results of the paper are found with a nonclassical approach. Applications of the found results to nonlinear optics are shown.