
Luka GrubisicUniversity of Zagreb · Department of Mathematics
Luka Grubisic
dr.rer.nat.
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70
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449
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Introduction
Additional affiliations
May 2018 - present
October 2007 - April 2018
September 2005 - September 2007
Education
November 2001 - August 2005
Publications
Publications (70)
In this paper we study eigenvalue estimates for the solutions of Lyapunov equations with a noncompact (but relatively Hilbert Schmidt) control operator. We compute eigenvalue estimates from Galerkin discretizations of Lyapunov equations and discuss the appearance of a spurious branch in the discrete spectrum. This phenomenon is called the spectral...
Various iterative eigenvalue solvers have been developed to compute parts of the spectrum for a large sparse matrix, including the power method, Krylov subspace methods, contour integral methods, and preconditioned solvers such as the so called LOBPCG method. All of these solvers rely on random matrices to determine, e.g., starting vectors that hav...
We propose and study a method for finding quasi-resonances for a linear acoustic transmission problem in frequency domain. Starting from an equivalent boundary-integral equation we perform Galerkin boundary element discretization and look for the minima of the smallest singular value of the resulting matrix as a function of the wave number k. We de...
In this paper, we present an adaptive spectral projection based finite element method to numerically approximate the solution of the wave equation with memory. The adaptivity is not restricted to the mesh (hp-adaptivity), but it is also applied to the size of the computed spectrum (k-adaptivity). The meshes are refined using a residual based error...
We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding eigenvalues are separated from the rest of the spectrum for all values of the parameters. We show that such eigenspaces...
In this paper, we consider the numerical inverse Laplace transform for distributed order time-fractional equations, where a discontinuous Galerkin scheme is used to discretize the problem in space. The success of Talbot’s approach for the computation of the inverse Laplace transform depends critically on the problem’s spectral properties and we pre...
In this manuscript we present a mathematical theory and a computational algorithm to study optimal design of mesh-like structures such as metallic stents by changing the stent strut thickness and width to optimize the overall stent compliance. The mathematical constrained optimization problem is to minimize the “compliance functional” over a closed...
We present a low-rank greedily adapted hp -finite element algorithm for computing an approximation to the solution of the Lyapunov operator equation. We show that there is a hidden regularity in eigenfunctions of the solution of the Lyapunov equation which can be utilized to justify the use of high order finite element spaces. Our numerical experim...
We present an algorithm for the fully automatic generation of a class-compliant mesh for ship structural analysis. Our algorithm is implemented as an end-to-end solution. It starts from a description of a geometry and produces a class conforming surface mesh as a result. The algorithm consists of two parts, the automatic geometry refinement and the...
This paper presents an algorithm for the fully automatic mesh generation for the finite element analysis of ships and offshore structures. The quality requirements on the mesh generator are imposed by the acceptance criteria of the classification societies as well as the need to avoid shear locking when using low degree shell elements. The meshing...
This paper presents an algorithm for the fully automatic mesh generation for the finite element analysis of ships and offshore structures. The quality requirements on the mesh generator are imposed by the acceptance criteria of the classification societies as well as the need to avoid shear locking when using low degree shell elements. The meshing...
In this paper, we compare two approaches to numerically approximate the solution of second-order Gurtin-Pipkin type of integro-differential equations. Both methods are based on a high-order Discontinous Galerkin approximation in space and the numerical inverse Laplace transform. In the first approach, we use functional calculus and the inverse Lapl...
We propose a method to accelerate the solution of 3D FEM-discretized nonlinear eigenvalue problems by drastically reducing the problem dimension. Our method yields a reduced order model (ROM) via a projection onto a suitable subspace, with eigenpairs identical to the full problem in a region of the complex plane. The subspace is automatically const...
In this paper we consider a constrained parabolic optimal control problem. The cost functional is quadratic and it combines the distance of the trajectory of the system from the desired evolution profile together with the cost of a control. The constraint is given by a term measuring the distance between the final state and the desired state toward...
Contour integration methods are claimed to be the methods of choice for computing many (several hundred) eigenvalues of a nonlinear eigenvalue problem inside a closed region of the complex plane. Typically, contour integration methods are designed for circular (or more generally elliptic) shaped contours and rely on the exponential convergence of t...
We propose an a posteriori error estimator for high-order p- or hp-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant subspace. The estimator is based on the computation of approximate error funct...
Discontinuous Galerkin composite finite element methods (DGCFEM) are designed to tackle approximation problems on complicated domains. Partial differential equations posed on complicated domain are common when there are mesoscopic or local phenomena which need to be modeled at the same time as macropscopic phenomena. In this paper, an optical latti...
We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is motivated by the analysis of inexact (perturbed) inverse iteration algorithms in numerical linear algebra. We...
We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is motivated by the analysis of inexact (perturbed) inverse iteration algorithms in numerical linear algebra. We...
We study eigenmode localization for a class of elliptic reaction-diffusion operators. As the prototype model problem we use a family of Schrödinger Hamiltonians parametrized by random potentials and study the associated effective confining potential. This problem is posed in the finite domain and we compute localized bounded states at the lower end...
A new model description for the numerical simulation of elastic frame structures is proposed. Instead of resolving algebraic constraints at frame nodes and incorporating them into the finite element spaces, the constraints are included explicitly in the model via new variables and enforced via Lagrange multipliers. Based on the new formulation, an...
We propose an a posteriori error estimator for high-order $p$- or $hp$-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant subspace. The estimator is based on the computation of approximate error f...
We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-indefinite symmetric sesquilinear forms as well as the associated operators that are semi-bounded neither from below nor from above. In the semi-bounded case, we refine the obtained results and, as an example, revisit the block Stokes operator from flui...
We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding eigenvalues are separated from the rest of the spectrum for all values of the parameters. We show that such eigenspaces...
A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as “FEAST”, has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the pa...
A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known by the acronym FEAST, has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent...
We are interested in numerical methods for approximating vector‐valued functions on a metric graph. As a model problem, we formulate and analyze a numerical method for the solution of the stationary problem for the one‐dimensional elastic stent model. The approximation is built using the mixed finite element method. The discretization matrix is a s...
A new model description for the numerical simulation of elastic stents is proposed. Based on the new formulation an inf-sup inequality for the finite element discretization is proved and the proof of the inf-sup inequality for the continuous problem is simplified. The new formulation also leads to faster simulation times despite an increased number...
This paper covers the analysis of (reference) position estimation procedures in
the navigation application domain, discovering of potential weaknesses and
suggesting the improvement. The algorithm analysis involves practical
performance and rating of the algorithm performance quality using measured
pseudorange. For the empirical comparison, we used...
This paper presents a design of a system for monitoring and recording the influence of a running sea on a vessel in motion. Our approach is based on machine learning techniques that relate measured wave parameters (encounter angle, wave height and wave amplitude) with measured motion characteristics of the vessel. High quality GRIB data for wave me...
We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-indefinite symmetric sesquilinear forms as well as the associated operators that are semi-bounded neither from below nor from above. In the semi-bounded case, we refine the obtained results and, as an example, revisit the block Stokes Operator from flui...
We consider the problem of computing a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. A rational function of the operator is constructed such that the eigenspace of interest is its dominant eigenspace, and a subspace iteration procedure is used to approximate this eigenspace...
We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating t...
We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating t...
We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asympt...
In this note, we study the eigenvalue problem for a class of block operator matrix pairs. Our study is motivated by an analysis of abstract
differential algebraic equations. Such problems frequently appear in the study of complex systems, e.g. differential equations posed on metric
graphs, in mixed variational formulation.
In this paper we formulate and analyze the mixed formulation of the one-dimensional equilibrium model of elastic stents. The model is based on the curved rod model for the inextensible and ushearable struts and is formulated in the weak form in \v{C}ani\'{c} and Tamba\v{c}a, 2012. It is given by a system of ordinary differential equations at the gr...
In this paper we formulate and analyze the mixed formulation of the one-dimensional equilibrium model of elastic stents. The model is based on the curved rod model for the inextensible and ushearable struts and is formulated in the weak form in \v{C}ani\'{c} and Tamba\v{c}a, 2012. It is given by a system of ordinary differential equations at the gr...
We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we...
We present an hp-adaptive continuous Galerkin (hp-CG) method for approximating eigenvalues of elliptic operators, and demonstrate its utility on a collection of benchmark problems having features seen in many important practical applications—for example, high-contrast discontinuous coefficients giving rise to eigenfunctions with reduced regularity....
In this paper we present new double angle theorems for the rotation of the eigenspaces of Hermitian matrix pairs (H;M), where H is a non-singular matrix which can be factorized as H = GJG∗, J = diag(±1); andM is non-singular. The rotation of the eigenspaces is measured in the matrix-dependent scalar product, and the bounds belong to relative pertur...
We present an algorithm for approximating an eigensubspace of a spectral component of an analytic Fredholm valued function. Our approach is based on numerical contour integration and the analytic Fredholm theorem. The presented method can be seen as a variant of the FEAST algorithm for infinite dimensional nonlinear eigenvalue problems. Numerical e...
We present new residual estimates based on Kato’s square root theorem for spectral approximations of non-self-adjoint differential operators of convection–diffusion–reaction type. It is not assumed that the eigenvalue/vector approximations are obtained from any particular numerical method, so these estimates may be applied quite broadly. Key eigenv...
We present perturbation estimates for eigenvalue and eigenvector approximations for a class of Fredholm operator-valued functions. Our approach is based on perturbation estimates for the generalized resolvents and the exponential convergence of the contour integration by the trapezoidal rule. We use discrete residual functions to estimate the resol...
This paper is concerned with the eigenvalue decay of the solution to operator Lyapunov equations with right-hand sides of finite rank. We show that the k-th (generalized) eigenvalue decays exponentially in sqrt(k), provided that the involved operator A generates an exponentially stable analytic semigroup, and A is either self-adjoint or diagonaliza...
This paper studies the perturbation theory for spectral projections of Hermitian matrix pairs (H,M) , where H is a non-singular Hermitian matrix which can be factorized as H=GJG^*, J=diag(\pm 1) and M is positive definite. The class of allowed perturbations is so restricted that the corresponding perturbed pair (tH,tM)=(H+dH,M+dM) must have the for...
A version of the Davis–Kahan Tan 2Θ theorem for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a result by Motovilov and Selin.
The main contribution of this paper is an error representation formula for eigenvalue approximations for positive definite operators defined as quadratic forms. The formula gives an operator theoretic framework for treating discrete eigenvalue approximation/estimation problems for unbounded positive definite operators independent of the multiplicit...
We present reliable
$\alpha $
-posteriori error estimates for
$hp$
-adaptive finite element approximations of semi-definite eigenvalue/eigenvector problems. Our model problems are motivated by applications in photonic crystal eigenvalue computations. We present detailed numerical experiments confirming our theory and give several benchmark resu...
A discontinuous Galerkin method, with hp-adaptivity based on the approximate solution of appropriate dual problems, is employed for highly-accurate eigenvalue
computations on a collection of benchmark examples. After demonstrating the effectivity
of our computed error estimates on a few well-studied examples, we present results for several exampl...
We present a general framework for the a posteriori estimation and enhancement of error in eigenvalue/eigenvector computations for symmetric and elliptic eigenvalue problems, and provide detailed analysis of a specific and important example within this framework—finite element methods with continuous, affine elements. A distinguishing feature of th...
We present reliable a-posteriori error estimates for $hp$-adaptive finite
element approximations of eigenvalue/eigenvector problems. Starting from our
earlier work on $h$ adaptive finite element approximations we show a way to
obtain reliable and efficient a-posteriori estimates in the $hp$-setting. At
the core of our analysis is the reduction of t...
Given a two dimensional signal, e.g. surface recorded 2-D wavefield representing marine seismic data, we consider the task of removing interference due to multiple reflection (the so-called multiple events). This problem has been successfully modeled in the previous work by the third author as a matrix optimization problem. Seen from an abstract pe...
We present new sin Theta theorems for perturbations of positive definite matrix pairs. The rotation of eigenspaces is measured in the matrix dependent scalar product. We assess the sharpness of the new estimates in terms of effectivity quotients (the quotient of the measure of the perturbation and the estimator). Known relative sin Theta theorems f...
The first and second representation theorems for sign-indefinite, not
necessarily semi-bounded quadratic forms are revisited. New straightforward
proofs of these theorems are given. A number of necessary and sufficient
conditions ensuring the second representation theorem to hold is proved. A new
simple and explicit example of a self-adjoint operat...
We prove optimal convergence estimates for eigenvalues and eigenvectors of a class of singular/stiff perturbed problems. Our
profs are constructive in nature and use (elementary) techniques which are of current interest in computational Linear Algebra
to obtain estimates even for eigenvalues which are in gaps of the essential spectrum. Further, we...
We prove optimal convergence estimates for eigenvalues and eigenvectors of a class of singular/stiff perturbed problems. Our profs are constructive in nature and use (elementary) techniques which are of current interest in computational Linear Algebra to obtain estimates even for eigenvalues which are in gaps of the essential spectrum. Further, we...
We consider a large class of residuum based a posteriori
eigenvalue/eigenvector estimates and present an abstract framework for
proving their asymptotic exactness. Equivalence of the estimator and the
error is also established. To demonstrate the strength of our abstract
approach we present a detailed study of hierarchical error estimators
for Lapl...
We are concerned with singularly perturbed spectral problems which appear in engineering sciences. Typically under the influence of a singular perturbation the model can be approximated by a simpler, perturbation independent model. Such reduced model is usually better amenable to analytic or numeric analysis. However, the question of the quality of...
We use a “weakly formulated” Sylvester equation
H1/2TM-1/2 - H-1/2TM1/2 = F \rm{{\bf H}}^{1/2}{\it T}{\rm {\bf M}}^{-1/2} - \rm{{\bf H}}^{-1/2}{\it T}{\rm {\bf M}}^{1/2} = {\it F}
to obtain new bounds for the rotation of spectral subspaces of a nonnegative selfadjoint operator in a Hilbert space. Our
bound extends the known results of Davis and...
We combine abstract eigenvalue/eigenvector estimates (from our earlier work) with a saturation assumption for finite element solution of associated stationary problem to obtain a posteriori estimates of the accuracy of finite element Rayleigh–Ritz approximations. Attention will be payed to the interplay between the accuracy estimate for the finite...
We give new lower bounds on the Rayleigh–Ritz approximations of a part of the spectrum of an elliptic operator. Furthermore, we present bounds for the accompanying Ritz vectors. The bounds include a form of a relative gap between the Ritz values and the rest of the spectrum of the operator. A model example shows that the obtained bounds may be very...
In this article we further develop a perturbation approach to the Rayleigh--Ritz approximations from our earlier work. We both sharpen the estimates and extend the applicability of the theory to nonnegative definite operators . The perturbation argument enables us to solve two problems in one go: We determine which part of the spectrum of the opera...
We give both lower and upper estimates for eigenvalues of unbounded positive definite operators in an arbitrary Hilbert space. We show scaling robust relative eigenvalue estimates for these operators in analogy to such estimates of current interest in Numerical Linear Algebra. Only simple matrix theoretic tools like Schur complements have been used...
We use a ``weakly formulated'' Sylvester equation
$$A^{1/2}TM^{-1/2}-A^{-1/2}TM^{1/2}=F$$ to obtain new bounds for the
rotation of spectral subspaces of a nonnegative selfadjoint operator in
a Hilbert space. Our bound extends the known results of Davis and Kahan.
Another application is a bound for the square root of a positive
selfadjoint operator...
In this article we further develop a perturbation approach to the Rayleigh--Ritz approximations from our earlier work. We both sharpen the estimates and extend the applicability of the theory to nonnegative definite operators . The perturbation argument enables us to solve two problems in one go: We determine which part of the spectrum of the opera...