Showing papers in "Numerische Mathematik in 2013"
TL;DR: It is proved that convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution is convergence.
Abstract: We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in Charrier et al. (SIAM J Numer Anal, 2013), to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Frechet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen---Loeve expansion of the random coefficient. Numerical results complete the paper.
235 citations
TL;DR: The reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries are extended, focusing on the role played by the Brezzi’s and Babuška's stability constants.
Abstract: In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi's and Babuska's stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi's saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuska's inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle.
142 citations
TL;DR: This work proposes a discontinuous Galerkin method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain, and proves the crucial inverse inequality that leads to a coercive bilinear form.
Abstract: We present a discontinuous Galerkin method, based on the classical method of Nitsche, for elliptic problems with an immersed boundary representation on a structured grid. In such methods very small elements typically occur at the boundary, leading to breakdown of the discrete coercivity as well as numerical instabilities. In this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain. This construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates. Furthermore, we prove a bound of the condition number of the stiffness matrix. All the results are valid for polynomials of arbitrary order. We also discuss the implementation of the method and present numerical examples in three dimensions.
131 citations
TL;DR: It is shown that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations in Lyapunov equations arising in bilinear and stochastic control.
Abstract: In this paper, we study possible low rank solution methods for generalized Lyapunov equations arising in bilinear and stochastic control. We show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations. Since the theoretical tools strongly make use of a connection to the standard linear Lyapunov equation, we can even extend the result to the $$d$$ -dimensional case described by a tensorized linear system of equations. We further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit (ADI) iteration and the Krylov-Plus-Inverted-Krylov (K-PIK) method. By means of some standard numerical examples used in the area of bilinear model order reduction, we will show the efficiency of the new methods.
95 citations
TL;DR: A new numerical method for the compressible Navier–Stokes equations governing the flow of an ideal isentropic gas convergence to a global weak solution as discretization parameters go to zero is presented.
Abstract: This paper presents a new numerical method for the compressible Navier---Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix---Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax---Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.
74 citations
TL;DR: This paper introduces a class of incremental displacement-correction schemes for the explicit coupling of a thin-structure with an incompressible fluid and provides a general stability and convergence analysis that covers both the incremental and the non-incremental variants.
Abstract: In this paper we introduce a class of incremental displacement-correction schemes for the explicit coupling of a thin-structure with an incompressible fluid. These methods enforce a specific Robin---Neumann explicit treatment of the interface coupling. We provide a general stability and convergence analysis that covers both the incremental and the non-incremental variants. Their stability properties are independent of the added-mass effect. The superior accuracy of the incremental schemes (with respect to the original non-incremental variant) is highlighted by the error estimates, and then confirmed in a benchmark by numerical experiments.
72 citations
TL;DR: The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback–Leibler divergence and convergence and convergence rates as the noise level tends to 0 are proved.
Abstract: We study Newton type methods for inverse problems described by nonlinear operator equations $$F(u)=g$$ in Banach spaces where the Newton equations $$F^{\prime }(u_n;u_{n+1}-u_n) = g-F(u_n)$$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss---Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to $$0$$ both for an a priori stopping rule and for a Lepskia?-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback---Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performance of the proposed method for these problems is illustrated in numerical examples.
67 citations
TL;DR: A rigorous convergence analysis is provided and it is shown that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables, therefore, the rate of convergence may only be algebraic.
Abstract: In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.
66 citations
TL;DR: The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.
Abstract: This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.
65 citations
TL;DR: This paper derives new trace inequalities for NURBS-mapped domains with Sobolev-type inequalities, and compares the bounding constants appearing in the explicit trace inequalities with numerically computed optimalbounding constants.
Abstract: We derive new trace inequalities for NURBS-mapped domains. In addition to Sobolev-type inequalities, we derive discrete trace inequalities for use in NURBS-based isogeometric analysis. All dependencies on shape, size, polynomial degree, and the NURBS weighting function are precisely specified in our analysis, and explicit values are provided for all bounding constants appearing in our estimates. As hexahedral finite elements are special cases of NURBS, our results specialize to parametric hexahedral finite elements, and our analysis also generalizes to T-spline-based isogeometric analysis. We compare the bounding constants appearing in our explicit trace inequalities with numerically computed optimal bounding constants, and we discuss application of our results to a Laplace problem. We finish this paper with a brief exploration of so-called patch-wise trace inequalities for isogeometric analysis.
61 citations
TL;DR: A nonlinear technique to correct a general finite volume scheme for anisotropic diffusion problems, which provides a discrete maximum principle and two specific corrections proving, under numerical assumptions, that the corresponding approximate solutions converge to the continuous one as the size of the mesh tends to zero.
Abstract: We present a nonlinear technique to correct a general finite volume scheme for anisotropic diffusion problems, which provides a discrete maximum principle. We point out general properties satisfied by many finite volume schemes and prove the proposed corrections also preserve these properties. We then study two specific corrections proving, under numerical assumptions, that the corresponding approximate solutions converge to the continuous one as the size of the mesh tends to zero. Finally we present numerical results showing that these corrections suppress local minima produced by the original finite volume scheme.
TL;DR: It is proved that one can use the T-coercivity theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma.
Abstract: To solve variational indefinite problems, one uses classically the Banach---Nea?as---Babuska theory. Here, we study an alternate theory to solve those problems: T-coercivity. Moreover, we prove that one can use this theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma. We apply this theory to solve the indefinite problem $$\text{ div}\sigma
abla u=f$$ set in $$H^1_0$$ , with $$\sigma $$ exhibiting a sign change.
TL;DR: Low order superconvergence error estimates are obtained for linear hyperbolic conservation laws with singular source terms by applying Duhamel’s principle and negative-order norm error estimates for the accuracy of DG approximations are investigated.
Abstract: In this paper, we develop and analyze discontinuous Galerkin (DG) methods to solve hyperbolic equations involving $$\delta $$ -singularities. Negative-order norm error estimates for the accuracy of DG approximations to $$\delta $$ -singularities are investigated. We first consider linear hyperbolic conservation laws in one space dimension with singular initial data. We prove that, by using piecewise $$k$$ th degree polynomials, at time $$t$$ , the error in the $$H^{-(k+2)}$$ norm over the whole domain is $$(k+1/2)$$ th order, and the error in the $$H^{-(k+1)}(\mathbb R \backslash \mathcal R _t)$$ norm is $$(2k+1)$$ th order, where $$\mathcal R _t$$ is the pollution region due to the initial singularity with the width of order $$\mathcal O (h^{1/2} \log (1/h))$$ and $$h$$ is the maximum cell length. As an application of the negative-order norm error estimates, we convolve the numerical solution with a suitable kernel which is a linear combination of B-splines, to obtain $$L^2$$ error estimate of $$(2k+1)$$ th order for the post-processed solution. Moreover, we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying Duhamel's principle. Numerical examples including an acoustic equation and the nonlinear rendez-vous algorithms are given to demonstrate the good performance of DG methods for solving hyperbolic equations involving $$\delta $$ -singularities.
TL;DR: In the finite-dimensional Hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented, which are helpful for the convergence analysis of common Krylov subspace methods.
Abstract: For a general class of saddle point problems sharp estimates for Babuska's inf-sup stability constants are derived in terms of the constants in Brezzi's theory. In the finite-dimensional Hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented, which are helpful for the convergence analysis of common Krylov subspace methods. The theoretical results are applied to two model problems from optimal control with time-periodic state equations. Numerical experiments with the preconditioned minimal residual method are reported.
TL;DR: In this article, pathwise Hölder convergence with optimal rates of the implicit Euler scheme for the abstract stochastic Cauchy problem 1.1 is proved.
Abstract: In this article we prove pathwise Holder convergence with optimal rates of the implicit Euler scheme for the abstract stochastic Cauchy problem $$\begin{aligned} \left\{ \begin{aligned} dU(t)&= AU(t)\,dt + F(t,U(t))\,dt + G(t,U(t))\,dW_H(t);\quad t\in [0,T],\\ U(0)&=x_0. \end{aligned}\right. \end{aligned}$$ Here $$A$$ is the generator of an analytic $$C_0$$ -semigroup on a umd Banach space $$X,\,W_H$$ is a cylindrical Brownian motion in a Hilbert space $$H$$ , and the functions $$F:[0,T]\times X\rightarrow X_{\theta _F}$$ and $$G:[0,T]\times X\rightarrow {\fancyscript{L}}(H,X_{\theta _G})$$ satisfy appropriate (local) Lipschitz conditions. The results are applied to a class of second order parabolic SPDEs driven by multiplicative space-time white noise.
TL;DR: The exponential stability of the solution of the stationary Navier–Stokes equations is proven and the optimal error estimates uniform in time are provided for the Euler implicit/explicit scheme based on the mixed finite element.
Abstract: In this paper, a new uniqueness assumption (A2) of the solution for the stationary Navier---Stokes equations is presented. Under assumption (A2), the exponential stability of the solution $$(\bar{u},\bar{p})$$ for the stationary Navier---Stokes equations is proven. Moreover, the Euler implicit/explicit scheme based on the mixed finite element is applied to solve the stationary Navier---Stokes equations. Finally, the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme.
TL;DR: An optimal nonconforming adaptive finite element algorithm is presented and its quasi-optimal complexity for the Stokes equations with respect to natural approximation classes is proved.
Abstract: This paper presents an optimal nonconforming adaptive finite element algorithm and proves its quasi-optimal complexity for the Stokes equations with respect to natural approximation classes. The proof does not explicitly involve the pressure variable and follows from a novel discrete Helmholtz decomposition of deviatoric functions.
TL;DR: In this paper, error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation, and the explicit form of all constants in a computable form under the same assumptions of the existing theorems.
Abstract: Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated Giving more explicit estimates, ie, evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of "verified numerical computations" In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region We also improve some formulas themselves to decrease their computational costs Numerical examples that confirm the theory are also given
TL;DR: A finite volume method on general meshes for the discretization of a degenerate parabolic convection–reaction–diffusion equation and obtains a convergence result based upon a priori estimates and the Fréchet–Kolmogorov compactness theorem.
Abstract: We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection---reaction---diffusion equation. Equations of this type arise in many contexts, such as for example the modeling of contaminant transport in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized using a recently developed hybrid mimetic mixed framework. We construct a family of discretizations for the convection term, which uses the hybrid interface unknowns. We consider a wide range of unstructured possibly nonmatching polyhedral meshes in arbitrary space dimension. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Peclet number. We obtain a convergence result based upon a priori estimates and the Frechet---Kolmogorov compactness theorem. We implement the scheme both in two and three space dimensions and compare the numerical results obtained with the upwind and the centered discretizations of the convection term numerically.
TL;DR: The aim of this paper is to propose some suitable discretization schemes of these ABCs and to prove some semi-discrete stability results.
Abstract: Four families of ABCs where built in Antoine et al. (Math Models Methods Appl Sci, 22(10), 2012) for the two-dimensional linear Schrodinger equation with time and space dependent potentials and for general smooth convex fictitious surfaces. The aim of this paper is to propose some suitable discretization schemes of these ABCs and to prove some semi-discrete stability results. Furthermore, the full numerical discretization of the corresponding initial boundary value problems is considered and simulations are provided to compare the accuracy of the different ABCs.
TL;DR: This work considers retarded boundary integral formulations of the three-dimensional wave equation in unbounded domains and proposes new types of finite-dimensional spaces for the time discretization that allow variable time-stepping, variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially.
Abstract: We consider retarded boundary integral formulations of the three-dimensional wave equation in unbounded domains. Our goal is to apply a Galerkin method in space and time in order to solve these problems numerically. In this approach the computation of the system matrix entries is the major bottleneck. We will propose new types of finite-dimensional spaces for the time discretization. They allow variable time-stepping, variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially. The reason is that the basis functions of these spaces are globally smooth and compactly supported. In order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere. In this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments.
TL;DR: In this paper, an extension of the iteratively regularized Gauss-Newton method to the Banach space setting by defining the iterates via convex optimization problems is proposed.
Abstract: In this paper we propose an extension of the iteratively regularized Gauss---Newton method to the Banach space setting by defining the iterates via convex optimization problems. We consider some a posteriori stopping rules to terminate the iteration and present the detailed convergence analysis. The remarkable point is that in each convex optimization problem we allow non-smooth penalty terms including $$L^1$$ and total variation like penalty functionals. This enables us to reconstruct special features of solutions such as sparsity and discontinuities in practical applications. Some numerical experiments on parameter identification in partial differential equations are reported to test the performance of our method.
TL;DR: An efficient and easy to implement approach to solving the semidiscrete equation systems resulting from time discretization of nonlinear parabolic problems with discontinuous Galerkin methods of order r, based on applying Newton’s method and decoupling the Newton update equation.
Abstract: We present an efficient and easy to implement approach to solving the semidiscrete equation systems resulting from time discretization of nonlinear parabolic problems with discontinuous Galerkin methods of order $$r$$ . It is based on applying Newton's method and decoupling the Newton update equation, which consists of a coupled system of $$r+1$$ elliptic problems. In order to avoid complex coefficients which arise inevitably in the equations obtained by a direct decoupling, we decouple not the exact Newton update equation but a suitable approximation. The resulting solution scheme is shown to possess fast linear convergence and consists of several steps with same structure as implicit Euler steps. We construct concrete realizations for order one to three and give numerical evidence that the required computing time is reduced significantly compared to assembling and solving the complete coupled system by Newton's method.
TL;DR: It is shown that the inexact algorithms can achieve the same order of convergence as the exact methods if appropriate sequences of tolerances are applied to the inner solves, and the use of a nonlinear Rayleigh functional is shown to be fundamental in achieving higher order of converge rates.
Abstract: We study the local convergence of several inexact numerical algorithms closely related to Newton's method for the solution of a simple eigenpair of the general nonlinear eigenvalue problem $$T(\lambda )v=0$$ . We investigate inverse iteration, Rayleigh quotient iteration, residual inverse iteration, and the single-vector Jacobi---Davidson method, analyzing the impact of the tolerances chosen for the approximate solution of the linear systems arising in these algorithms on the order of the local convergence rates. We show that the inexact algorithms can achieve the same order of convergence as the exact methods if appropriate sequences of tolerances are applied to the inner solves. We discuss the connections and emphasize the differences between the standard inexact Newton's method and these inexact algorithms. When the local symmetry of $$T(\lambda )$$ is present, the use of a nonlinear Rayleigh functional is shown to be fundamental in achieving higher order of convergence rates. The convergence results are illustrated by numerical experiments.
TL;DR: This is the first success in constructing the second-order convergent nonconforming element for the fourth order elliptic problem.
Abstract: In this paper, a theoretical framework is constructed on how to develop $$C^0$$ -nonconforming elements for the fourth order elliptic problem. By using the bubble functions, a simple practical method is presented to construct one tetrahedral $$C^{0}$$ -nonconforming element and two cuboid $$C^{0}$$ -nonconforming elements for the fourth order elliptic problem in three spacial dimensions. It is also proved that one element is of first order convergence and other two are of second order convergence. From the best knowledge of us, this is the first success in constructing the second-order convergent nonconforming element for the fourth order elliptic problem.
TL;DR: By making use of the block Meany inequality, this paper improves existing results and establishes new convergence theorems for row-action iteration schemes such as the block Kaczmarz and the Householder–Bauer methods used to solve large linear systems and least-squares problems.
Abstract: The Meany inequality gives an upper bound in the Euclidean norm for a product of rank-one projection matrices. In this paper we further derive a lower bound related to this inequality. We discuss the internal relationship between the upper bounds given by the Meany inequality and by the inequality in Smith et al. (Bull Am Math Soc 83:1227---1270, 1977) in the finite dimensional real linear space. We also generalize the Meany inequality to the block case. In addition, by making use of the block Meany inequality, we improve existing results and establish new convergence theorems for row-action iteration schemes such as the block Kaczmarz and the Householder---Bauer methods used to solve large linear systems and least-squares problems.
TL;DR: A multi-level Monte Carlo (MLMC) discretization of the stochastic Brinkman problem is developed and it is proved that the MLMC-MFEM allows the estimation of the statistical mean field with the same asymptotical accuracy versus work as the MFEM for a single instance of the Brink man problem.
Abstract: We present the formulation and the numerical analysis of the Brinkman problem derived in Allaire (Arch Rational Mech Anal 113(3): 209---259,1990. doi: 10.1007/BF00375065 , Arch Rational Mech Anal 113(3): 261---298, 1990. doi: 10.1007/BF00375066 ) with a lognormal random permeability. Specifically, the permeability is assumed to be a lognormal random field taking values in the symmetric matrices of size $$d\times d$$ , where $$d$$ denotes the spatial dimension of the physical domain $$D$$ . We prove that the solutions admit bounded moments of any finite order with respect to the random input's Gaussian measure. We present a Mixed Finite Element discretization in the physical domain $$D$$ , which is uniformly stable with respect to the realization of the lognormal permeability field. Based on the error analysis of this mixed finite element method (MFEM), we develop a multi-level Monte Carlo (MLMC) discretization of the stochastic Brinkman problem and prove that the MLMC-MFEM allows the estimation of the statistical mean field with the same asymptotical accuracy versus work as the MFEM for a single instance of the stochastic Brinkman problem. The robustness of the MFEM implies in particular that the present analysis also covers the Darcy diffusion limit. Numerical experiments confirm the theoretical results.
TL;DR: In this paper, the condition number of the TLS problem with the coefficient matrix and the right-hand side of the value matrix was studied and lower and upper bounds for the condition numbers were derived.
Abstract: This paper concerns singular value decomposition (SVD)-based computable formulas and bounds for the condition number of the total least squares (TLS) problem. For the TLS problem with the coefficient matrix $$A$$ and the right-hand side $$b$$ , a new closed formula is presented for the condition number. Unlike an important result in the literature that uses the SVDs of both $$A$$ and $$[A,\ b]$$ , our formula only requires the SVD of $$[A,\ b]$$ . Based on the closed formula, both lower and upper bounds for the condition number are derived. It is proved that they are always sharp and estimate the condition number accurately. A few lower and upper bounds are further established that involve at most the smallest two singular values of $$A$$ and of $$[A,\ b]$$ . Tightness of these bounds is discussed, and numerical experiments are presented to confirm our theory and to demonstrate the improvement of our upper bounds over the two upper bounds due to Golub and Van Loan as well as Baboulin and Gratton. Such lower and upper bounds are particularly useful for large scale TLS problems since they require the computation of only a few singular values of $$A$$ and $$[A, \ b]$$ other than all the singular values of them.
TL;DR: This paper addresses consistency and stability of W-methods up to order three for nonlinear ODE-constrained control problems with possible restrictions on the control using the transformed adjoint system and the control uniqueness property.
Abstract: This paper addresses consistency and stability of W-methods up to order three for nonlinear ODE-constrained control problems with possible restrictions on the control. The analysis is based on the transformed adjoint system and the control uniqueness property. These methods can also be applied to large-scale PDE-constrained optimization, since they offer an efficient way to compute gradients of the discrete objective function.
TL;DR: This paper discretizes the time-domain electric field integral equation using Runge–Kutta convolution quadrature in time and a Galerkin method in space and obtains convergence results for the fully discrete scheme.
Abstract: In this paper we consider time-dependent electromagnetic scattering problems from conducting objects. We discretize the time-domain electric field integral equation using Runge---Kutta convolution quadrature in time and a Galerkin method in space. We analyze the involved operators in the Laplace domain and obtain convergence results for the fully discrete scheme. Numerical experiments indicate the sharpness of the theoretical estimates.