Factorization Of Operators Research Articles

Effective masses for Laplacians on periodic graphs

We consider Laplacians on both periodic discrete and periodic metric equilateral graphs. Their spectrum consists of an absolutely continuous part (which is a union of non-degenerate spectral bands) and flat bands, i.e., eigenvalues of infinite multiplicity. We estimate effective masses associated with the ends of each spectral band in terms of geometric parameters of the graphs. Moreover, in the case of the bottom of the spectrum we determine two-sided estimates on the effective mass in terms of geometric parameters of the graphs. The proof is based on Floquet theory, factorization of fiber operators, perturbation theory and the relation between effective masses for Laplacians on discrete and metric graphs obtained in our paper.

THE NON-COMMUTATIVE KHINTCHINE INEQUALITIES FOR

We give a proof of the Khintchine inequalities in non-commutative $L_{p}$-spaces for all $0<p<1$. These new inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, for example, for the analogues of such random variables in free probability. We also prove a factorization for operators from a Hilbert space to a non-commutative $L_{p}$-space, which is new for $0<p<1$. We end by showing that Mazur maps are Hölder on semifinite von Neumann algebras.

Factorization of Operators Through Orlicz Spaces

We study factorization of operators between quasi-Banach spaces. We prove the equivalence between certain vector norm inequalities and the factorization of operators through Orlicz spaces. As a consequence, we obtain the Maurey–Rosenthal factorization of operators into \(L_p\)-spaces. We give several applications. In particular, we prove a variant of Maurey’s Extension Theorem.

Completion, extension, factorization, and lifting of operators

The well-known results of M. G. Kreĭn concerning the description of selfadjoint contractive extensions of a hermitian contraction \(T_1\) and the characterization of all nonnegative selfadjoint extensions \({{\widetilde{A}} }\) of a nonnegative operator A via the inequalities \(A_K\le {{\widetilde{A}} } \le A_F\), where \(A_K\) and \(A_F\) are the Kreĭn–von Neumann extension and the Friedrichs extension of A, are generalized to the situation, where \({{\widetilde{A}} }\) is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the negative index of the operators \(I-T_1^*T_1\) and A, respectively; these conditions are automatically satisfied if \(T_1\) is contractive or A is nonnegative, respectively. The approach developed in this paper starts by establishing first a generalization of an old result due to Yu. L. Shmul’yan on completions of nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M. G. Kreĭn and, in addition, to solve some related lifting problems for J-contractive operators in Hilbert, Pontryagin and Kreĭn spaces in a simple manner. Also some new factorization results are derived, for instance, a generalization of the well-known Douglas factorization of Hilbert space operators. In the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non-contractive selfadjoint operators under Cayley transforms properly.

A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.

The factorization of the linear differential operator

Abstract In this paper, necessary and sufficient conditions for factorization of a linear differential operator are presented. As a consequence of the factorization result some criterion of polynomial factorization is obtained. As a special case of the main result we have got Polynomial Remainder Theorem.

Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations

AbstractThis paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF‐DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL decomposition that facilitates the efficient inversion of the discretized operator. HIF‐DE is based on the nested dissection multifrontal method but uses skeletonization on the separator fronts to sparsify the dense frontal matrices and thus reduce the cost. We conjecture that this strategy yields linear complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity in 3D can be achieved by skeletonizing the compressed fronts themselves, which amounts geometrically to a recursive dimensional reduction scheme. Numerical experiments support our claims and further demonstrate the performance of our algorithm as a fast direct solver and preconditioner. MATLAB® codes are freely available.© 2016 Wiley Periodicals, Inc.

Hierarchical Interpolative Factorization for Elliptic Operators: Integral Equations

AbstractThis paper introduces the hierarchical interpolative factorization for integral equations (HIF‐IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIF‐IE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higher‐dimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIF‐IE is compatible with geometric adaptivity and can handle both boundary and volume problems.© 2016 Wiley Periodicals, Inc.

On a factorization of operators as a product of an essentially unitary operator and a strongly irreducible operator

As it is well-known, for any operator T on a complex separable Hilbert space, T has the polar decomposition T=U|T|, where U is a partial isometry and |T| is the non-negative operator (T⁎T)12. In this paper, we will give a decomposition theorem in a new sense that |T| will be replaced by a strongly irreducible operator. More precisely, for any operator T and any ε>0, there exists a decomposition T=(U+K)S, where U is a partial isometry, K is a compact operator with ||K||<ε and S is strongly irreducible.

Efficient preconditioning of variational time discretization methods for parabolic Partial Differential Equations

This paper is concerned with the design of efficient preconditioners for systems arising from variational time discretization methods for parabolic partial differential equations. We consider the first order discontinuous Galerkin method (dG(1)) and the second order continuous Galerkin Petrov method (cGP(2)). The time-discrete formulation of these methods leads to a coupled 2 × 2 block system whose efficient solution strongly depends on efficient preconditioning strategies. The preconditioner proposed in this paper is based on a Schur complement formulation for the so called essential unknown. By introducing an inexact factorization of this ill-conditioned fourth order operator, we are able to circumvent complex arithmetic and prove uniform bounds for the condition number of the preconditioned system. In addition, the resulting preconditioned operator is symmetric and positive definite, therefore allowing for the usage of efficient Krylov subspace solvers such as the conjugate gradient method. For both the dG(1) and cGP(2) method, we provide optimal choices for the sole parameter of the preconditioner and deduce corresponding upper bounds for the condition number of the resulting preconditioned system. Several numerical experiments including the heat equation and a convection-diffusion example confirm the theoretical findings.

Inside-Outside Duality and the Determination of Electromagnetic Interior Transmission Eigenvalues

We introduce an inside-outside duality approach for the determination of interior transmission eigenvalues of a possibly anisotropic dielectric electromagnetic scattering object using time-harmonic electromagnetic far field data. To this end, we exploit a self-adjoint factorization of the far field operator to link the electromagnetic interior transmission eigenvalues to the maximal or minimal phase of the eigenvalues of the corresponding far field operator, depending on whether the sign of the contrast function is positive or negative.

Regularized seismic velocity inversion based on inverse scattering theory

In this paper, the regularized finite difference contrast source inversion algorithm based on inverse scattering theory is utilized to invert the seismic velocity distribution in the underground medium, and this algorithm is a waveform inversion method in frequency domain based on wave equation. Velocity distribution is updated iteratively by minimizing the cost function using the nonlinear conjugate gradient method. The geophysical inversion problem has the characters of ill-posedness and instability, we handle this problem by adding an additional regularization item based on total variation of inversion parameter, making the inversion problem become a well-posed problem, and the algorithm has the ability of anti-noise interference. In the inversion process we use the frequency-space domain 9-point difference propagation operator under PML absorption boundary condition. Compared with other inversion algorithm, the construction of forward modeling operator and other matrix factorization calculation process can be avoided in each iterative step because the background medium remains unchanged throughout the inversion process, this makes the algorithm more suitable for large-scale three-dimensional inversion. In addition, the MPI parallel computation is applied, by which the efficiency of inversion process is improved greatly. The two-dimensional CSEG model inversion results show that this algorithm can obtain high resolution seismic velocity reconstruction, and provide an accurate velocity information for seismic data processing and interpretation.

Path factorization approach to stochastic simulations.

The computational efficiency of stochastic simulation algorithms is notoriously limited by the kinetic trapping of the simulated trajectories within low energy basins. Here we present a new method that overcomes kinetic trapping while still preserving exact statistics of escape paths from the trapping basins. The method is based on path factorization of the evolution operator and requires no prior knowledge of the underlying energy landscape. The efficiency of the new method is demonstrated in simulations of anomalous diffusion and phase separation in a binary alloy, two stochastic models presenting severe kinetic trapping.

Wiener–Hopf Factorization Through an Intermediate Space

An operator factorization conception is investigated for a general Wiener–Hopf operator \({W = P_2 A |_{P_1 X}}\) where X, Y are Banach spaces, \({P_1 \in \mathcal{L}(X), P_2 \in \mathcal{L}(Y)}\) are projectors and \({A \in \mathcal{L}(X,Y)}\) is boundedly invertible. Namely we study a particular factorization of \({A = A_- C A_+ \,\,{\rm where}\,\, A_+ : X \rightarrow Z \,\,{\rm and} \,\,A_- : Z \rightarrow Y}\) have certain invariance properties and \({C : Z \rightarrow Z}\) splits the “intermediate space” Z into complemented subspaces closely related to the kernel and cokernel of W, such that W is equivalent to a “simpler” operator, \({W \sim P C|_{P X}}\). The main result shows equivalence between the generalized invertibility of the Wiener–Hopf operator and this kind of factorization (provided \({P_1 \sim P_2}\)) which implies a formula for a generalized inverse of W. It embraces I.B. Simonenko’s generalized factorization of matrix measurable functions in L p spaces, is significantly different from the cross factorization theorem and more useful in numerous applications. Various connected theoretical questions are answered such as: How to transform different kinds of factorization into each other? When is W itself the truncation of a cross factor?

Rotation operator propagators for time-varying radiofrequency pulses in NMR spectroscopy: Applications to shaped pulses and pulse trains

The propagator for trains of radiofrequency pulses can be directly integrated numerically or approximated by average Hamiltonian approaches. The former provides high accuracy and the latter, in favorable cases, convenient analytical formula. The Euler-angle rotation operator factorization of the propagator provides insights into performance that are not as easily discerned from either of these conventional techniques. This approach is useful in determining whether a shaped pulse can be represented over some bandwidth by a sequence τ1–Rϕ(β)–τ2, in which Rϕ(β) is a rotation by an angle β around an axis with phase ϕ in the transverse plane and τ1 and τ2 are time delays, allowing phase evolution during the pulse to be compensated by adjusting time periods prior or subsequent to the pulse. Perturbation theory establishes explicit formulas for τ1 and τ2 as proportional to the average transverse magnetization generated during the shaped pulse. The Euler-angle representation of the propagator also is useful in iterative reduction of pulse-interrupted-free precession schemes. Application to Carr–Purcell–Meiboom–Gill sequences identifies an eight-pulse phase alternating scheme that generates a propagator nearly equal to the identity operator.

Near-field imaging of scattering obstacles with the factorization method

In this paper we establish a factorization method for recovering the location and shape of an acoustic bounded obstacle with using the near-field data, corresponding to infinitely many incident point sources. The obstacle is allowed to be an impenetrable scatterer of sound-soft, sound-hard or impedance type or a penetrable scatterer. An outgoing-to-incoming operator is constructed for facilitating the factorization of the near-field operator, which can be easily implemented numerically. Numerical examples are presented to demonstrate the feasibility and effectiveness of our inversion algorithm, including the case where limited aperture near-field data are available only.

Remark on the Helmholtz decomposition in domains with noncompact boundary

Let $$\varOmega $$ be a domain in $$\mathbb {R}^{d+1}$$ whose boundary is given as a uniform Lipschitz graph $$x_{d+1}=\eta (x)$$ for $$x \in \mathbb {R}^d$$ . For such a domain, it is known that the Helmholtz decomposition is not always valid in $$L^p(\varOmega )$$ except for the energy space $$L^2 (\varOmega )$$ . In this paper we show that the Helmholtz decomposition still holds in certain anisotropic spaces which include vector fields decaying slowly in the $$x_{d+1}$$ variable. In particular, these classes include some infinite energy vector fields. For the purpose, we develop a new approach based on a factorization of divergence form elliptic operators whose coefficients are independent of one variable.

Majorization, range inclusion, and factorization for unbounded operators on Banach spaces

In this paper, we aim to study the relationships among the notions of majorization, factorization and range inclusion in the context of closed densely defined operators on a Banach space. In fact, our results extend Douglas range factorization theorem for unbounded operators on a Hilbert space to the setting of unbounded operators on a Banach space.

Factorization of Non-Symmetric Operators and Exponential H-Theorem

We present an abstract method for deriving decay estimates on the resolvents and semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as a regularizing part, plus a dissipative part. The core of the method is a high-order quantitative factorization argument on the resolvents and semigroups. We then apply this approach to the Fokker-Planck equation, to the kinetic Fokker- Planck equation in the torus, and to the linearized Boltzmann equation in the torus. We finally use this information on the linearized Boltzmann semi- group to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy method to prove nonlinear stability in this context in $L^1_v L^\infty _x (1 + |v|^k)$, $k > 2$, with sharp rate of decay in time. As a consequence of these results we obtain the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. This improves the result in [32] where polynomial rates at any order were obtained, and solves the conjecture raised in [91, 29, 86] about the optimal decay rate of the relative entropy in the H-theorem.

The inside–outside duality for scattering problems by inhomogeneous media

This paper investigates the relationship between interior transmission eigenvalues k0 > 0 and the accumulation point 1 of the eigenvalues of the scattering operator when k approaches k0. As is well known, the spectrum of is discrete, the eigenvalues μn(k) lie on the unit circle in and converge to 1 from one side depending on the sign of the contrast. Under certain (implicit) conditions on the contrast it is shown that interior transmission eigenvalues k0 can be characterized by the fact that one eigenvalue of converges to 1 from the opposite side if k tends to k0 from below. The proof uses the Cayley transform, Courant’s maximum–minimum principle, and the factorization of the far field operator. For constant contrasts that are positive and large enough or negative and small enough, we show that the conditions necessary to prove this characterization are satisfied at least for the smallest transmission eigenvalue.