Search results for "Elliptic operator"
showing 10 items of 40 documents
Elliptic convolution operators on non-quasianalytic classes
2001
For those nonquasianalytic classes in which an extension of the classical Borel's theorem holds we show that every elliptic convolution operator is the composition of a translation and an invertible ultradifferential operator. This answers a question asked by Chou in: La transformation de Fourier complexe et l'equation de convolution, LNM 325, Berlin-Heidelberg-New York (1973).
The factorization method for real elliptic problems
2006
The Factorization Method localizes inclusions inside a body from mea- surements on its surface. Without a priori knowing the physical parameters inside the inclusions, the points belonging to them can be characterized using the range of an auxiliary operator. The method relies on a range characterization that relates the range of the auxiliary operator to the measurements and is only known for very particular applications. In this work we develop a general framework for the method by considering sym- metric and coercive operators between abstract Hilbert spaces. We show that the important range characterization holds if the difference between the inclusions and the background medium satisfi…
Computation of a few smallest eigenvalues of elliptic operators using fast elliptic solvers
2001
The computation of a few smallest eigenvalues of generalized algebraic eigenvalue problems is studied. The considered problems are obtained by discretizing self-adjoint second-order elliptic partial differential eigenvalue problems in two- or three-dimensional domains. The standard Lanczos algorithm with the complete orthogonalization is used to compute some eigenvalues of the inverted eigenvalue problem. Under suitable assumptions, the number of Lanczos iterations is shown to be independent of the problem size. The arising linear problems are solved using some standard fast elliptic solver. Numerical experiments demonstrate that the inverted problem is much easier to solve with the Lanczos…
Operator splitting methods for American option pricing
2004
Abstract We propose operator splitting methods for solving the linear complementarity problems arising from the pricing of American options. The space discretization of the underlying Black-Scholes Scholes equation is done using a central finite-difference scheme. The time discretization as well as the operator splittings are based on the Crank-Nicolson method and the two-step backward differentiation formula. Numerical experiments show that the operator splitting methodology is much more efficient than the projected SOR, while the accuracy of both methods are similar.
Fundamental solutions for general anisotropic multi-field materials based on spherical harmonics expansions
2016
Abstract A unified method to evaluate the fundamental solutions for generally anisotropic multi-field materials is presented. Based on the relation between the Rayleigh expansion and the three-dimensional Fourier representation of a homogenous partial differential operator, the proposed technique allows to obtain the fundamental solutions and their derivatives up to the desired order as convergent series of spherical harmonics. For a given material, the coefficients of the series are computed only once, and the derivatives of the fundamental solutions are obtained without any term-by-term differentiation, making the proposed approach attractive for boundary integral formulations and efficie…
Homology of pseudodifferential operators on manifolds with fibered cusps
2003
The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the 0 0 -dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.
Nonlinear elliptic equations having a gradient term with natural growth
2006
Abstract In this paper, we study a class of nonlinear elliptic Dirichlet problems whose simplest model example is: (1) { − Δ p u = g ( u ) | ∇ u | p + f , in Ω , u = 0 , on ∂ Ω . Here Ω is a bounded open set in R N ( N ⩾ 2 ), Δ p denotes the so-called p-Laplace operator ( p > 1 ) and g is a continuous real function. Given f ∈ L m ( Ω ) ( m > 1 ), we study under which growth conditions on g problem (1) admits a solution. If m ⩾ N / p , we prove that there exists a solution under assumption (3) (see below), and that it is bounded when m > N p ; while if 1 m N / p and g satisfies the condition (4) below, we prove the existence of an unbounded generalized solution. Note that no smallness condit…
On the operators which are invertible modulo an operator ideal
2001
Atkinson [3] studied the operators which are left invertible $i(X, Y) or right invertible $T{X, Y) modulo /C, with K. the compact operators. He proved that an operator T € C(X, Y) belongs to <£/ or $ r if and only if the kernel and the range of T are complemented and additionally, the kernel is finite dimensional or the range is finite codimensional, respectively. Yood [19] obtained some perturbation results for these classes and Lebow and Schechter [12] proved that the inessential operators form the perturbation class for $,(A") and $r{X). Yang [18] extended some results of ^3, 19] to operators invertible modulo W, with W the weakly compact operators. His aim was to study a generalised Fre…
Maximal regularity for Kolmogorov operators in L2 spaces with respect to invariant measures
2006
Abstract We prove an optimal embedding result for the domains of Kolmogorov (or degenerate hypoelliptic Ornstein–Uhlenbeck) operators in L 2 spaces with respect to invariant measures. We use an interpolation method together with optimal L 2 estimates for the space derivatives of T ( t ) f near t = 0 , where T ( t ) is the Ornstein–Uhlenbeck semigroup and f is any function in L 2 .
Abstract Estimates of the Rate of Convergence for Optimal Control Problems
1997
A method for solving optimal control problems with general elliptic operators is presented and analyzed. Especially, estimates of the rate of convergence for the control problems with the proposed approach are derived independently of the underlying approximation method. Some numerical experiments with the proposed method are included.