APPROXIMATION OF NEGATIVE PLURISUBHARMONIC FUNCTIONS WITH GIVEN BOUNDARY VALUES

In this thesis we study approximation of negative plurisubharmonic functions by functions defined on strictly larger domains. We show that, under certain conditions, every function u that is defined on a bounded hyperconvex domain Ω in Cn and has essentially boundary values zero and bounded Monge-Ampere mass, can be approximated by an increasing sequence of functions {uj} that are defined on strictly larger domains, has boundary values zero and bounded Monge-Ampere mass. We also generalize this and show that, under the same conditions, the approximation property is true if the function u has essentially boundary values G, where G is a plurisubharmonic functions with certain properties. To show these approximation theorems we use subextension. We show that if Ω_1 and Ω_2 are hyperconvex domains in Cn and if u is a plurisubharmonic function on Ω_1 with given boundary values and with bounded Monge-Ampere mass, then we can find a plurisubharmonic function u defined on Ω_2, with given boundary values, such that u <= u on Ω and with control over the Monge-Ampere mass of u.

Approximation of plurifinely plurisubharmonic functions

On the approximation of weakly plurifinely plurisubharmonic functions

Given a negative plurisubharmonic function $\phi $ in a bounded pseudoconvex domain of $\mathbb C^n$, we introduce and study its <em>residual function</em> $g_\phi $ determined by the asymptotic behavior of $\phi $ near its singularity points, both inside

Dirac’s formalism combined with complex Fourier operational matrices to solve initial and boundary value problems

Our aim in this article is the study of subextension and approximation of plurisubharmonic functions in $\mathcal {E}_\chi (\varOmega ,H)$, the class of functions with finite $\chi $-energy and given boundary values. We show that, under certain conditions

We present an effective numerical method for the recovery of biharmonic functions in a disk from continuous boundary values of these functions and of their normal derivatives using wavelets that are harmonic in the disk and interpolating on its boundary on dyadic rational grids. The expansions of solutions of boundary value problems into cumbersome interpolation series in the wavelet basis are replaced by sequences of their partial sums that are compactly presentable in the subspace bases of the corresponding multiresolution analysis (MRA) of Hardy spaces $$h_{\infty}(K)$$ of functions harmonic in the disk. Effective estimates are obtained for the approximation of solutions by partial sums of any order in terms of the best approximation of the boundary functions by trigonometric polynomials of a slightly smaller order. As a result, to provide the required accuracy of the representation of the unknown biharmonic functions, one can choose in advance the scaling parameter of the corresponding MRA subspace such that the interpolation projection to this space defines a simple analytic representation of the corresponding partial sums of interpolation series in terms of appropriate compressions and shifts of the scaling functions, skipping complicated iterative procedures for the numerical construction of the coefficients of expansion of the boundary functions into series in interpolating wavelets. We write solutions using interpolating and interpolating-orthogonal wavelets based on modified Meyer wavelets; the latter are convenient to apply if the boundary values of the boundary value problem are given approximately, for example, are found experimentally. In this case, one can employ the usual, well-known procedures of discrete orthogonal wavelet transformations for the analysis and refinement (correction) of the boundary values.

The biharmonic equation in Cartesian coordinates is considered for the case of the upper half-plane. The solution of such a fourth-order partial differential equation for given boundary conditions is represented in the form of an integral of the product of the function and the delta-shaped kernel, which in this paper plays the role of an approximating aggregate. In the paper, we found an exact equality for the upper bound of the deviation of Hölder class functions from the considered biharmonic Poisson operator in the uniform metric.

One considers the spectral problem with boundary conditions , , for functions on . It is assumed that is a linear bounded operator from the Hölder space , , into and the are bounded linear functionals on with . Let be the linear span of the root functions of the problem , , , corresponding to the eigenvalues with , and let . An estimate of is obtained in terms of the -functional for (the direct theorem) and an estimate of this -functional is obtained in terms of for (the inverse theorem). In several cases two-sided bounds of the -functional are found in terms of appropriate moduli of continuity, and then the direct and the inverse theorems are stated in terms of moduli of continuity. For the spectral problem with periodic boundary conditions these results coincide with Jackson's and Bernstein's direct and inverse theorems on the approximation of functions by a trigonometric system.

Artificial boundary conditions, which can be identified as Robin boundary conditions positioned at a complex space coordinate, are introduced in order to obtain pertinent approximations for the Green's functions in grounded layered media. These artificial boundary conditions include perfectly matched layers backed by perfectly electric or magnetic conductors. As a first result, we obtain analytical expressions for the differences of Green's functions subject to different boundary conditions. Since weighted sums of Green's functions are again Green's functions, the need arises to solve an optimization problem, in the sense of obtainingthe optimal weig hted mixture of Green's functions, as compared to the exact Green's function. Comprehensive eigenexpansions for the Green's functions are given in the general case, and a few examples illustrate the goodness of fit between the approximate Green's functions and the exact Green's function.

This paper addresses the efficient solution of acoustic problems in which the primary interest is obtaining the solution only on restricted portions of the domain but over a wide range of frequencies. The exterior acoustics boundary value problem is approximated using the finite element method in combination with the Dirichlet-to-Neumann (DtN) map. The restriction domain problem is formally posed in transfer function form based on the finite element solution. In order to obtain the solution over a range of frequencies, a matrix-valued Padé approximation of the transfer function is employed, using a two-sided block Lanczos algorithm. This approach provides a stable and efficient representation of the Padé approximation. In order to apply the algorithm, it is necessary to reformulate the transfer function due to the frequency dependency in the nonreflecting boundary condition. This is illustrated for the case of the DtN boundary condition, but there is no restriction on the approach which can also be applied to other radiation boundary conditions. Numerical tests confirm that the approach offers significant computational speed-up.

One aspect of implementing a limited-area ensemble Kalman filter (EnKF) involves the specification of a suitable ensemble of lateral boundary conditions. Two classes of methods to populate a boundary condition ensemble are proposed. In the first class, the ensemble of boundary conditions is provided by an EnKF on a larger domain and is approximately a random draw from the probability distribution function for the forecast (or analysis) on the limited-area domain boundary. The second class perturbs around a deterministic estimate of the state using assumed spatial and temporal covariance relationships. Methods in the second class are relatively flexible and easy to implement. Experiments that test the utility of these methods are performed for both an idealized low-dimensional model and limited-area simulations using the Weather Research and Forecasting (WRF) model; all experiments employ simulated observations under the perfect model assumption. The performance of the ensemble boundary condition methods is assessed by comparing the results of each experiment against a control “global” EnKF that extends beyond the limited-area domain. For all methods tested, results show that errors for the limited-area EnKF are larger near the lateral boundaries than those from a control EnKF, but decay inside the limited-area domain so that errors there are comparable to the control case. The relatively larger errors near the boundaries in the limited-area EnKF originate from not assimilating observations outside the limited-area domain and, in the second class of methods, from deficiencies in boundary spatial and temporal covariances. Overall, these experiments suggest that for observation densities typical in numerical weather prediction models, ensemble boundary conditions can be specified in the absence of a global ensemble without significant penalty in the domain interior by perturbing around an ensemble mean.

Computer simulation of the spin-echo spatial distribution in the case of restricted self-diffusion.

Large amplitude vibrations of thin elastic plates by the method of conformal transformation

Laplace's equation with any boundary conditions can be solved by means of the Parametric Integral Equation System (PIES). For modelling of the boundary geometry in 3D problems Bezier and Coons surfaces are used. A numerical solution of the PIES requires no boundary discretization and is reduced only to the approximation of boundary functions. For its solving a collocation method with Chebyshev polynomials was used. An arrangement of collocation points has high influence on the accuracy of obtained results. Genetic algorithms are applied for searching most optimal arrangement of collocation points.