Browsing by Author "Dallas, A.G."
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Item Basis Properties of Traces and Normal Derivatives of Spherical-Separable Solutions of the Helmholtz Equation(Department of Mathematical Sciences, 2000) Dallas, A.G.The classical solutions of the Heimholtz equation resulting from the separation-of-variables procedure in spherical coördinates are frequently used in one way or another to approximate other solutions. In particular, traces and/or normal derivatives of certain sequences of these spherical-separable solutions are commonly used as trial-and-test-functions in Galerkin procedures for the approximate solution of boundary-operator problems arising from the reformulation of exterior or interior boundary-value problems and set on the boundary Γ of the domain where a solution is wanted. While the completeness properties of these traces and normal derivatives in the usual Hilbert space L2( Γ) are well known, their basis properties are not. We show that such sequences of traces or normal derivatives of the outgoing spherical-separable solutions form bases for L2( Γ) only when Γ is a sphere centered at the pole of the spherical solutions; corresponding results are given for the entire solutions, accounting for the possibility of an interior eigenvalue. We identify other Hilbert spaces, connected with the far-field pattern, for which these functions do provide bases. We apply the results to discuss some aspects of the Waterman schemes for approximate solutions of scattering problems (the so-called “T-matrix method”), including the previous article of KRISTENSSON, RAMM, and STRÖM (J. Math. Phys.24 (1983), 2619-2631) on the convergence of such methods.Item Numerical Experiments with Isometric Mapping and Back-Projection for Conditioning Families in a Simple Sobolev Space(Department of Mathematical Sciences, 2001) Dallas, A.G.Numerically stable Galerkin procedures can be constructed by ensuring that the families of trial- and test-functions are well conditioned in the respective Hilbert spaces between which the operator is an (appropriate) isomorphism. We explain an idea for constructing a family that may be well conditioned in a given Sobolev space of nonzero fractional order from a family that is well conditioned in the corresponding zero-order space, by using a naturally occurring isometric operator followed by projection back onto the original subspace. Effectively, the construction results in "preconditioning matrices," to be used in transforming the original Galerkin matrix to produce new ones which may be of much smaller condition number. The underlying geometric setting must be sufficiently simple, so that the Sobolev structures can me "manipulated numerically." While we have not yet proven the well-conditioning of the constructed families, the use of the scheme is illustrated numerically in applications to the approximate solution of a first-kind integral equation arising in two-dimensional acoustic scattering, where a pronounced stabilizing effect is observed.Item On the Convergence and Numerical Stability of the Second Waterman Scheme for Approximation of the Acoustic Field Scattered by a Hard Object(Department of Mathematical Sciences, 2000) Dallas, A.G.The numerical schemes of P.C. Waterman (J. Acoust. Soc. Am.45 (1969), 1417-1429), frequently referred to under the name of "the T-Matrix method," have formed the basis for many scattering computations in many settings. However, no successful analyses of the algorithms have been published, so the limitations on their range of applicability and numerical stability remain largely unknown; this is of particular importance because of the apparently inconsistent success achieved in numerical experiments. Here, we give an operator condition that guarantees the viability of the algorithm and mean-square convergence of the far-field patterns generated by the second Waterman scheme for the case of time-harmonic acoustic scattering by a hard obstacle; we prove further that the operator condition holds at least whenever the scattering obstacle is ellipsoidal. For the convergence proof, we also assume that the square of the wavenumber is not an interior Dirichlet eigenvalue for the negative Laplacianl in the contrary case, we show that the algorithm is at best numerically ill-coordinated. With this and previous experience in numerical applications, it appears that the performance of the algorithm is markedly shape-dependent; for certain obstacles, e.g., ellipsoids, instabilities are so localized in wavenumber that they are practically numerically irrelevant, while it is not clear whether the erratic results found in applications to various other shapes arise from a failure of convergence or form numerical instability.Item Reformulating a Boundary-Integral Equation in Three Dimensions as as Integral-Operator Problem in a Plane Region(Department of Mathematical Sciences, 2001) Dallas, A.G.Motivated by a desire to simplify the design of numerically stable and efficient approximation schemes for boundary-operator problems, we develop a framework in which an integral equation on the boundary of a domain ${I\kern-.30em R}^3$ can be systematically reformulated as an integral-operator problem set in a region in the plane; some geometric restrictions are imposed on the shape of the (smooth) boundary. When the plane region is chosen to be a rectangle, the necessary Sobolev-space structures can be handled numerically rather easily in the new simpler geometry, in contrast to the situation on the original boundary. Moreover, familiar trial- and test- functions can then be employed in the construction of approximate solutions of the reformulated problems. We show for two examples how a well-posed problem can be transferred from the domain-boundary setting to the plane-region setting. We describe a numerical implementation of these ideas to a lower-dimensional example involving the approximate solution of a first-kind integral equation associated with the Helmholtz equation that is originally set on the boundary of a domain in ${I\kern-.30em R}^2$.Item Toward the Direct Analytical Determination of the Pareto Optima of a Differentiable Mapping, I: Domains in Finite-Dimensional Spaces(Department of Mathematical Sciences, 2000) Dallas, A.G.The problem of locating the Pareto-optimal points of a differentiable mapping $F: {\mathcal M}^N \to {I\kern-.30em R}^n$ is studied, with the domain ${\cal M}^N$ a differentiable N-dimensional submanifold-without-boundary in a euclidean space ${I\kern-.30em R}^{N_{0}}$ and $N_0 \ge N \ge n$. The case in which the domain is the closure of a bounded, regular, open subset of ${I\kern-.30em R}^N$ is also discussed. The search is initiated from these observations: for a manifold-domain, (1) the image of any Pareto optimum lies in the boundary of the range of F; (2) a point of the boundary of the range of F that also lies in the range must be the image of a singular point of F, i.e., must appear amongst the singular values of the map. Further conditions are then needed to distinguish which of the singular values should be discarded because they belong to the interior of the range; local tests of this sort are given for the bicriterial case (n = 2). A search procedure based on the present developments can systematically determine all of the Pareto optima for sufficiently simple F. The conditions established here may be regarded as analogues of the classical ones for the determination of the global extrema of a real-valued differentiable function. The results proven are illustrated with single examples, including plots of the ranges, singular points, and singular values.