The Method of Least Squares for Boundary Value Problems

An approximate solution for linear boundary-value problems with slowly varying coefficients

Computation of vertical velocity within the confines of a three-dimensional, finite element model is a difficult but important task. This paper examines four approaches to the solution of the overdetermined system of equations arising when the first-order continuity equation is solved in conjunction with two boundary conditions. The traditional (TRAD) method neglects one boundary condition, solving the continuity equation with the remaining boundary condition. The vertical derivative of continuity (VDC) method involves solution of the second-order equation obtained by differentiation of the continuity equation with respect to the vertical co-ordinate. The least squares (LS) method minimizes the residuals of the continuity equation (in discrete form) and the two boundary conditions. The adjoint (ADJ) method minimizes the residuals of the continuity equation (in continuous form) and the two boundary conditions. Two domains are considered: a quarter-annular harbour and the southwest coast of Vancouver Island. Results indicate that the highest-quality solution is obtained with both LS and ADJ. Furthermore, ADJ requires less CPU and memory than LS. Therefore the optimal method for computation of vertical velocity in a three-dimensional finite element model is the adjoint (ADJ) method. © 1997 John Wiley & Sons, Ltd.

Surface reflection tomography is an inversion method that attempts to determine simultaneously the subsurface elastic parameters and reflector depths from pre-stack seismic data. To date, this technique has been applied to acoustic and isotropic elastic media. In this paper, we study surface reflection tomography in the anisotropic elastic case. Anisotropic media are described by many more elastic parameters than are needed to describe isotropic media; these additional parameters are useful for obtaining rock properties such as crack density, pore shape and fracture strike that cannot be found with isotropic methods. Similar to isotropic surface reflection tomography, many features of an anisotropic model are poorly resolved due to limited ray path coverage; the vertical smearing of isotropic tomography occurs in each elastic parameter in the anisotropic problem. Unlike the isotropic problem, however, there is additional indeterminacy in the solution of the anisotropic tomography problem because of ambiguity amongst the several elastic parameters needed to describe anisotropic media. Also unlike the isotropic problem, the reflector depths participate very strongly in this ambiguity. We investigate the nature of this indeterminacy by studying the null space for linearized tomography, that is, the class of model perturbations of a background medium which, to first order, cause no perturbation at all in the surface reflection traveltimes. Such model perturbations cannot be determined from the traveltime perturbations; describing these null space model perturbations gives insight into the indeterminacy in the anisotropic problem. Complementary to computational approaches towards identifying the null space for discrete formulations of tomography, we study a continuum formulation. Our first set of results concerns the ‘elastic’ null space: perturbations of the elastic parameters (with zero depth perturbation) which cause no perturbation in the traveltimes. The results here are very similar to previous results for cross-well transmission tomography. As expected, the ‘elastic’ null space is larger than the isotropic null space due to the ambiguity amongst the elastic parameters. We identify three categories of model perturbations in the ‘elastic’ null space. The first category consists of model perturbations for which the perturbation in each of the individual elastic parameters is itself either in the isotropic null space, or in a closely related set of model perturbations which we call the odd isotropic null space. Elements in the second category are anisotropic versions of the most well-known isotropic null space elements: perturbations which are polynomials in the horizontal variable with coefficients which are functions of the depth variable satisfying certain linear integral constraints; unlike the isotropic problem, the integral constraints in the anisotropic problem couple together the several elastic parameters. The third category consists of model perturbations satisfying zero boundary conditions at the surface and at the reflector for which a specific linear combination of integrals and derivatives of the several elastic parameters is in the isotropic null space. In particular, there are model perturbations in this third category which represent anomalies that are completely contained in the interior of the model and yet are in the ‘elastic’ null space; this behaviour is different from the isotropic problem. These categories are sufficient to describe the ‘elastic’ null space completely. We demonstrate that every model perturbation in the ‘elastic’ null space is the sum of an element in the first category (indicating an indeterminacy of the same nature as in the isotropic problem in each of the elastic parameters separately) and an element in the third category (indicating an ambiguity amongst the parameters). The second category gives a rich family of examples of sums of null space elements in the first and third categories, and thereby gives a sense of just how large the ‘elastic’ null space is. Moreover, we show that the traveltime perturbations caused by an elastic perturbation determine only a small number of features of the elastic perturbation which distinguish between the several elastic parameters. We identify these features pre-cisely: they are functions of the horizontal variable representing vertical averages of combinations of the elastic parameters and their derivatives. Elastic parameters influencing the vertical velocity appear more prominently in these features than those influencing the horizontal velocity, and these features are closely related to the zero-offset traveltimes and the normal moveout velocities. Our second set of results concerns the so-called velocity–depth ambiguity, more precisely here the elastic parameter versus the depth parameter ambiguity. We demonstrate that for any smooth depth perturbation, there is a perturbation in the elastic parameters for which the combined elastic and depth perturbations cause zero traveltime perturbations; moreover, this can be accomplished by an elliptically anisotropic elastic perturbation with a vertical axis of symmetry, perturbing the vertical velocity alone. This implies that the velocity–depth ambiguity cannot be resolved here as it can be in the isotropic case. We show an example of an anticlinal structure embedded in a homogeneous background whose reflection traveltimes exactly match those from a model with elliptically anisotropic elastic perturbations overlying a flat reflector. We also draw a number of conclusions from these results for the special case of transversely isotropic media with a vertical symmetry axis.

Numerical solution of sixth order boundary value problems with sixth degree B-spline functions

Here we consider the initial-boundary value problems in a homogeneous cylindrical domain YI Ω ×+ ( Ω+ is an open two-dimensional bounded simply connected domain with a boundary 5 ∂Ω ∈C , 2 \ Ω≡ Ω − + R is the open exterior of the domain Ω+ , [0, ] YI ≡ Y is the height of the cylinder) on a time interval [0, ] TI ≡ T . The initial conditions and the boundary conditions on the bases of the cylinder are zero, and the boundary conditions on the lateral surface of the cylinder are given by the function 1 2 wx x yt ( , , ,) ( 1 2 (, ) x x ∈∂Ω , Y y ∈ I , T t I ∈ ). An approximate solution of such problems is obtained through the combined use of the Fourier method and the collocation boundary element method based on piecewise quadratic interpolation (PQI). The solution to the problem in the cylinder is expanded in a Fourier series in terms of eigenfunctions of the operator 2 By yy ≡ ∂ with the corresponding zero boundary conditions. The coefficients of such a Fourier series are solutions of problems for two-dimensional heat equations 2 2 t ∇ =∂ + u u ku . With a low smoothness of the functions w in the variable y, the weight of solutions at large values of k increases and the accuracy of solving the problem in the cylinder decreases. To maintain accuracy on a uniform grid, the step of discretization of the boundary function w with respect to the variable y is decreased by a factor of j. Here j is an averaged value of the quantity Y k π depending on the function w. In addition, the steps of discretization of functions ( ) 2 exp − τ k with respect to the variable τ in domains τ≤πT k are reduced by a factor of 2 2 k π . The steps in the remaining ranges of values τ and the steps by the other variables remain unchanged. The approximate solutions obtained on the basis of this procedure converge stably to exact solutions in the 2 ( ) LI I Y T × -norm with a cubic velocity uniformly with respect to sets of functions w, bounded by norm of functions with low smoothness in the variable y, uniformly along the length of the generatrix of the cylinder Y , and uniformly in the domain Ω . The latter is also associated with the use of PQI along the curve ∂Ω over the variable 2 2 ρ≡ − r d , which is carried out at small values of r ( d and r are the distances from the observed point of the domain Ω to the boundary ∂Ω and to the current point of integration along ∂Ω , respectively). The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder, where the dependence of the boundary functions w on y is given by the normalized eigenfunctions of the differential operator By which vary in a sufficiently large range of values of k .

The conditions of existence and the construction of pseudosolutions, being the best by the method of least squares, of a Noetherian differential-algebraic boundary-value problem with concentrated delay, respectively, are determined.

The conditions of existence and the construction of pseudosolutions, being the best by the method of least squares, of a Noetherian differential-algebraic boundary-value problem with concentrated delay, respectively, are determined.

The least-squares meshfree method for the steady incompressible viscous flow

In the present work, we propose a two-phase fourth-order method for the approximate numerical solution for second-order non-linear two-point boundary value problems with Dirichlet boundary conditions. Our numerical approach is based on a finite difference and the solution of the problem at discrete points. Our method generates a system of equations, and the solution of the system of equations is considered an approximate solution to the problem. An essential analysis of the method is considered to ensure the performance of the method. A numerical experiment is carried out with model problems to test the performance in terms of efficiency and accuracy of the proposed method.

Approximate solution to the generalized radial Boussinesq equation

A numerical method for a magnetostatic inverse problem using the edge element

Two problems of scattering of surface water waves involving a semi-infinite elastic plate and a pair of semi-infinite elastic plates, separated by a gap of finite width, floating horizontally on water of finite depth, are investigated in the present work for a two-dimensional time-harmonic case. Within the frame of linear water wave theory, the solutions of the two boundary value problems under consideration have been represented in the forms of eigenfunction expansions. Approximate values of the reflection and transmission coefficients are obtained by solving an over-determined system of linear algebraic equations in each problem. In both the problems, the method of least squares as well as the singular value decomposition have been employed and tables of numerical values of the reflection and transmission coefficients are presented for specific choices of the parameters for modelling the elastic plates. Our main aim is to check the energy balance relation in each problem which plays a very important role in the present approach of solutions of mixed boundary value problems involving Laplace equations. The main advantage of the present approach of solutions is that the results for the values of reflection and transmission coefficients obtained by using both the methods are found to satisfy the energy-balance relations associated with the respective scattering problems under consideration. The absolute values of the reflection and transmission coefficients are presented graphically against different values of the wave numbers.

Thermal spreading resistance is one of the key factors for designing the thermal management systems in microelectronic devices. This type of thermal resistance occurs in most of the microelectronic devices and causes some difficulties for thermal engineers to model the system. One of the common geometries in these devices is the flux channel. Different boundary conditions can be applied on the flux channel based on the designing criteria of the system including the arbitrary distribution of heat sinks over the sink plane. This boundary condition is usually simplified as a constant heat transfer coefficient to facilitate the modeling of the system. In this paper, a flux channel with an arbitrary distributed heat transfer coefficient over the sink plane is studied without simplification of the sink boundary condition. Both adiabatic and convective cooling over the edges of the flux channel are considered. Due to the complexity of the sink boundary condition, the conventional analytical solutions are not applicable and the method of least squares is used. By employing this approach, the effect of a non-uniform heat transfer coefficient on thermal spreading resistance is investigated. The solution is presented in form of a Fourier series expansion which can be used to obtain the temperature all over the channel. Results are validated with Finite Element Models, FEM. This approach is useful for thermal engineers who have some difficulty for modeling complex boundary conditions and presents an effective solution for thermal resistance in the flux channels.

Under simply supported plate (SSP) boundary conditions, a numerical method based on the higher-order Legendre polynomial approximation was studied and developed for fourth-order problems with variable coefficients. We first divide the SSP boundary conditions into two types, namely, forced boundary conditions and natural boundary conditions. According to the forced boundary conditions, an appropriate Sobolev space is defined, and a variational formulation and a discrete scheme associated with the original problem are established. Then, the existence and uniqueness of this weak solution and approximate solution are proved. By using the Céa lemma and the tensor Jacobian polynomial approximation, we further obtain the error estimation for the numerical solutions. In addition, we use the orthogonality of Legendre polynomials to construct a set of effective basis functions and derive the equivalent tensor product linear system associated with the discrete scheme, respectively. Finally, some numerical tests were carried out to validate our algorithm and theoretical analysis.

Functional identities that hold for deviations from the exact solutions of boundary value and initial boundary value problems with monotone operators are obtained. These identities hold for any function from the corresponding functional class that contains the exact solution of the problem. The left-hand side of an identity is the sum of terms that measure deviation of the approximate solution from the exact one. It is shown that these measures are natural characteristics of the accuracy of approximate solutions. In some cases, the right-hand side of the identity contains only known data of the problem and functions that characterize the approximate solution. Such an identity can be directly used for error control. In other cases, the right-hand side includes unknown functions. However, they can be eliminated to obtain fully computable two-sided bounds. In this case, it is necessary to use special functional inequalities relating the deviation measures to the properties of the monotone operator under consideration. As an example, such bounds and the exact values of the corresponding constants are obtained for a class of problems with the-Laplacian operator. It is shown that the identities and the resulting bounds make it possible to estimate the error of any approximation regardless of the method used to obtain it. In addition, they open a way for comparing exact solutions of problems with different data, which makes it possible to evaluate the errors of mathematical models, e.g., those that arise when the coefficients of a differential equation are simplified. In the first part of the paper, the theory and applications concern stationary models, and then the main results are extended to evolutionary models with monotone spatial operators.