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Solution Set (solution + set)
Selected AbstractsQuantitative Comparison of Approximate Solution Sets for Bi-criteria Optimization Problems,DECISION SCIENCES, Issue 1 2003W. Matthew Carlyle ABSTRACT We present the Integrated Preference Functional (IPF) for comparing the quality of proposed sets of near-pareto-optimal solutions to bi-criteria optimization problems. Evaluating the quality of such solution sets is one of the key issues in developing and comparing heuristics for multiple objective combinatorial optimization problems. The IPF is a set functional that, given a weight density function provided by a decision maker and a discrete set of solutions for a particular problem, assigns a numerical value to that solution set. This value can be used to compare the quality of different sets of solutions, and therefore provides a robust, quantitative approach for comparing different heuristic, a posteriori solution procedures for difficult multiple objective optimization problems. We provide specific examples of decision maker preference functions and illustrate the calculation of the resulting IPF for specific solution sets and a simple family of combined objectives. [source] A hybrid-Trefftz finite element model for Helmholtz problemINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2008K. Y. Sze Abstract In this paper, a hybrid-Trefftz four-node quadrilateral element model is formulated for Helmholtz problem. In this model, two Helmholtz approximations are defined. The first approximation is obtained by nodal interpolation, and the second approximation is truncated from a Trefftz solution set. A hybrid variational functional is employed to enforce the equality of and other necessary conditions on the two approximations. From the numerical tests, it can be seen that the hybrid model is markedly more accurate than the conventional finite element model. Copyright © 2008 John Wiley & Sons, Ltd. [source] POTENTIAL IMPACTS OF CLIMATE CHANGE ON CALIFORNIA HYDROLOGY,JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION, Issue 4 2003Norman L. Miller ABSTRACT: Previous reports based on climate change scenarios have suggested that California will be subjected to increased wintertime and decreased summertime streamflow. Due to the uncertainty of projections in future climate, a new range of potential climatological future temperature shifts and precipitation ratios is applied to the Sacramento Soil Moisture Accounting Model and Anderson Snow Model in order to determine hydrologic sensitivities. Two general circulation models (GCMs) were used in this analysis: one that is warm and wet (HadCM2 run 1) and one that is cool and dry (PCM run B06.06), relative to the GCM projections for California that were part of the Third Assessment Report of the Intergovernmental Panel on Climate Change. A set of specified incremental temperature shifts from 1.5°C to 5.0°C and precipitation ratios from 0.70 to 1.30 were also used as input to the snow and soil moisture accounting models, providing for additional scenarios (e.g., warm/dry, cool/wet). Hydrologic calculations were performed for a set of California river basins that extend from the coastal mountains and Sierra Nevada northern region to the southern Sierra Nevada region; these were applied to a water allocation analysis in a companion paper. Results indicate that for all snow-producing cases, a larger proportion of the streamflow volume will occur earlier in the year. The amount and timing is dependent on the characteristics of each basin, particularly the elevation. Increased temperatures lead to a higher freezing line, therefore less snow accumulation and increased melting below the freezing height. The hydrologic response varies for each scenario, and the resulting solution set provides bounds to the range of possible change in streamflow, snowmelt, snow water equivalent, and the change in the magnitude of annual high flows. An important result that appears for all snowmelt driven runoff basins, is that late winter snow accumulation decreases by 50 percent toward the end of this century. [source] On the domain dependence of solutions to the Navier,Stokes equations of a two-dimensional compressible flowMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2009Fei Jiang Abstract We consider the Navier,Stokes equations for compressible, barotropic flow in two space dimensions, with pressure satisfying p(,)=a,logd(,) for large ,, here d>1 and a>0. After introducing useful tools from the theory of Orlicz spaces, we prove a compactness result for the solution set of the equations with respect to the variation of the underlying bounded spatial domain. Especially, we get a general existence theorem for the system in question with no restrictions on smoothness of the bounded spatial domain. Copyright © 2009 John Wiley & Sons, Ltd. [source] On the domain dependence of solutions to the compressible Navier,Stokes equations of a barotropic fluidMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2002Eduard Feireisl We prove a general compactness result for the solution set of the compressible Navier,Stokes equations with respect to the variation of the underlying spatial domain. Among various corollaries, we then prove a general existence theorem for the system in question with no restrictions on smoothness of the spatial domain. Copyright © 2002 John Wiley & Sons, Ltd. [source] Asymptotic convergence of p -Laplace equationswith constraint as p tends to 1MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2002Ken Shirakawa In this paper we treat the Euler,Lagrange equation of a functional including the p -Laplacian for 1 The matricial Schur problem in both nondegenerate and degenerate casesMATHEMATISCHE NACHRICHTEN, Issue 2 2009Bernd Fritzsche Abstract The principal object of this paper is to present a new approach simultaneously to both nondegenerate and degenerate cases of the matricial Schur problem. This approach is based on an analysis of the central matrixvalued Schur functions which was started in [24],[26] and then continued in [27]. In the nondegenerate situation we will see that the parametrization of the solution set obtained here coincides with the well-known formula of D. Z. Arov and M. G. Kre,n for that case (see [1]). Furthermore, we give some characterizations of the situation that the matricial Schur problem has a unique solution (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Nonlinear Riemann,Hilbert problems with circular target curvesMATHEMATISCHE NACHRICHTEN, Issue 9 2008Christer Glader Abstract The paper gives a systematic and self-contained treatment of the nonlinear Riemann,Hilbert problem with circular target curves |w , c | = r, sometimes also called the generalized modulus problem. We assume that c and r are Hölder continuous functions on the unit circle and describe the complete set of solutions w in the disk algebra H, , C and in the Hardy space H, of bounded holomorphic functions. The approach is based on the interplay with the Nehari problem of best approximation by bounded holomorphic functions. It is shown that the considered problems fall into three classes (regular, singular, and void) and we give criteria which allow to classify a given problem. For regular problems the target manifold is covered by the traces of solutions with winding number zero in a schlicht manner. Counterexamples demonstrate that this need not be so if the boundary condition is merely continuous. Paying special attention to constructive aspects of the matter we show how the Nevanlinna parametrization of the full solution set can be obtained from one particular solution of arbitrary winding number. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraintNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2007An-ping Liao Abstract An n × n real matrix A = (aij)n × n is called bi-symmetric matrix if A is both symmetric and per-symmetric, that is, aij = aji and aij = an+1,1,n+1,i (i, j = 1, 2,..., n). This paper is mainly concerned with finding the least-squares bi-symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright © 2007 John Wiley & Sons, Ltd. [source] The inverse problem of bisymmetric matrices with a submatrix constraintNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2004Zhen-yun Peng Abstract An n×n real matrix A is called a bisymmetric matrix if A=AT and A=SnASn, where Sn is an n×n reverse unit matrix. This paper is mainly concerned with solving the following two problems: Problem I Given n×m real matrices X and B, and an r×r real symmetric matrix A0, find an n×n bisymmetric matrix A such that where A([1: r]) is a r×r leading principal submatrix of the matrix A. Problem II Given an n×n real matrix A*, find an n×n matrix  in SE such that where ,·, is Frobenius norm, and SE is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided. Copyright © 2003 John Wiley & Sons, Ltd. [source] Quantitative Comparison of Approximate Solution Sets for Bi-criteria Optimization Problems,DECISION SCIENCES, Issue 1 2003W. Matthew Carlyle ABSTRACT We present the Integrated Preference Functional (IPF) for comparing the quality of proposed sets of near-pareto-optimal solutions to bi-criteria optimization problems. Evaluating the quality of such solution sets is one of the key issues in developing and comparing heuristics for multiple objective combinatorial optimization problems. The IPF is a set functional that, given a weight density function provided by a decision maker and a discrete set of solutions for a particular problem, assigns a numerical value to that solution set. This value can be used to compare the quality of different sets of solutions, and therefore provides a robust, quantitative approach for comparing different heuristic, a posteriori solution procedures for difficult multiple objective optimization problems. We provide specific examples of decision maker preference functions and illustrate the calculation of the resulting IPF for specific solution sets and a simple family of combined objectives. [source]
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