iterative solution of multiple radiation and scattering problems in structural acoustics using a block quasi minimal residual algorithm abstract finite element discretizations of time harmonic acoustic wave problems in exterior domains result in large sparse systems of linear equations with complex symmetric coefficient matrices in many situations these matrix problems need to be solved repeatedly for different right hand sides but with the same coefficient matrix recently freund and malhotra have proposed a block quasi minimal residual bl qmr algorithm 13 for the iterative solution of non hermitian linear systems with multiple right hand sides the bl qmr algorithm is a block krylov subspace iterative method that incorporates deflation to delete linearly and almost linearly dependent vectors in the underlying block krylov sequences in this paper we describe a j symmetric variant of the bl qmr algorithm that introduces important simplifications for the case when the coefficient matrix is symmetric with respect to a bilinear form induced by a certain matrix j in particular the j symmetric variant includes the complex symmetric form of bl qmr as a special case we identify suitable preconditioners for the bl qmr algorithm applied to multiple radiation and scattering problems our numerical tests with the preconditioned bl qmr algorithm for such multiple linear systems show that instead of solving each of the linear systems individually it is significantly more efficient to employ the block version of the iterative method moreover the numerical results clearly illustrate the importance of deflation and its effect on iterative convergence

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we

flexible and deflated variants of the block shifted gmres method abstract the solution of linear systems with multiple shifts and multiple right hand sides given simultaneously is required in many large scale scientific and engineering applications in this paper we introduce new flexible and deflated variants of the shifted block gmres method for this problem class the proposed methods solve the whole sequence of linear systems simultaneously detecting effectively the linear systems convergence and allowing the use of variable preconditioning which may be particularly useful in some applications numerical experiments are illustrated to show the overall significant robustness of the iterative method for solving general sparse multi shifted and multiple right hand side systems and in realistic pagerank calculations to the best of our knowledge this is the first krylov subspace method that combines deflation techniques and variable preconditioning for solving sequences of multi shifted linear systems with multiple right hand sides simultaneously

a block qmr method for computing multiple simultaneous solutions to complex symmetric systems the solution of complex symmetric indefinite systems of equations where multiple solutions are required is considered the quasi minimum residual qmr method ideally suited for these matrices is generalized using the block lanczos algorithm to solve multiple solutions simultaneously this modification alone is shown through numerical examples involving large sparse matrices from finite element discretization of maxwell s equations to accelerate the convergence by a factor almost as great as the number of simultaneous solutions a natural convergence criterion for this method is presented that is shown to be as effective as and easier to compute than the usual equation residual finally a numerical comparison of the classical incomplete cholesky and a variant of the ilu t preconditioners is given showing superior performance by the latter

viable three dimensional medical microwave tomography theory and numerical experiments three dimensional microwave tomography represents a potentially very important advance over 2d techniques because it eliminates associated approximations which may lead to more accurate images however with the significant increase in problem size computational efficiency is critical to making 3d microwave imaging viable in practice in this paper we present two 3d image reconstruction methods utilizing 3d scalar and vector field modeling strategies respectively finite element fe and finite difference time domain fdtd algorithms are used to model the electromagnetic field interactions in human tissue in 3d image reconstruction techniques previously developed for the 2d problem such as the dual mesh scheme iterative block solver and adjoint jacobian method are extended directly to 3d reconstructions speed improvements achieved by setting an initial field distribution and utilizing an alternating direction implicit adi fdtd are explored for 3d vector field modeling the proposed algorithms are tested with simulated data and correctly recovered the position size and electrical properties of the target the adjoint formulation and the fdtd method utilizing initial field estimates are found to be significantly more effective in reducing the computation time finally these results also demonstrate that cross plane measurements are critical for reconstructing 3d profiles of the target

singular points of harmonic maps from 4 dimensional domains into 3 spheres in this article we show that if a stable stationary harmonic map u from a domain in r4 into s3 has an isolated singular point then the mapping degree deg u of u at is 1 or 1 furthermore the complete characterization of stable stationary tangent maps from b4 to s3 is given

galerkin projection methods for solving multiple linear systems in this paper we consider using conjugate gradient cg methods for solving multiple linear systems a i x i b i for 1 le i le s where the coefficient matrices a i and the right hand sides b i are different in general in particular we focus on the seed projection method which generates a krylov subspace from a set of direction vectors obtained by solving one of the systems called the seed system by the cg method and then projects the residuals of other systems onto the generated krylov subspace to get the approximate solutions the whole process is repeated until all the systems are solved most papers in the literature t f chan and w l wan it siam j sci comput 18 1997 pp 1698 1721 b parlett it linear algebra appl 29 1980 pp 323 346 y saad it math comp 48 1987 pp 651 662 v simoncini and e gallopoulos it siam j sci comput 16 1995 pp 917 933 c smith a peterson and r mittra it ieee trans antennas and propagation 37 1989 pp 1490 1493 considered only the case where the coefficient matrices a i are the same but the right hand sides are different we extend and analyze the method to solve multiple linear systems with varying coefficient matrices and right hand sides a theoretical error bound is given for the approximation obtained from a projection process onto a krylov subspace generated from solving a previous linear system finally numerical results for multiple linear systems arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

adaptive array detection of uncertain rank one waveforms adaptive array detection of known within a complex scaling rank one space time waveforms in unknown spatial noise has received considerable attention the two published solutions are the adaptive matched filter and the glrt we expand on this work to consider the case of rank one waveforms that are uncertain i e only partially known more precisely we model the space time steering vector as the kronecker product of two vectors each of which is unknown but is known to lie in a known subspace applications for such a model include detection in the presence of multipath and spectral or polarization diversity in both radar processing and wireless communication using the principle of invariance we construct detectors based on the maximal invariant we show that the snr required to achieve a given detection probability for a given false alarm rate is only weakly impacted by waveform uncertainty thus our detector approaches the performance of earlier detectors which entail known waveforms

microwave image reconstruction from 3 d fields coupled to 2 d parameter estimation an efficient gauss newton iterative imaging technique utilizing a three dimensional 3 d field solution coupled to a two dimensional 2 d parameter estimation scheme 3 d 2 d is presented for microwave tomographic imaging in medical applications while electromagnetic wave propagation is described fully by a 3 d vector field a 3 d scalar model has been applied to improve the efficiency of the iterative reconstruction process with apparently limited reduction in accuracy in addition the image recovery has been restricted to 2 d but is generalizable to three dimensions image artifacts related primarily to 3 d effects are reduced when compared with results from an entirely two dimensional inversion 2 d 2 d important advances in terms of improving algorithmic efficiency include use of a block solver for computing the field solutions and application of the dual mesh scheme and adjoint approach for jacobian construction methods which enhance the image quality such as the log magnitude unwrapped phase minimization were also applied results obtained from synthetic measurement data show that the new 3 d 2 d algorithm consistently outperforms its 2 d 2 d counterpart in terms of reducing the effective imaging slice thickness in both permittivity and conductivity images over a range of inclusion sizes and background medium contrasts

accurate solutions of maxwell s equations around pec corners and highly curved surfaces using nodal finite elements a method is presented for computing accurate solutions of maxwell s equations in the presence of perfect electrical conductors pecs with sharp corners and highly curved surfaces using conventional nodal finite elements and a scalar vector s v potential formulation this technique approximates the pec with an impedance boundary condition ibc where the impedance is small critically it couples both potentials through this boundary condition rather than setting the scalar potential to zero this permits cancellation of the tangential components of the vector potential resulting in an accurate normal electric field the cause for the inaccuracies that nodal methods experience in the presence of sharp pec corners or highly curved pec surfaces is elucidated it is then shown how the inclusion of the scalar potential cures these deficiencies permitting accurate solutions spectral analysis of the resulting finite element matrices are shown validating the boundary conditions used examples are presented comparing a benchmark solution conventional pec and ibc boundary conditions and the new s v potential ibc on a pec wedge and pec ellipse in both cases the new s v ibc produces superior results

complex solution to nonideal contaminant transport through porous media nonideal or nonequilibrium transport through porous media is described by a convection diffusion equation coupled to a first order kinetics accounting for mass transfer between the solid and the fluid phases the overall mathematical model may be formulated using an integro differential approach and very effectively laplace transformed with complex parameterspk k 0 1 2m 1 solution in laplace space may be addressed by finite elements fe the resulting complex valued fe equations can be solved with either a complex or an equivalent real arithmetic operating on a problem twice as large as the original one for both approaches preconditioned projection or conjugate gradient like methods are used particularly difficult problems with high peclet numbers are investigated as well the results from three representative test cases totalling up to 15 000 equations show that the complex solution approach is superior to the real approach by up to almost two orders of magnitude depending on the problem it is also shown that while the solver performance vspkis stable in complex arithmetic this does not hold true for the solver in real arithmetic and an argument based on the quality of preconditioning is offered to account for the observed different computational behavior

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

a new modified goldstein levitin polyakprojection method for variational inequality problems in this paper we first show that the adjustment parameter in the step size choice strategy of the modified goldstein levitin polyak projection method proposed by he et al for asymmetric strongly monotone variational inequality problems can be bounded away from zero by a positive constant under this observation we propose a new step size rule which seems to be more practical and robust than the original one we show that the new modified method is globally convergent under the same conditions and report some computational results to illustrate the method

some developments in general variational inequalities general variational inequalities provide us with a unified natural novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences in this paper we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection wiener hopf equations updating the solution auxiliary principle inertial proximal penalty function dynamical system and well posedness we also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection type algorithms our proofs of convergence are very simple as compared with other methods our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems since the general variational inequalities include quasi variational inequalities and quasi implicit complementarity problems as special cases results presented here continue to hold for these problems several open problems have been suggested for further research in these areas

an iterative scheme for dense complex symmetric linear systems in acoustics boundary element computations we consider the indirect variational boundary integral formulation for the helmholtz equation in acoustics applications this formulation leads after discretization to a dense complex symmetric system of linear equations which are typically solved by direct elimination methods using the bunch kaufman pivoting strategy for matrices in the symmetric packed storage format as implemented in the lapack library anderson et al lapack users guide siam philadelphia 1995 in this paper we describe an iterative solution procedure which uses an adaptation of the orthogonal factorization method gill and murray in numerical methods for constrained optimization academic press new york 1974 pp 29 65 in conjunction with the qmr iteration freund siam j sci statist comput 13 1992 pp 425 448 this iterative procedure which achieves rapid convergence by exploiting the structure of the matrix generated by the discretization is competitive with the lapack direct elimination approach for small but difficult industrial applications on which it has been tested so far and for larger problems the iterative procedure is expected to provide significant advantages in terms of robustness computational cost and parallelizability

new step lengths in projection method for variational inequality problems in this paper we consider projection method for variational inequality problems first we give a new modification of recently proposed self adaptive step length rules with possibly minimal parameters then the resulting self adaptive projection method is proven to converge globally at a r linear rate provided that the underlying mapping is strongly monotone and lipschitz continuous preliminary numerical results confirm its flexibility and effectiveness

qmr based projection techniques for the solution of non hermitian systems with multiple right hand sides in this work we consider the simultaneous solution of large linear systems of the form ax j b j j 1 k where a is sparse and non hermitian we describe single seed and block seed projection approaches to these multiple right hand side problems that are based on the qmr and block qmr algorithms respectively we use block qmr to solve the block seed system and generate the relevant biorthogonal subspaces approximate solutions to the nonseed systems are simultaneously generated by minimizing their appropriately projected block residuals after the initial block seed has converged the process is repeated by choosing a new block seed from among the remaining nonconverged systems and using the previously generated approximate solutions as initial guesses for the new seed and nonseed systems we give theory for the single seed case that helps explain the convergence behavior under certain conditions implementation details for both the single seed and block seed algorithms are discussed and advantages of the block seed algorithm in cache based serial and parallel environments are noted the computational savings of our methods over using qmr to solve each system individually are illustrated in two examples

a block qmr algorithm for non hermitian linear systems with multiple right hand sides abstract many applications require the solution of multiple linear systems that have the same coefficient matrix but differ in their right hand sides instead of applying an iterative method to each of these systems individually it is often more efficient to employ a block version of the method that generates iterates for all the systems simultaneously in this paper we propose a block version of freund and nachtigal s quasi minimal residual qmr method for the iterative solution of non hermitian linear systems the block qmr method uses a novel lanczos type process for multiple starting vectors which was recently developed by aliaga boley freund and hernandez to compute suitable basis vectors for the underlying block krylov subspaces we describe the basic block qmr method and also give important implementation details in particular we show how to incorporate deflation to drop converged linear systems and to delete linearly and almost linearly dependent vectors in the underlying block krylov sequences numerical results are reported that illustrate typical features of the block qmr method

efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering in this study we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions we show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution we also show how to generate a variety of statistical quantities from the computed solution finally we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac