a quadratic convergence yielding iterative method for the implementation of lavrentiev regularization method for ill posed equations george and elmahdy 2012 considered an iterative method which converges quadratically to the unique solution is a nonlinear monotone operator defined on a real hilbert space x the convergence analysis of the method was based on a majorizing sequence in this paper we are concerned with the problem of expanding the applicability of the method considered by george and elmahdy 2012 by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular lipschitz type hypotheses considered in earlier studies such as george and elmahdy 2012 mahale and nair 2009 mathe and perverzev 2003 nair and ravishankar 2008 semenova 2010 and tautanhahn 2002 we show that the adaptive scheme considered by perverzev and schock 2005 for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate in the concluding section the method is applied to numerical solution of the inverse gravimetry problem

a new l curve for ill posed problems the truncated singular value decomposition is a popular method for the solution of linear ill posed problems the method requires the choice of a truncation index which affects the quality of the computed approximate solution this paper proposes that an l curve which is determined by how well the given data right hand side can be approximated by a linear combination of the first few left singular vectors or functions be used as an aid for determining the truncation index

determining conductivity by boundary measurements ii interior results abstract in a recent paper the authors showed that an unknown real analytic conductivity gamma may be determined from static boundary measurements in this document they extend this analysis by demonstrating that a similar result holds for piecewise real analytic conductivities in addition for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements originator supplied keywords include inversion convergence algorithms estimates coefficients

relative canonical sheaves of a family of curves abstract in this paper we study the relative canonical sheaf of a relatively minimal fibration of curves of genus g 2 over a one dimensional regular scheme using the configurations of 2 chains we show that its m tensored product is base point free for any m 2 we utilize koszul cohomology to prove that the relative canonical algebra of the fibration is generated in degree up to five it is a generalization of k konno s work on the 1 2 3 conjecture of m reid

on converse and saturation results for tikhonov regularization of linear ill posed problems in this paper we prove some new converse and saturation results for tikhonov regularization of linear ill posed problems tx y where t is a linear operator between two hilbert spaces

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

an analysis of lavrentiev regularization method and newton type process for nonlinear ill posed problems abstract in this paper we consider the lavrentiev regularization method and a modified newton method for obtaining stable approximate solution to nonlinear ill posed operator equations f x y where f d f x x is a nonlinear monotone operator or f x 0 is nonnegative selfadjoint operator defined on a real hilbert space x we assume that only a noisy data y x with y y are available further we assume that frechet derivative f of f satisfies center type lipschitz condition a priori choice of regularization parameter is presented we proved that under a general source condition on x 0 x the error x x n between the regularized approximation x n x 0 x 0 and the solution x is of optimal order in the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem

numerical solution method for the dbar equation in the plane a fast method for solving equations of the form v tv is presented where v and t are complex valued functions of two real variables the multigrid method of vainikko int soc anal appl comput 5 2000 is adapted to the problem with a fft implementation convergence with rate o h is proved for the method applied to equations of the form above one grid and two grid versions of the method are implemented and their effectiveness is demonstrated on an application arising in electrical impedance tomography eit

an a posteriori parameter choice for ordinary and iterated tikhonov regularization of ill posed problems leading to optimal convergence rates on propose un choix a posteriori du parametre pour la regularisation de tikhonov ordinaire et iteree des problemes mal poses conduisant des vitesses de convergence optimales

on a class of frozen regularized gauss newton methods for nonlinear inverse problems in this paper we consider a class of regularized gauss newton methods for solving nonlinear inverse problems for which an a posteriori stopping rule is proposed to terminate the iteration such methods have the frozen feature that they require only the computation of the frechet derivative at the initial approximation thus the computational work is considerably reduced under certain mild conditions we give the convergence analysis and derive various estimates including the order optimality on these methods

inverse geometry heat transfer problem based on a radial basis functions geometry representation summary we present a methodology for solving a nonlinear inverse geometry heat transfer problem where the observations are temperature measurements at points inside the object and the unknown is the geometry of the volume where the problem is de ned the representation of the geometry is based on radial basis functions rbfs and the nonlinear inverse problem is solved using the iteratively regularized gauss newton method in our work we consider not only the problem with no geometry restrictions but also the bound constrained problem the methodology is used for the industrial application of estimating the location of the 1150 c isotherm in a blast furnace hearth based on measurements of the thermocouples located inside it we validate the solution of the algorithm against simulated measurements with different levels of noise and study its behavior on different regularization matrices finally we study the error behavior of the solution copyright c 2004 john wiley sons ltd

level set method for the inverse elliptic problem in nonlinear electromagnetism an inverse problem of inhomogeneity identification inside a nonlinear magnetic material from the local measurements of the magnetic induction is investigated the representation of the shape of the inhomogeneity and its evolution during an iterative reconstruction process is achieved by the level set method the reconstruction is realized by the minimization of a cost function using the steepest descent method the gradient directions are evaluated using the sensitivity equation and the adjoint variable method simulations has been performed showing the robustness of the algorithm and its ability to reconstruct single inhomogeneities convex and non convex as well as multiple inhomogeneities

multi parameter tikhonov regularization and model function approach to the damped morozov principle for choosing regularization parameters in this paper we study the multi parameter tikhonov regularization method which adds multiple different penalties to exhibit multi scale features of the solution an optimal error bound of the regularization solution is obtained by a priori choice of multiple regularization parameters some theoretical results of the regularization solution about the dependence on regularization parameters are presented then an a posteriori parameter choice i e the damped morozov discrepancy principle is introduced to determine multiple regularization parameters five model functions i e two hyperbolic model functions a linear model function an exponential model function and a logarithmic model function are proposed to solve the damped morozov discrepancy principle furthermore four efficient model function algorithms are developed for finding reasonable multiple regularization parameters and their convergence properties are also studied numerical results of several examples show that the damped discrepancy principle is competitive with the standard one and the model function algorithms are efficient for choosing regularization parameters

a discontinuous galerkin based immersed boundary method with non homogeneous boundary conditions and its application to elasticity we propose a discontinuous galerkin based immersed boundary method for elasticity problems the resulting numerical scheme does not require boundary fitting meshes and avoids boundary locking by switching the elements intersected by the boundary to a discontinuous galerkin approximation special emphasis is placed on the construction of a method that retains an optimal convergence rate in the presence of non homogeneous essential and natural boundary conditions the role of each one of the approximations introduced is illustrated by analyzing an analog problem in one spatial dimension finally extensive two and three dimensional numerical experiments on linear and nonlinear elasticity problems verify that the proposed method leads to optimal convergence rates under combinations of essential and natural boundary conditions

on the solution of an inverse natural convection problem using various conjugate gradient methods the inverse problem of determining the time varying strength of a heat source which causes natural convection in a two dimensional cavity is considered the boussinesq equation is used to model the natural convection induced by the heat source the inverse natural convection problem is solved through the minimization of a performance function utilizing the conjugate gradient method the gradient of the performance function needed in the minimization procedure of the conjugate gradient method is obtained by employing either the adjoint variable method or the direct differentiation method the accuracy and efficiency of these two methods are compared and a new method is suggested that exploits the advantageous aspects of both methods while avoiding the shortcomings of them copyright 2000 john wiley sons ltd

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd

extrapolation techniques for ill conditioned linear systems in this paper the regularized solutions of an ill conditioned system of linear equations are computed for several values of the regularization parameter lambda then these solutions are extrapolated at lambda 0 by various vector rational extrapolations techniques built for that purpose these techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition numerical results illustrate the effectiveness of the procedures

on the discrepancy principle for some newton type methods for solving nonlinear inverse problems we consider the computation of stable approximations to the exact solution x dagger of nonlinear ill posed inverse problems f x y with nonlinear operators f x y between two hilbert spaces x and y by the newton type methods x k 1 delta x 0 g alpha k left f x k delta f x k delta right f x k delta left f x k delta y delta f x k delta x k delta x 0 right of y satisfying y delta y le delta with a given small noise level delta 0 we terminate the iteration by the discrepancy principle in which the stopping index k delta is determined as the first integer such that f x k delta delta y delta le tau delta f x k delta y delta quad 0 le k k delta converges to x dagger as delta rightarrow 0 and establish various order optimal convergence rate results it is remarkable that we even can show the order optimality under merely the lipschitz condition on the frechet derivative f of f if x 0 x dagger is smooth enough

universal meshes a method for triangulating planar curved domains immersed in nonconforming meshes summary we introduce a new method to triangulate planar curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary in the process no new vertices are introduced and connectivities of triangles are left unaltered the method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection to guarantee its robustness we require that the domain be c2 regular the background mesh be sufficiently refined near the boundary and that specific angles in triangles near the boundary be strictly acute the method can render both straight edged and curvilinear triangulations for the immersed domain the latter includes curved triangles that conform exactly to the immersed boundary and ones constructed with isoparametric mappings to interpolate the boundary at select points high order finite elements constructed over these curved triangles achieve optimal accuracy which has customarily proven difficult in numerical schemes that adopt nonconforming meshes aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains with no conformity requirements the same background mesh can be adopted to triangulate a large family of domains immersed in it including ones realized over several updates during the coarse of simulating problems with moving boundaries we term such a background mesh as a universal mesh for the family of domains it can be used to triangulate universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes furthermore because the evolving geometry can be approximated with any desired order numerical solutions can be computed with high order accuracy we present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with stokesian fluids copyright 2014 john wiley sons ltd

simplified gsvd computations for the solution of linear discrete ill posed problems the generalized singular value decomposition gsvd often is used to solve tikhonov regularization problems with a regularization matrix without exploitable structure this paper describes how the standard methods for the computation of the gsvd of a matrix pair can be simplified in the context of tikhonov regularization also other regularization methods including truncated gsvd are considered we compare the computational efforts required by the simplified gsvd method and the a weighted generalized inverse introduced by elden

flat top footprint pattern synthesis through the design of arbitrary planar shaped apertures the problem of generating a flat top main beam with an arbitrary footprint for array elements placed in an arbitrary planar aperture is considered in this paper some simplifying properties of the bessel functions encourages the general framework of the paper to encompass patterns produced by circular aperture and eventually generalize it to arbitrary aperture geometries in this regard two synthesis methods are presented the first method is based on the use of the rayleigh quotient to obtain constant phase array patterns hence a class of generally linear phase patterns can be considered the second approach is based on power pattern synthesis where there is no restriction on the phase of the pattern hence it provides us with greater flexibility the nonlinear problem is appropriately modeled and formulated for amenable performance these two new methods can exhibit a significant reduction in the number of unknown parameters and high flexibility in shaping the desired main beam by arbitrary lattice geometry

completeness of products of solutions and some inverse problems for pde abstract in this paper we give conditions which guarantee that products of solutions of partial differential equations pu au 0 are complete in l 2 here p is a linear partial differential operator with constant coefficients and a is a function in l we check these conditions for elliptic parabolic and hyperbolic equations of second order and give applications to inverse problems for related equations where a is to be found

a tree of generalizations of the ordinary singular value decomposition abstract it is shown how to generalize the ordinary singular value decomposition of a matrix into a combined factorization of any number of matrices we propose to call these factorizations generalized singular value decompositions for two matrices this reduces to the product and quotient singular value decompositions one of the factorizations for three matrices is the restricted singular value decomposition these generalizations form a tree of factorizations where at level k for k matrices there are 2 k factorizations not all of which are independent the different levels are related to each other in a recursive fashion any generalized singular value decomposition for k matrices can be constructed from a decomposition for k 1 matrices this results in an inductive proof which uses only the ordinary singular value decomposition several examples are analysed in detail

solving variational problems and partial differential equations mapping into general target manifolds a framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper we discuss the framework for arbitrary target manifolds while the domain manifold problem was addressed in j comput phys 174 2 2001 759 the key idea is to implicitly represent the target manifold as the level set of a higher dimensional function and then implement the equations in the cartesian coordinate system where this embedding function is defined in the case of variational problems we restrict the search of the minimizing map to the class of maps whose target is the level set of interest in the case of partial differential equations we re write all the equation s geometric characteristics with respect to the embedding function we then obtain a set of equations that while defined on the whole euclidean space are intrinsic to the implicitly defined target manifold and map into it this permits the use of classical numerical techniques in cartesian grids regardless of the geometry of the target manifold the extension to open surfaces and submanifolds is addressed in this paper as well in the latter case the submanifold is defined as the intersection of two higher dimensional hypersurfaces and all the computations are restricted to this intersection examples of the applications of the framework here described include harmonic maps in liquid crystals where the target manifold is a hypersphere probability maps where the target manifold is a hyperplane chroma enhancement texture mapping and general geometric mapping between high dimensional manifolds

on clifford s theorem for singular curves let c be a 2 connected projective curve either reduced with planar singularities or contained in a smooth algebraic surface and let s be a subcanonical cluster that is a zero dimensional scheme such that the space h 0 c iskc contains a generically invertible section under some general assumptions on s or c we show that h 0 c iskc pa c 1 deg s and if equality holds then either s is trivial or c is honestly hyperelliptic or 3 disconnected as a corollary we give a generalization of clifford s theorem for reduced curves with planar singularities

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

on a generalization of reginska s parameter choice rule and its numerical realization in large scale multi parameter tikhonov regularization abstract a crucial problem concerning tikhonov regularization is the proper choice of the regularization parameter this paper deals with a generalization of a parameter choice rule due to reginska 1996 31 analyzed and algorithmically realized through a fast fixed point method in bazan 2008 3 which results in a fixed point method for multi parameter tikhonov regularization called mfp like the single parameter case the algorithm does not require any information on the noise level further combining projection over the krylov subspace generated by the golub kahan bidiagonalization gkb algorithm and the mfp method at each iteration we derive a new algorithm for large scale multi parameter tikhonov regularization problems the performance of mfp when applied to well known discrete ill posed problems is evaluated and compared with results obtained by the discrepancy principle the results indicate that mfp is efficient and competitive the efficiency of the new algorithm on a super resolution problem is also illustrated

a shifted complex global lanczos method and the quasi minimal residual variant for the stein conjugate matrix equation x ax b c abstract in this paper based on a complex global lanczos process a shifted complex global lanczos method is proposed to solve the stein conjugate matrix equation x a x b c the new method adopts the shifted technique and is implemented by the original coefficient matrices the direct version of the proposed method is derived by exploiting the special structure of the l u factorization moreover the quasi minimal residual variant is developed to stabilize the convergence behavior finally some numerical examples are given to illustrate the effectiveness with comparison to some existing methods

algebraically constrained extended edge element method exfem ac for resolution of multi material cells abstract surface effects are critical to the predictive simulation of electromagnetics as current tends to concentrate near material interfaces there are two principal difficulties in the accurate representation of these effects in discrete models first many applications of interest operate at large deformations where body fitted meshes are impractical second physics compatible discretizations of the governing equations require curl conforming edge elements for which no practical alternatives to body fitted meshes exist the main purpose of this paper is to develop such an alternative that avoids remeshing the problem our approach uses the existing edge element basis to dynamically construct an interface conforming basis we show that in the case of triangular grids in two dimensions our approach generates a basis that spans the same space as edge elements on an interface fitted mesh we also demonstrate the efficacy of the approach computationally

a convergence analysis of the landweber iteration for nonlinear ill posed problems in this paper we prove that the landweber iteration is a stable method for solving nonlinear ill posed problems for perturbed data with noise level delta we propose a stopping rule that yields the convergence rate o delta 1 2 under appropriate conditions we illustrate these conditions for a few examples

simulating curvilinear crack propagation in two dimensions with universal meshes summary we formulate a class of delicately controlled problems to model the kink free evolution of quasistatic cracks in brittle isotropic linearly elastic materials in two dimensions the evolving crack satisfies familiar principles griffith s criterion local symmetry and irreversibility a novel feature of the formulation is that in addition to the crack path the loading is also treated as an unknown specifically a scaling factor for prescribed dirichlet and neumann boundary conditions is computed as part of the solution to yield an always propagating and ostensibly kink free crack and a continuous loading history beyond the initial step a dimensionless statement of the problem depends only on the poisson s ratio of a homogeneous material and is in particular independent of its young s modulus and fracture toughness numerical resolution of the formulated problem relies on two new ideas the first is an algorithm to compute triangulations conforming to cracked domains by locally deforming a given background mesh in the vicinity of evolving cracks the algorithm is robust under mild assumptions on the sizes and angles of triangles near the crack and its smoothness hence a subset if not the entire family of cracked domains realized during the course of a simulation can be discretized with the same background mesh we term the latter a universal mesh for such a family of domains universal meshes facilitate adopting a discrete representation for the crack as splines in our examples preclude the need for local splitting retriangulation operations liberate the crack from following directions prescribed by the mesh and enable the adoption of standard fems to compute the elastic fields in the cracked solid second we employ a method specifically designed to approximate the stress intensity factors for curvilinear cracks we examine the performance of the resulting numerical method with detailed examples including comparisons with an exact solution of a crack propagating along a circular arc which we construct and comparisons with experimental fracture paths in all cases we observe convergence of computed paths their derivatives and loading histories with refinement of the universal mesh copyright 2014 john wiley sons ltd

multi parameter regularization techniques for ill conditioned linear systems when a system of linear equations is ill conditioned regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties the ill conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen in this paper we consider the case of several regularizations terms added simultaneously thus overcoming the problem of the best choice of the regularization matrix the error of this procedure is analyzed and numerical results prove its efficiency

multi parameter regularization and its numerical realization in this paper we propose and analyse a choice of parameters in the multi parameter regularization of tikhonov type a modified discrepancy principle is presented within the multi parameter regularization framework an order optimal error bound is obtained under the standard smoothness assumptions we also propose a numerical realization of the multi parameter discrepancy principle based on the model function approximation numerical experiments on a series of test problems support theoretical results finally we show how the proposed approach can be successfully implemented in laplacian regularized least squares for learning from labeled and unlabeled examples

shape identification for natural convection problems using the adjoint variable method an inverse geometry problem is investigated to identify the boundary shape of a domain from temperature measurements on the other boundary where the temperature field is dominated by natural convection the potential applications of the present investigation are the determination of a phase change isotherm in the bridgman crystal growth or the thermal tomography which detects flaws in materials nondestructively the inverse problem is posed as a minimization problem of the performance function which is the sum of square residuals between calculated and observed temperature by means of a conjugate gradient method employing the adjoint variable method the present method is found to identify the domains reasonably accurately even with noisy temperature measurements

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

a modified tikhonov regularization method tikhonov regularization and truncated singular value decomposition tsvd are two elementary techniques for solving a least squares problem from a linear discrete ill posed problem based on these two techniques a modified regularization method is proposed which is applied to an arnoldi based hybrid method theoretical analysis and numerical examples are presented to illustrate the effectiveness of the method

weighted least squares criteria for electrical impedance tomography methods are developed for the design of electrical impedance tomographic reconstruction algorithms with specified properties assuming a starting model with constant conductivity or some other specified background distribution an algorithm with the following properties is found 1 the optimum constant for the starting model is determined automatically 2 the weighted least squares error between the predicted and measured power dissipation data is as small as possible 3 the variance of the reconstructed conductivity from the starting model is minimized 4 potential distributions with the largest volume integral of gradient squared have the least influence on the reconstructed conductivity and therefore distributions most likely to be corrupted by contact impedance effects are deemphasized 5 cells that dissipate the most power during the current injection tests tend to deviate least from the background value for a starting model with nonconstant conductivity the reconstruction algorithm has analogous properties

optimal a posteriori parameter choice for tikhonov regularization for solving nonlinear ill posed problems the authors propose an a posteriors strategy for choosing the regularization parameter in tikhonov regularization for solving nonlinear ill posed problems and show that under certain conditions the convergence rate obtained with this strategy is optimal as a by product a new stability estimate for the regularized solutions is given which applies to a class of parameter identification problems the authors compare the parameter choice strategy with morozov s discrepancy principle finally numerical results are presented

on a variational formulation of the qsvd and the rsvd abstract recently m t chu r f funderlic and g h golub siam j matrix anal appl 18 1997 1082 1092 presented a variational formulation for the quotient singular value decomposition qsvd of two matrices a r n m c r p m which is a generalization to two matrices of the ordinary singular value decomposition svd and characterizes the role of two orthogonal matrices in qsvd in this paper we give an alternative derivation of this variational formulation and extend it to establish an analogous variational formulation for the restricted singular value decomposition rsvd of a matrix triplet a r n m b r n l c r p m which provides a new understanding of the orthogonal matrices appearing in this decomposition

on an a posteriori parameter choice strategy for tikhonov regularization of nonlinear ill posed problems in the study of the choice of the regularization parameter for tikhonov regularization of nonlinear ill posed problems scherzer engl and kunisch proposed an a posteriori strategy in 1993 to prove the optimality of the strategy they imposed many very restrictive conditions on the problem under consideration their results are difficult to apply to concrete problems since one can not make sure whether their assumptions are valid in this paper we give a further study on this strategy and show that tikhonov regularization is order optimal for each 0 nu le 2 with the regularization parameter chosen according to this strategy under some simple and easy checking assumptions this paper weakens the conditions needed in the existing results and provides a theoretical guidance to numerical experiments

on a regularized levenberg marquardt method for solving nonlinear inverse problems we consider a regularized levenberg marquardt method for solving nonlinear ill posed inverse problems we use the discrepancy principle to terminate the iteration under certain conditions we prove the convergence of the method and obtain the order optimal convergence rates when the exact solution satisfies suitable source wise representations

numerical solution of the ℝ linear beltrami equation the r linear beltrami equation appears in applications such as in the inverse problem of recovering the electrical conductivity distribution in the plane in this paper a new way to discretize the r linear beltrami equation is considered this gives rise to large and dense r linear systems of equations with structure for their iterative solution norm minimizing krylov subspace methods are devised in the numerical experiments these improvements combined are shown to lead to speed ups of almost two orders of magnitude in the electrical conductivity problem

elastic wave identification of penetrable obstacles using shape material sensitivity framework this study deals with elastic wave identification of discrete heterogeneities inclusions in an otherwise homogeneous reference solid from limited aperture waveform measurements taken on its surface on adopting the boundary integral equation bie framework for elastodynamic scattering the inverse query is cast as a minimization problem involving experimental observations and their simulations for a trial inclusion that is defined through its boundary elastic moduli and mass density for an optimal performance of the gradient based search methods suited to solve the problem explicit expressions for the shape i e boundary and material sensitivities of the misfit functional are obtained via the adjoint field approach and direct differentiation of the governing bies making use of the message passing interface the proposed sensitivity formulas are implemented in a data parallel code and integrated into a nonlinear optimization framework based on the direct bie method and an augmented lagrangian whose inequality constraints are employed to avoid solving forward scattering problems for physically inadmissible or overly distorted trial inclusion configurations numerical results for the reconstruction of an ellipsoidal defect in a semi infinite solid show the effectiveness of the proposed shape material sensitivity formulation which constitutes an essential computational component of the defect identification algorithm

the implicit closest point method for the numerical solution of partial differential equations on surfaces many applications in the natural and applied sciences require the solutions of partial differential equations pdes on surfaces or more general manifolds the closest point method is a simple and accurate embedding method for numerically approximating pdes on rather general smooth surfaces however the original formulation is designed to use explicit time stepping this may lead to a strict time step restriction for some important pdes such as those involving the laplace beltrami operator or higher order derivative operators to achieve improved stability and efficiency we introduce a new implicit closest point method for surface pdes the method allows for large stable time steps while retaining the principal benefits of the original method in particular it maintains the order of accuracy of the discretization of the underlying embedding pde it works on sharply defined bands without degrading the accuracy of the method and it applies to general smooth surfaces it also is very simple and may be applied to a rather general class of surface pdes convergence studies for the in surface heat equation and a fourth order biharmonic problem are given to illustrate the accuracy of the method we demonstrate the flexibility and generality of the method by treating flows involving diffusion reaction diffusion and fourth order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension

a convergence analysis of nonlinear implicit iterative method for nonlinear ill posed problems abstract implicit iterative method acquires good effect in solving linear ill posed problems we have ever applied the idea of implicit iterative method to solve nonlinear ill posed problems under the restriction that is appropriate large we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence numerical results show that the implicit iterative method for nonlinear ill posed problems is efficient in this paper we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when k are determined by hanke criterion

high order finite element methods for moving boundary problems with prescribed boundary evolution abstract we introduce a framework for the design of finite element methods for two dimensional moving boundary problems with prescribed boundary evolution that have arbitrarily high order of accuracy both in space and in time at the core of our approach is the use of a universal mesh a stationary background mesh containing the domain of interest for all times that adapts to the geometry of the immersed domain by adjusting a small number of mesh elements in the neighborhood of the moving boundary the resulting method maintains an exact representation of the prescribed moving boundary at the discrete level or an approximation of the appropriate order yet is immune to large distortions of the mesh under large deformations of the domain the framework is general making it possible to achieve any desired order of accuracy in space and time by selecting a preferred and suitable finite element space on the universal mesh for the problem at hand and a preferred and suitable time integrator for ordinary differential equations we illustrate our approach by constructing a particular class of methods and apply them to a prescribed boundary variant of the stefan problem we present numerical evidence for the order of accuracy of our schemes in one and two dimensions

more comments on c p we discuss the interpretation of c p plots and show how they can be calibrated in several ways we comment on the practice of using the display as a basis for formal selection of a subset regression model and extend the range of application of the device to encompass arbitrary linear estimates of the regression coefficients for example ridge estimates

direct reconstructions of conductivities from boundary measurements the problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging geophysics and nondestructive testing a nachman ann of math 2 143 1996 pp 71 96 proved global uniqueness for the two dimensional inverse conductivity problem using a constructive method of proof based on this proof siltanen mueller and isaacson inverse problems 16 2000 pp 681 699 presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration the method was verified with nonnoisy rotationally symmetric examples in this paper the method is extended by introducing a new regularization scheme which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise

constructing the physical parameters of a damped vibrating system from eigendata abstract in this paper we consider the inverse problem for a discrete damped mass spring system where the mass damping and stiffness matrices are all symmetric tridiagonal we first show that the model can be constructed from two real eigenvalues and three real eigenvectors or two complex conjugate eigenpairs and a real eigenvector then we study the general under determined and over determined problems in particular we provide the sufficient and necessary conditions on the given two real or complex conjugate eigenpairs so that the under determined problem has a physical solution however for large model order the construction from these data may be sensitive to perturbations to reduce the sensitivity we propose the minimum norm solution over the under determined noisy data and the least squares solution to the over determined measured data we also discuss the physical realizability of the required model by the positivity constrained regularization method for the ill posed under determined problem and the least squares optimization problems with positivity constraints for the ill posed over determined problem finally we give simple numerical examples to illustrate the effectiveness of our methods

global uniqueness for a two dimensional inverse boundary value problem we show that the coefficient y x of the elliptic equation vie qyvu 0 in a two dimensional domain is uniquely determined by the corresponding dirichlet to neumann map on the boundary and give a reconstruction pro

adjoint divisors on algebraic curves abstract let c be a numerically connected curve lying on a smooth algebraic surface we show that if a is an ample invertible sheaf satisfying some technical numerical hypotheses then c a is normally generated as a corollary we show that the sheaf c 2 on a numerically connected curve c of arithmetic genus pa 3 is normally generated if c is ample and does not exist a subcurve b c such that pa b 1 b c b

parameter choice strategies for multipenalty regularization the widespread applicability of the multipenalty regularization is limited by the fact that theoretically optimal rate of reconstruction for a given problem can be realized by a one parameter counterpart provided that relevant information on the problem is available and taken into account in the regularization in this paper we explore the situation where no such information is given but still accuracy of optimal order can be guaranteed by employing multipenalty regularization our focus is on the analysis and the justification of an a posteriori parameter choice rule for such a regularization scheme first we present a modified version of the discrepancy principle within the multipenalty regularization framework as a consequence we provide a theoretical justification to the multipenalty regularization scheme equipped with the a posteriori parameter choice rule we then establish a fast numerical realization of the proposed discrepancy principle based on a model function approximation finally we pro

stability of the inverse conductivity problem in the plane for less regular conductivities abstract the dirichlet to neumann map or voltage to current map takes dirichlet data u f to the conormal derivatives u at the boundary where u is a solution to the elliptic equation div u 0 on a plane lipchitz domain r2 we consider the inverse conductivity problem that consists of the recovery of in the interior of from the knowledge of we prove stability for this inverse map assuming an isotropic conductivity c1 e we reduce the equation div u 0 to an equivalent system and use the scattering transform this approach relaxes the number of derivatives of required in former works

a functional optimization formulation and implementation of an inverse natural convection problem abstract a formulation and a fem implementation for the solution of an ill posed inverse design natural convection problem is proposed in particular we consider an incompressible viscous liquid material occupying a given domain where convection is driven by buoyancy thermal boundary conditions are only prescribed in the part h 0 of the boundary in addition the temperature distribution is also prescribed in the part i of the boundary h 1 where the heat flux is known i e i is a boundary with over specified thermal boundary conditions the inverse convection problem is posed as an optimization problem in l 2 i 0 t max for the calculation of the boundary heat flux q o x t with x t h 0 0 t max the optimization scheme minimizes the discrepancy m x t x t q o l 2 1 0 t max between the temperature x t q o calculated from the solution of a direct problem for each flux q 0 and the measured or desired temperature m x t at the boundary i the gradient of the cost functional is obtained from the adjoint fields the adjoint problem is defined from the sensitivity operators obtained by linearization of the equations governing the direct problem the conjugate gradient method is employed to perform the optimization process the fem is used for the calculation of the direct adjoint and sensitivity thermal and fluid flow fields finally the method is demonstrated with the solution of some two dimensional inverse problems with known solutions the numerical results are discussed and future work with potential applications is addressed

singular solutions of elliptic equations and the determination of conductivity by boundary measurements abstract we improve some results on uniqueness and stability in the determination of the coefficient a in the equation i div a grad u 0 in when all possible pairs of dirichlet and neumann data on are known we also treat cases of anisotropic equations our method relies on the construction of solutions to i having an isolated singularity with prescribed asymptotic behaviour

nonlinear fourier analysis for discontinuous conductivities computational results abstract two reconstruction methods of electrical impedance tomography eit are numerically compared for nonsmooth conductivities in the plane based on the use of complex geometrical optics cgo solutions to d bar equations involving the global uniqueness proofs for calderon problem exposed in nachman 1996 43 and astala and paivarinta 2006 6 the astala paivarinta theory based low pass transport matrix method implemented in astala et al 2011 3 and the shortcut method which considers ingredients of both theories the latter method is formally similar to the nachman theory based regularized eit reconstruction algorithm studied in knudsen et al 2009 34 and several references from there new numerical results are presented using parallel computation with size parameters larger than ever leading mainly to two conclusions as follows first both methods can approximate piecewise constant conductivities better and better as the cutoff frequency increases and there seems to be a gibbs like phenomenon producing ringing artifacts second the transport matrix method loses accuracy away from a freely chosen pivot point located outside of the object to be studied whereas the shortcut method produces reconstructions with more uniform quality

parameterization of planar curves immersed in triangulations with application to finite elements we construct a method for the parameterization of a class of planar piecewise c2 curves over a collection of edges in an ambient triangulation the map from the collection of edges to the curve is the closest point projection a distinguishing feature of the method is that edges in the ambient triangulation need not interpolate the curve we formulate conditions on the ambient triangulations so that the resulting parameterization over its selected edges is i bijective ii maps simple connected collection of edges to simple connected components of the curve and iii is c1 within each edge of the collection these properties of the parameterization make it particularly useful in the construction of high order finite element approximation spaces on planar curves immersed in triangulations we discuss this application and illustrate it with numerical examples the parameterization method applies to a large class of planar curves including most ones of interest in engineering and computer graphics applications and to a large family of triangulations including acute angled triangulations copyright 2011 john wiley sons ltd

material and shape derivative method for quasi linear elliptic systems with applications in inverse electromagnetic interface problems we study a shape optimization problem for quasi linear elliptic systems the state equations describe an interface problem and the ultimate goal of our research is to determine the interface between two materials with different physical properties the interface is identified by the minimization of the shape or the cost functional representing the misfit between the data and the simulations for shape sensitivity of the shape functional we elaborate the material and the shape derivative method in this concept a vector field is introduced that deforms the unknown shape toward the optimum we characterize the elliptic interface problems whose solutions give the material and the shape derivatives in particular we show the existence of weak as well as strong material derivatives further we employ the adjoint variable method to obtain an explicit expression for the gradient of the shape functional this gradient is then used for the actual implementation of the minimization algorithm in simulations we

on the iteratively regularized gauss newton method for solving nonlinear ill posed problems the iteratively regularized gauss newton method is applied to compute the stable solutions to nonlinear ill posed problems f x y when the data y is given approximately by y with y y in this method the iterative sequence x k is defined successively by x k 1 x k k i f x k f x k 1 f x k f x k y k x k x 0 where x 0 x 0 is an initial guess of the exact solution x and k is a given decreasing sequence of positive numbers admitting suitable properties when x k is used to approximate x the stopping index should be designated properly in this paper an a posteriori stopping rule is suggested to choose the stopping index of iteration and with the integer k determined by this rule it is proved that x k x c inf x k x k k 0 1 with a constant c independent of where x k denotes the iterative solution corresponding to the noise free case as a consequence of this result the convergence of x k is obtained and moreover the rate of convergence is derived when x 0 x satisfies a suitable source wise representation the results of this paper suggest that the iteratively regularized gauss newton method combined with our stopping rule defines a regularization method of optimal order for each 0 v 1 numerical examples for parameter estimation of a differential equation are given to test the theoretical results