the faces of the unit balls of c norms and c spectral norms abstract we give a complete description of the faces of the closed unit ball of an arbitrary c norm on r n and we determine the barycenter the volume and the symmetries of each face as a consequence of a previous result we characterize the faces of the closed unit ball of the corresponding c spectral norm on the space of n m real or complex matrices

aerodynamic shape optimization for natural laminar flow using a discrete adjoint approach a framework for the design of natural laminar flow airfoils is developed based on multipoint aerodynamic shape optimization capable of efficiently incorporating and exploiting laminar turbulent transition a two dimensional reynolds averaged navier stokes flow solver making use of the spalart allmaras turbulence model is extended to incorporate an iterative laminar turbulent transition prediction methodology the natural transition locations due to tollmien schlichting instabilities are predicted using a simplified en method or the compressible form of the arnal habiballah delcourt criterion the reynolds averaged navier stokes solver is subsequently used in a gradient based sequential quadratic programming shape optimization framework the transition criteria are tightly coupled into the objective and gradient evaluations the gradients are obtained using an augmented discrete adjoint formulation for nonlocal transition criteria robust design over a range of cruise flight conditions is demonstrated thro

parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

isometries for the induced c norm on square matrices and some related results abstract let fn be the linear space of column vectors with n coordinates over f r or c denote by gp n the group of n n generalized permutation matrices i e matrices with exactly one nonzero entry with magnitude 1 in each row and column given a nonzero vector c c1 cn t fn define the c norm of x x1 xn t fn by x c max c t p x p gp n and the induced c norm of a fn n by a c max a y c y c 1 we characterize the isometries for induced c norms to achieve our goal we study the geometric properties of fn n and obtain some inequalities related to the induced c norm as by products these results are of independent interest

efficient numerical differentiation of implicitly defined curves for sparse systems a numerical technique is developed for the efficient numerical differentiation of regular implicitly defined curves existing in high dimensional real space such as those representing homotopies where the system of equations which defines the curve implicitly is assumed to be sparse the calculation is verified numerically through its application to the curve defined implicitly by a homotopy constructed based on a discretization of the equations governing compressible aerodynamic fluid flow consideration is given to computational cost data storage and accuracy this method is applicable to any implicitly defined curves or trajectories which can occur for example in dynamical systems analysis or control applications also exist in the area of homotopy continuation where implicitly defined curves are approximately traced numerically such applications include the analysis of curve traceability and the construction of higher order predictors the latter is investigated numerically and it is found that increasing the order of accuracy of the predictor can significantly improve the curve tracing accuracy within a limited radius

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

l subinvariant norms abstract let l be a compact semigroup of operators on f n where f r or f c a norm p on f n is said to be l subinvariant if p s x p x for all s l x f n if l is a closed subgroup of unitary operators on f n the concept of l subinvariant norm coincides with the concept of l invariant norm studied in c k li n k tsing linear algebra appl 150 1991 179 194 in this paper some basic properties of l subinvariant norms are studied and several results of li and tsing c k li n k tsing linear algebra appl 150 1991 179 194 on group invariant norms are extended to semigroup subinvariant norms as an application necessary and sufficient conditions on a pair of vectors x y f n are obtained such that p x p y holds for each weakly monotonic norm p

the isometries of certain maximum norms abstract let a 1 a m be nonsingular n n matrices over f r or c and let 1 p j for j 1 m we characterize the linear isometries for the norm n on f n defined by n x max l p j a j x 1 j m where l p y denotes the l p norm of y f n as an application we study the conditions on matrices a 1 a m for which n is absolute or a symmetric gauge function

pde constrained optimization with error estimation and control the paper describes an algorithm for pde constrained optimization that controls numerical errors using error estimates and grid adaptation during the optimization process a key aspect of the algorithm is the use of adjoint variables to estimate errors in the first order optimality conditions multilevel optimization is used to drive the optimality conditions and their estimated errors below a specified tolerance the error estimate requires two additional adjoint solutions but only at the beginning and end of each optimization cycle moreover the adjoint systems can be formed and solved with limited additional infrastructure beyond that found in typical pde constrained optimization algorithms the approach is general and can accommodate both reduced space and full space formulations of the optimization problem the algorithm is illustrated using the inverse design of a nozzle constrained by the quasi one dimensional euler equations

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

review and unification of methods for computing derivatives of multidisciplinary computational models this paper presents a review of all existing discrete methods for computing the derivatives of computational models within a unified mathematical framework this framework hinges on a new equation the unifying chain rule from which all the methods can be derived the computation of derivatives is described as a two step process the evaluation of the partial derivatives and the computation of the total derivatives which are dependent on the partial derivatives finite differences the complex step method and symbolic differentiation are discussed as options for computing the partial derivatives it is shown that these are building blocks with which the total derivatives can be assembled using algorithmic differentiation the direct and adjoint methods and coupled analytic methods for multidisciplinary systems each of these methods is presented and applied to a common numerical example to demonstrate and compare the various approaches the paper ends with a discussion of current challenges and possib

certain isometries on rn abstract let r n be the linear space of all real column vectors with n coordinates given x x 1 x n t r n its p k norm is defined by x p k max x i 1 p x i k p 1 p 1 i 1 k n where k is a positive integer satisfying 1 k n and p satisfies 1 p whereas its c norm is defined by x c max c t px p is a generalized permutation matrix for any nonzero c r n this paper characterizes the isometries for c norms and p k norms some related results are also considered

lieb functions and m positivity of norms abstract some properties of 2 or 3 positive maps on matrix spaces are proved where special attention is given to norms it is shown that for a locally bounded function the definition of a lieb function can be simplified somewhat unitarily invariant norms are characterized by a certain monotonicity property the set of 2 positive numerical radii is described and an example of a 3 positive norm is obtained

extensions of the first and second complex step derivative approximations a general framework for the first and second complex step derivative approximation to compute numerical derivatives is presented for first derivatives the complex step approach does not suffer roundoff errors as in standard numerical finite difference approaches therefore since an arbitrarily small step size can be chosen the complex step approach can achieve near analytical accuracy however for second derivatives straight implementation of the complex step approach does suffer from roundoff errors therefore an arbitrarily small step size cannot be chosen in this paper the standard complex step approach is expanded by using general complex step sizes to provide a wider range of accuracy for both the first and second derivative approximations even higher accuracy formulations are obtained by repetitively applying richardson extrapolations the new extensions can allow the use of one step size to provide optimal accuracy for both derivative approximations

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

operator norms multiplicativity factors and c numerical radii abstract let v be a normed vector space over c let b v denote the algebra of linear bounded operators on v and let n be an arbitrary seminorm or norm on b v in this paper we discuss multiplicativity factors for n i e constants 0 for which n n is submultiplicative we find that while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors in the infinite dimensional case n may or may not have such factors our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius called c numerical radii this is done with the help of a combinatorial inequality which seems to be of independent interest

duality between some linear preserver problems ii isometries with respect to c special norms and matrices with fixed singular values let fm n m n denote the linear space of all m n complex or real matrices according as f c or r let c c1 cm 0 be such that c1 cm 0 the c spectral norm of a matrix a fm n is the quantity a c i imci i a where 1 a m a are the singular values of a let d d1 dm 0 where d1 dm 0 we consider the linear isometries between the normed spaces fm n c and fm n d and prove that they are dual transformations of the linear operators which map l d onto l c where l c x fm n x has singular values c1 cm

minimal curvature trajectories riemannian geometry concepts for slow manifold computation in chemical kinetics in dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories model reduction methods exploit this for simplifying chemical kinetics via a time scale separation into fast and slow modes the aim is to approximate the system dynamics with a dimension reduced model after eliminating the fast modes by enslaving them to the slow ones via computation of a slow attracting manifold we present a novel method for computing approximations of such manifolds using trajectory based optimization we discuss riemannian geometry concepts as a basis for suitable optimization criteria characterizing trajectories near slow attracting manifolds and thus provide insight into fundamental geometric properties of multiple time scale chemical kinetics the optimization criteria correspond to a suitable mathematical formulation of minimal relaxation of chemical forces along reaction trajectories under given constraints we present various geometrically motivated criteria and the results of their application to four test case reaction mechanisms serving as examples we demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained

aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement an efficient gradient based algorithm for aerodynamic shape optimization is presented the algorithm consists of several components including a novel integrated geometry parameterization and mesh movement a parallel newton krylov flow solver and an adjoint based gradient evaluation to integrate geometry parameterization and mesh movement generalized b spline volumes are used to parameterize both the surface and volume mesh the volume mesh of b spline control points mimics a coarse mesh a linear elasticity mesh movement algorithm is applied directly to this coarse mesh and the fine mesh is regenerated algebraically using this approach mesh movement time is reduced by two to three orders of magnitude relative to a node based movement the mesh adjoint system also becomes smaller and is thus amenable to complex step derivative approximations when solving the flow adjoint equations using restarted krylov subspace methods a nested subspace strategy is shown to be more robust than truncating the entire subspace optimization is accomplished using a sequential quadratic programming algorithm the effectiveness of the complete algorithm is demonstrated using a lift constrained induced drag minimization that involves large changes in geometry

on the characterization of the extremal points of the unit sphere of matrices abstract we give a characterization of the extremal points of the unit sphere of matrices for the unitarily invariant norms we also investigate the properties of the dual matrices moreover we show that a result given by lau and riha is not correct

g invariant norms and g c radii abstract let v be a finite dimensional inner product space over f r or c and let g be a closed subgroup of the group of unitary operators on v a norm or a seminorm on v is said to be g invariant if g x x for all g g x v the concept of g invariant norm specializes to many interesting particular cases such as the absolute norms on f n symmetric gauge functions on r n unitarily invariant norms on f m n etc which are of wide research interest in this paper we study the general properties of g invariant norms our main strategy is to study g invariant norms via the g c radius rg c on v defined by r g c x max x g c g g where c v it is shown that the g c radii are very important g invariant seminorms because every g invariant norm or seminorm admits a representation in terms of them as a result one may focus attention on g c radii in order to get results on g invariant norms we study the norm properties of g c radii and obtain various results relating g invariant norms and g c radii the linear operators on v that preserve g invariant norms g invariant seminorms or g c radii are also investigated several open questions are mentioned

faces of the unit ball of a unitarily invariant norm abstract let be a unitarily invariant norm on the space of all n m complex matrices and let g r n r be the symmetric gauge function associated with that is to say we have a g s a for any a where s a is the nonincreasing sequence of singular values of a in this article we consider the relationship between the facial structures of the closed unit balls of g and which we denote by bg and b our main result gives a complete characterization of the faces of b in terms of the faces of bg for with each face e of b we associate a standard face f e of bg i e f e is a face of bg whose barycenter is a nonnegative nonincreasing n vector conversely each standard face f of bg originates a set of n m matrices say the set m f which is proven to be a face of b our main result also asserts that any face e of b is of the form q m fr where q and r are unitary matrices of appropriate orders and f f e we then explore the main result and draw some of its consequences for example we show that the singular values of the barycenter of a face e of b have the same multiplicities as the barycenter of f e formulas are given which relate the dimensions of e and f e the relative interior and the relative boundary of e are characterized in terms of the relative interior and the relative boundary of f e we completely describe the face of b generated by a matrix a our results also cover the case of real matrices

multiplicativity factors for c numerical radii abstract for matrices a c c n x n the c numerical radius of a is the nonnegative quantity r c a max tr cu au uunitary this generalizes the classical numerical radius r a it is known that r c constitutes a norm on c n x n if and only if c is nonscalar and tr c 0 for all such c we obtain multiplicativity factors for r c i e constant 0 for which r c is submultiplicative on c n x n

an adaptive discontinuous galerkin method for very stiff systems of ordinary differential equations abstract we present a discontinuous galerkin dg method for the numerical solution of stiff systems of ordinary differential equations odes we use a standard dg variational formulation with polynomials of degree k in each time interval we show that the method is a stable for every k we then introduce a hierarchical legendre finite element basis and we show that a whole family of approximations can be obtained simply by truncating the last p degrees of freedom from the computed solution we show that these approximations converge to order k 1 p in l2 norm and to order k 1 2 p in supremum norm we then show how this can be used to control the error and the time step length we present numerical examples of solutions on very stiff problems and on stiff problems with very long time integration where the time step length can vary on many orders of magnitude

linear maps relating different unitary similarity orbits or different generalized numerical ranges abstract let m be the complex linear space m n of n n complex matrices or the real linear space h n of n n hermitian matrices for c m its unitary similarity orbit is the set u c ucu u unitary and its circular unitary similarity orbit is the set v c x f 1 x u c where f is the scalar field c or r according as m m n or m h n related to u c and v c are the c numerical range and the c numerical radius of a m defined by w c a tr ax x u c and r c a max z z w c a respectively let c d h n we study the linear operators t on m satisfying one of the following properties i w d t a w c a for all a m ii r d t a r c a for all a m iii t u d u c iv t v d v c in particular we determine the conditions on c and d for the existence of a linear operator t on m satisfying any one of the conditions i iv and characterize such an operator if it exists

the singular value expansion of the volterra integral equation associated to a numerical differentiation problem abstract we consider the volterra integral equation of the first kind for the derivative of a given function with one side boundary conditions we give a method to obtain the singular value expansion for the corresponding integral kernel this singular value expansion can be used to give algorithms for the solution of the numerical differentiation problem a numerical experiment shows the results obtained by a simple version of such algorithms

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

permutation invariant norms abstract a norm or seminorm on on r n is permutation invariant if px x for all n n permutation matrices p and for all x r n we present a systematic study of permutation invariant seminorms their relations with other types of norms on matrices are discussed in addition we consider a special class of permutation invariant seminorms the c radii defined and denoted by r c x max c t px p a permutation for any given c r n it is shown that r c are the building blocks of all permutation invariant seminorms if the entries of c are not all equal and their sum is not 0 then r c is a norm on r n for such r c we study their norm properties and characterize their isometry groups

criteria for biholomorphic convex mappings on the unit ball in hilbert spaces in this paper we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball b in a complex hilbert space x is a biholomorphic convex mapping which improves some results of hamada and kohr and solves the problem which is posed by graham and kohr from this we derive some sufficient conditions for biholomorphic convex mapping we also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on b in hilbert spaces moreover we give some examples of biholomorphic convex mappings on b in hilbert spaces

unitary invariance and spectral variation abstract we call a norm on operators or matrices weakly unitarily invariant if its value at operator a is not changed by replacing a by u au provided only that u is unitary this class includes such norms as the numerical radius we extend to all such norms an inequality that bounds the spectral variation when a normal operator a is replaced by another normal b in terms of the arclength of any normal path from a to b computed using the norm in question related results treat the local metric geometry of the manifold of normal operators we introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball and identify this correspondence explicitly in certain cases

optimization of high order diagonally implicit runge kutta methods abstract this article presents constrained numerical optimization of high order linearly and algebraically stable diagonally implicit runge kutta methods after satisfying the desired order conditions undetermined coefficients are optimized with respect to objective functions which consider accuracy stability and computational cost constraints are applied during the optimization to enforce stability properties to ensure a well conditioned method and to limit the domain of the abscissa two promising third order methods are derived using this approach labelled sdirk 3 1 2 2 3 l 14 and sdirk 3 1 4 l sa 5 both optimized schemes have a good balance of properties the relative error norm of the latter the l 2 norm scaled by a function of the number of implicit stages is a factor of two smaller than comparable methods found in the literature variations on these methods are discussed relative to trade offs in their accuracy and stability properties a novel fifth order scheme sdirk 5 1 5 l 02 is derived with a significantly lower relative error norm than the comparable fifth order a stable reference method in addition the optimized scheme is l stable the accuracy and relative efficiency of the runge kutta methods are verified through numerical simulation of van der pol s equation as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder and in the turbulent wake of a naca 0012 airfoil these results demonstrate the value of numerical optimization for selecting undetermined coefficients in the construction of high order runge kutta methods with a balance between competing objectives

dual consistency and functional accuracy consider the discretization of a partial differential equation pde and an integral functional that depends on the pde solution the discretization is dual consistent if it leads to a discrete dual problem that is a consistent approximation of the corresponding continuous dual problem consequently a dual consistent discretization is a synthesis of the so called discrete adjoint and continuous adjoint approaches we highlight the impact of dual consistency on summation by parts sbp finite difference discretizations of steady state pdes specifically superconvergent functionals and accurate functional error estimates in the case of functional superconvergence the discrete adjoint variables do not need to be computed since dual consistency on its own is sufficient numerical examples demonstrate that dual consistent schemes significantly outperform dual inconsistent schemes in terms of functional accuracy and error estimate effectiveness the dual consistent and dual inconsistent discretizations have similar computational costs so dual consistency leads to improved efficiency to illustrate the dual consistency analysis of sbp schemes we thoroughly examine a discretization of the euler equations of gas dynamics including the treatment of the boundary conditions numerical dissipation interface penalties and quadrature by sbp norms

on the higher numerical radius and spectral norm abstract let c n n be the vector space of n n complex matrices and denote by u n the group of n n unitary matrices in c n n for 1 m n set jm im 0m n whereim is the m m identity matrix and 0k denotes a k k zero matrix with this for ain c n n and 1 m n we define r m a max tr j m uau u u n and a m max tr j m uav u v u n as the m numerical radius and m spectral norm of a respectively when m 1 these quantities are reduced to the classical numerical radius and spectral norm in this note we obtain inequalities concerning m and rm for 1 m n examples are given to show that the inequalities obtained cannot be improved we also study multiplicativity factors for rm 1 m n

an improved adaptive constraint aggregation for integrated layout and topology optimization abstract the purpose of this paper is to present a kreisselmeier steinhauser ks function based adaptive constraint aggregation approach it is implemented within the integrated layout and topology optimization of multi component structure systems to avoid using large numbers of non overlapping constraints defined by means of the previously proposed finite circle method fcm an improved adaptive approach is then put forward to obtain proper aggregation parameters for the ks function based constraint aggregation contributing to less numerical difficulties while meeting the same aggregation precision compared with the existing adaptive approach furthermore the complex step derivative approximation is utilized to yield better sensitivities for the aggregated constraint functions with high nonlinearity moreover during the integrated layout and topology optimization multi point constraints mpc are applied to establish the interconnections between movable components and supporting structures which can use fixed finite element meshes and analytical sensitivities finally some numerical examples are tested to demonstrate the validity and effectiveness of the proposed formulation

functional models in linear algebra abstract this paper examines the role of functional models in linear algebra and some related areas of application notably linear systems theory this approach adopts an intermidiate level of abstraction somewhere between abstract module theory on the one hand and matrix theory on the other it is a powerful method which simplifies many proofs provides insights and illuminates the less obvious connections between linear algebra and operator theory approximation theory moment problems interpolation and control theory

isometries for unitarily invariant norms a survey of linear isometries for unitarily invariant norms on real or complex rectangular matrices is given which includes some latest development on the topic a result on isometries for unitarily invariant norms without the linearity assumption is presented related results and problems are discussed

quaternion and octonion based finite element analysis methods for computing multiple first order derivatives abstract the complex taylor series expansion method for computing accurate first order derivatives is extended in this work to quaternion octonion and any order cayley dickson algebra the advantage of this new approach is that highly accurate multiple first order derivatives can be obtained in a single analysis quaternion and octonion based finite element analysis methods were developed in order to compute up to three quaternion and up to seven octonion first order derivatives of shape material properties and or loading conditions in a single analysis the traditional finite element formulation was modified such that each degree of freedom was augmented with three or seven additional imaginary nodes the quaternion and octonion based methods were integrated within the abaqus commercial finite element code through a user element subroutine numerical examples are presented for thermal conductivity and linear elasticity however the methodology is general the results indicate that the quaternion and octonion based methods provide derivatives of the same high accuracy as the complex finite element method but are significantly more efficient a fortran code to solve a simple seven variable quaternion example is given in the appendix

complex step derivative approximation in noisy environment abstract the complex step derivative approximation is a powerful method for derivative approximations which has been successfully implemented in deterministic numerical algorithms we explore and analyze its implementation in noisy environment through examples error analysis and application to optimization methods numerical results show a promising performance of the complex step gradient approximation in noisy environment

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

a new non intrusive polynomial chaos using higher order sensitivities abstract this paper proposes a new non intrusive method for uncertainty quantification called polynomial chaos decomposition with differentiation pcdd that uses higher order sensitivities of the response in pcdd the polynomial chaos expansion pce of the response is differentiated with respect to the basis random variables using multi indices this differentiation results in a system of linear equations which can then be solved to determine the expansion coefficients here the higher accuracy modified forward finite difference modffd that involves representation of the response using taylor expansion of order equal to the chaos order is used in combination with pce therefore the total number of samples required with this method is equal to the number of terms in the pce to verify the validity of this new technique two analytical problems and two stochastic composite laminate problems were studied the results of the analytical problems showed that the accuracy of pcdd using modffd is similar to that of pcdd using analytical sensitivities which in addition is comparable to the exact results for composite laminate problems the pcdd displayed very high accuracy comparable to 50 000 latin hypercube samples which underlines the computational efficiency of this proposed method

duality between some linear preserver problems iii c spectral norms and skew symmetric matrices with fixed singular values abstract let f denote either the complex field c or the real field r let v be sn f or kn f the vector spaces of all n n symmetric and skew symmetric matrices respectively over f for c c1 cn 0 with c1 cn 0 the c spectral norm of a matrix a v is the quantity a c i l n c i i a where 1 a n a are the singular values of a let d d1 dn 0 with d1 dn 0 we study the linear isometries between the normed spaces v c and v d by using the fact that they are dual transformations of the linear operators which map d onto c where c x v x has singular values c 1 c n it is shown that such isometries and hence their dual transformations exist if and only if c and d are scalar multiples of each other in such case we completely determine the structure of such isometries and prove that they and their dual transformations belong to a same class of operators in the proof we obtain characterizations of the extreme points of the unit ball in v for different cases with respect to c which is of independent interest

the isometries and the g invariance of certain seminorms abstract let f be the field of real or complex numbers and let g be a subgroup of the general linear group gl n f if a is an m n matrix over f let a be the seminorm on f n defined by x a ax for all x f n in this paper we characterize the linear isometries for the seminorm a and study the conditions on a for which a is g invariant that is sx a x a for all x f n and all s g as a special case we describe all matrices a for which a is absolute or a symmetric gauge function

isometries of norm ideals of compact operators abstract let j be a symmetric norm ideal of compact operators on hilbert space h and assume that the finite rank operators are dense in j and that j is not the ideal of hilbert schmidt operators a linear transformation on j is an isometry of j onto itself if and only if there are unitary operators u and v on h such that either x uxv or x uxtv where xt denotes the transpose of x with respect to a fixed orthonormal basis of h

mesh movement for a discrete adjoint newton krylov algorithm for aerodynamic optimization a grid movement algorithm based on the linear elasticity method with multiple increments is presented the methodisrelativelycomputationallyexpensivebutisexceptionallyrobust producinghigh qualityelementsevenfor large shape changes it is integrated with an aerodynamic shape optimization algorithm that uses an augmented adjoint approach for gradient calculation the discrete adjoint equations are augmented to explicitly include the sensitivities of the mesh movement resulting in an increase in efficiency and numerical accuracy this gradient computation method requires less computational time than a function evaluation and leads to significant computational savings as dimensionality is increased the results of the application of these techniques to several large deformation and optimization cases are presented

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

convergence error and higher order sensitivity estimations differential equation with respect to the design variables the accuracies of the convergence error and higher order sensitivity estimation methods are verified using laplace euler and navier stokes equations the developed methods are used to improve the accuracy of the finite difference sensitivity calculations in iteratively solved problems aboundonthenormvalueofthe finite differencesensitivityerrorinthestatevariablesisminimizedwith respect to the finite difference step size the optimum finite difference step size is formulated as a function of the norm values of both convergenceerror andhigher order sensitivities thesensitivities calculated with the analytical and the finite difference methods are compared the performance of the proposed methods on the convergence of inverse design optimization is evaluated

monotonicity properties of certain classes of norms let p be a norm on kn where k r or k c if s kn n is a nonsingular matrix let ps be the norm on kn defined by ps x p sx for all x kn this note gives some conditions on s for which ps has a certain monotonicity property and characterizes p by monotonicity properties of all norms ps with s in a given group of unitary orthogonal matrices in particular we obtain the following characterizations 1 if p is an lq norm 1 q all matrices s are described for which ps is weakly monotonic 2 if for each unitary orthogonal matrix s kn n the norm ps is quasimonotonic then p is a positive multiple of the l2 norm

convex subordination chains and injective mappings in cn abstract in this paper we continue the work related to convex subordination chains in c and c n and prove that if f z z k 2 a k z k is a holomorphic mapping on the euclidean unit ball b n in c n such that k 2 k 2 a k 1 a 0 1 0 is a function of class c 2 on 0 1 and continuous on 0 1 such that a 1 0 a t 0 t a t 1 2 for t 0 1 and if a satisfies a differential equation on 0 1 then f z t a t 2 d f t z t z f t z is a convex subordination chain over 0 1 and the mapping f z a z 2 d f z z f z is injective on b n we also present certain coefficient bounds which provide sufficient conditions for univalence quasiregularity and starlikeness for the chain f z t finally we give some examples of convex subordination chains over 0 1

dynamic sensitivity analysis based on sherman morrison woodbury formula the sensitivity analysis plays a significant role in structural optimization and repetitive calculations are necessary for the analysis procedure a reanalysis approach of frequency response funct

on the generalization of the complex step method we generalize the well known complex step method for computing derivatives by introducing a complex step in a strict sense exploring different combinations of terms we derive 52 approximations for computing the first order derivatives and 43 for the second order derivatives for an appropriate combination of terms and appropriate choice of the step size in the real and imaginary directions fourth order accuracy can be achieved in a very simple and efficient scheme on a compact stencil new different ways of computing second order derivatives in one single step are shown many of the first order derivative approximations avoid the problem of subtractive cancellation inherent to the classic finite difference approximations for real valued steps and the superior accuracy and stability of the generalized complex step approximations are demonstrated for an analytic test function

faces and traces of the unit ball of a symmetric gauge function abstract we investigate the faces of the closed unit ball bg x g x 1 of a symmetric gauge function g r n r firstly we describe the group of symmetries of a face f of bg in terms of its barycenter we introduce the concept of standard face of bg i e a face whose relative interior meets the set d x r n x 1 x n 0 called here the standard cone of r n and the concept of trace of a face f which is the intersection of f with d characterizations of standard faces are given and we show that the trace of a face if nonempty is the trace of a standard face it is proved that a face f and its trace have the same dimension iff f is a standard face a relation is found between the dimension of f and the dimension of its symmetry core i e the set of points of f that are fixed by all symmetries of f other results of this type are given the main result has a long and delicate proof it completely characterizes in several different ways the subsets of bg that are traces of faces of bg

translation invariant maps and applications in this paper we study translation invariant maps on a linear space a method for estimating quantities induced by such maps is presented applications for differences of operators commutators g contractive maps and gruss type inequalities are also given

two level free form and axial deformation for exploratory aerodynamic shape optimization an intuitive shape parameterization and control technique suitable for high fidelity aerodynamic shape optimization is presented it relies on the principles of free form and axial deformation enabling thorough exploration of the design space while keeping the number of design variables manageable surface sensitivities to the design variables are readily available their inclusion in a highly efficient and robust adjoint based optimization methodology involving linearly elastic volume mesh deformation and a newton krylov solver for the euler equations is described the flexibility of the proposed approach is demonstrated through the exploratory shape optimization of a three pronged feathered winglet leading to a span efficiency of 1 19 under a height to span ratio constraint of 0 1 and an optimization of a regional jet wing at transonic speed where a winglet is allowed to develop starting from a planar wingtip extension leading to an 18 8 reduction in drag

using complex variables to estimate derivatives of real functions a method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described numerical examples illustrating the power of the approximation are presented

on g invariant norms abstract a result of r mathias and horn cf linear algebra appl 142 1990 63 on the representation of the unitarily invariant norm is extended in the context of eaton triples and of real semisimple lie algebras the representation is related to a function criteria for being a norm is given in terms of and the longest element of the underlying finite reflection group in particular for the real simple lie algebras is a norm on its cartan subspace for a given nonzero if and only if 0 where 0 is the longest element of the weyl group this is the case if the weyl group contains id some related results are obtained

adjoint an approach for the rapid development of discrete adjoint solvers an automatic differentiation tool is used to develop the adjoint code for a three dimensional computational fluid dynamics solver rather than using automatic differentiation to differentiate the entire source code of the computational fluid dynamics solver we have applied it selectively to produce code that computes the flux jacobian matrix and the other partial derivatives that are necessary to compute total derivatives using an adjoint method the resulting linear discrete adjoint system is then solved using the portable extensible toolkit for scientific computation this selective application of automatic differentiation is the central idea behind the automatic differentiation adjoint adjoint approach this approach has the advantage that it is applicable to arbitrary sets of governing equations and cost functions and that it is exactly consistent with the gradients that would be computed by exact numerical differentiation of the original solver furthermore the approach is largely automatic thus avoiding the lengthy development times usually required to develop adjoint solvers for partial differential equations these significant advantages come at the cost of increased memory requirements for the adjoint solver derivatives of drag and lift coefficients are validated and the low computational cost and ease of implementation of the method are shown

properties of linear approximations of matrices in the spectral norm abstract the properties of linear approximations of a matrix are presented with respect to the spectrum norm we describe the connection between all spectral approximations of a given matrix and we give a sufficient condition under which the spectral approximation is unique

overgroups of some classical linear groups with applications to linear preserver problems abstract the study of linear operators on a matrix space that leave invariant certain functions subsets or relations is commonly referred to as the study of linear preserver problems and has attracted the attention of many mathematicians in the last few decades dynkin in his classic paper studied maximal subgroups of the classical groups and showed how his results may be used to study preserver problems the purpose of this paper is to further exploit this very powerful approach the subgroups g of the general linear group that we deal with are not necessarily maximal for the applications that we have in mind we obtained a description of all possible overgroups of g the results are then applied to various linear preserver problems shorter alternative proofs for various existing results are given and some open questions are answered

on the accuracy of the complex step finite difference method abstract the complex step finite difference method csfdm is a very simple methodology that can be implemented in well known numerical techniques helping to improve for instance in the wave propagation problem time and or space derivative based wavefields we clarify differences between the csfdm and previous implementations of the complex step cs derivative approximation in well known numerical techniques we study dispersion properties for the one way and two way wave equations using the finite difference method fdm the pseudospectral method psm the finite element method fem and the csfdm under the influence of a plane wave and ricker source time functions we show the gain in numerical accuracy offered by the methodology of the csfdm over the fdm psm and fem we finally discuss consequences of the csfdm in future scenarios and propose directions of study in this area

a generalization of spectral radius numerical radius and spectral norm abstract let c n be the linear space of complex column vectors with n coordinates associated with the usual inner product denote by k the set of all k tuples of orthonormal vectors in c n given a real number p 1and an n n complex matrix a with eigenvalues 1 n we define p k max j 1 k i j p 1 p 1 i 1 k n r p k max i 1 k x i ax i p 1 p x 1 x k k and a p k max i 1 k x i ay i p 1 p x 1 x k y 1 y k k as the p k spectral radius p k numerical radius and p k spectral norm of a respectively for k 1 they are the classical spectral radius numerical radius and spectral norm of a in this note we study the matrices for which p k a rp k a p k a a p k or rp k a a p k moreover the norm properties of rp k and p k are investigated

norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils abstract we start by proving a lower bound for the lp operator norm of a submatrix with sufficiently large dimensions and some relations between the lp norm and the lp operator norm we then study the lp norm and the lp operator norm of certain matrices arising in estimating the eigenvalue generalized eigenvalue singular value and generalized singular value variations of matrices matrix pencils and matrix pairs finally as applications of our bounds upon the lp norm and lp operator norm of certain matrices we give several new bounds for the variations of the spectra of matrices and matrix pencils these new bounds are generalizations of the weyl lidskii theorem for hermitian matrices and the bhatia davis theorem for unitary matrices and of some results obtained by the author recently for diagonalizable pencils with real spectra

norm hull of vectors and matrices let v be a real or complex finite dimensional vector space and let n be a set of norms on v the norm hull of a vector x v with respect to n is the set of vectors y v that satisfy y x for all n we study and give characterization of the norm hull for some sets of well known norms on general vector spaces and for the set of algebra norms and the set of induced norms on the algebra of n n real or complex matrices

complex variable method for eigensolution sensitivity analysis the application of complex variable method cvm for eigenvalue and eigenvector sensitivity analysis is presented gradi ent based methods usually used in structural optimization require accurate sensitivity information cvm is accurate robust and easy to implement as compared to other approximate sensitivity analysis methods by cvm the first order modal sensitivity can b e calculated using an existing structural analysis package this is achieved by making a small complex perturbation to a design variable the resulting eigensolutions would be complex the sensitivity of eigensolutions with respect to that design variable can then be obtained by taking the complex part of the eigensolutions divided by the perturbation since there is no subtraction involved in the sensitivity calculation the approximation is not sensitive to step size thus unlike finite difference method s cvm does not suffer the effect of subtraction cancellation error when a very small step size is used the computation of 1 st order modal sensitivity is not hampered by repeated eigenvalue problem as is the case with other sensitivity analysis methods for repeated eigenvalues cvm automatically provides correct eigenvalue sensitivity and the associated eigenvector sensitivity for a differentiable mode a 5 dof system with an eigenvalue of multiplicity 3 is used to illustrate the salient features of cvm in eigen solution sensitivity analysis nomenclature k stiffness matrix for spring mass system m mass matrix for spring mass system eigenvalue for spring mass system eigenvector for spring mass system derivative of w r t design variable

on the extension of the complex step derivative technique to pseudospectral algorithms the complex step derivative csd technique is a convenient and highly accurate strategy to perform a linearized perturbation analysis to determine a directional derivative via a minor modification of an existing nonlinear simulation code the technique has previously been applied to nonlinear simulation codes such as finite element codes which employ real arithmetic only the present note examines the suitability of this technique for extension to efficient pseudospectral simulation codes which nominally use the fast fourier transform fft to convert back and forth between the physical and transformed representations of the system it is found that if used carefully this extension retains the remarkable accuracy of the csd approach however to perform this extension without sacrificing this accuracy particular care must be exercised specifically the state real and perturbation imaginary components of the complexified system must be transformed separately and arranged in such a manner that they are kept distinct during the process of differentiation in the transformed space in order to avoid the linear combination of the large and small quantities in the analysis it is shown that this is relatively straightforward to implement even in complicated nonlinear simulation codes thus positioning the csd approach as an attractive and relatively simple alternative to hand coding a perturbation a k a tangent linear code for determining the directional derivative even when pseudospectral algorithms are employed

g invariant norms and bicircular projections abstract it is shown that for many finite dimensional normed vector spaces v over c a linear projection p v v will have nice structure if p i p is an isometry for some complex unit not equal to one from these results one can readily determine the structure of bicircular projections i e those linear projections p such that p i p is an isometry for every complex unit the key ingredient in the proofs is the knowledge of the isometry group of the given norm the proof techniques also apply to real vector spaces in such cases characterizations are given to linear projections p such that p i p 2p i is an isometry

polynomially bounded solutions to the loewner differential equation in several complex variables abstract we determine the form of polynomially bounded solutions to the loewner differential equation that is satisfied by univalent subordination chains of the form f z t e t a z where a l c n c n has the property m a 0 here m a min r a z z z 1 we also give sufficient conditions for g z t l f z t to be polynomially bounded where f z t is an a normalized polynomially bounded loewner chain solution to the loewner differential equation

on the norm property of g c radii and eaton triples abstract let v be a euclidean space and o v be the orthogonal group the norm criterion of the g c radius is studied in the framework of an eaton triple v g d

nonlinear robust performance analysis using complex step gradient approximation in this paper the complex step method is applied in the setting of numerical optimisation problems involving dynamical systems modelled as nonlinear differential equations the main advantage of the complex step method for gradient approximation is that it entails no subtractive cancellation error and therefore the truncation error can be made arbitrarily to machine precision small the method is applied to two robust performance analysis problems the accuracy and convergence rate of the solutions computed using the proposed approach are seen to be significantly better than those achieved using standard gradient approximation methods

scalable parallel approach for high fidelity steady state aeroelastic analysis and adjoint derivative computations aeroelastic systems achieve the best performance when the aerodynamic shape and structural sizing are optimized concurrently but such an optimization is challenging when high fidelity aerodynamic and structural models are required this paper addresses this challenge through several significant improvements fully coupled newton krylov methods are presented for the solution of aerostructural systems and for the corresponding adjoint systems the coupled adjoint method presented can compute gradients with respect to thousands of multidisciplinary design variables accurately and efficiently this is enabled by several improvements in the computation of the multidisciplinary terms in the coupled adjoint the parallel scalability of the methods is demonstrated for a full aircraft configuration using an euler computational fluid dynamics model with more than 8 106 state variables and a detailed structural finite element model of the wing with more than 1 106 degrees of freedom the coupled newton krylov metho

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

numerical implementation of complex orthogonalization parallel transport on stiefel bundles and analyticity abstract numerical integration of complex linear systems of odes depending analytically on an eigenvalue parameter are considered complex orthogonalization which is required to stabilize the numerical integration results in non analytic systems it is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non analytic characteristic function the orthonormal systems are constructed using the geometry of stiefel bundles different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one form on the stiefel bundle for the numerical integration gauss legendre runge kutta algorithms are the principal choice for preserving orthogonality and performance results are shown for a range of glrk methods the theory and methods are tested by application to example boundary value problems including the orr sommerfeld equation in hydrodynamic stability

numerical differentiation for local and global tangent operators in computational plasticity in this paper numerical differentiation is applied to integrate plastic constitutive laws and to compute the corresponding consistent tangent operators the derivatives of the constitutive equations are approximated by means of difference schemes these derivatives are needed to achieve quadratic convergence in the integration at gauss point level and in the solution of the boundary value problem numerical differentiation is shown to be a simple robust and competitive alternative to analytical derivatives quadratic convergence is maintained provided that adequate schemes and stepsizes are chosen this point is illustrated by means of some numerical examples