the optimal multilevel monte carlo approximation of the stochastic drift diffusion poisson system abstract existence and local uniqueness theorems for weak solutions of a system consisting of the drift diffusion poisson equations and the poisson boltzmann equation all with stochastic coefficients are presented for the numerical approximation of the expected value of the solution of the system we develop a multi level monte carlo mlmc finite element method fem and we analyze its rate of convergence and its computational complexity this allows to find the optimal choice of discretization parameters finally numerical results show the efficiency of the method applications are among others noise and fluctuations in nanoscale transistors in field effect bio and gas sensors and in nanopores

the higson roe exact sequence and l2 eta invariants abstract the goal of this paper is to solve the problem of existence of an l 2 relative eta morphism on the higson roe structure group using the cheeger gromov l 2 eta invariant we construct a group morphism from the higson roe maximal structure group constructed in 35 to the reals when we apply this morphism to the structure class associated with the spin dirac operator for a metric of positive scalar curvature we get the spin l 2 rho invariant when we apply this morphism to the structure class associated with an oriented homotopy equivalence we get the difference of the l 2 rho invariants of the corresponding signature operators we thus get new proofs for the classical l 2 rigidity theorems of keswani obtained in 41

realizing the analytic surgery group of higson and roe geometrically part iii higher invariants we construct an isomorphism between the geometric model and higson roe s analytic surgery group reconciling the constructions in the previous papers in the series on realizing the analytic surgery group of higson and roe geometrically with their analytic counterparts following work of lott and wahl we construct a chern character on the geometric model for the surgery group it is a delocalized chern character from which lott s higher delocalized rho invariants can be retrieved following work of piazza and schick we construct a geometric map from stolz positive scalar curvature sequence to the geometric model of higson roe s analytic surgery exact sequence

spectral flow eta invariants and von neumann algebras abstract in this paper we prove that the spectral flow of a path of tangential signature operators on a closed odd dimensional manifold parametrized by unitary representations of the fundamental group is a homotopy invariant of the manifold and the path in the space of unitary representations for a large class of fundamental groups with the help of results due to weinberger proc nat acad sci u s a 85 1988 5362 5363 we also prove that an eta invariant for normal covering spaces introduced by cheeger and gromov j differential geom 21 1985 1 34 and defined using the von neumann trace is a homotopy invariant for bieberbach groups finally we compute this invariant for closed locally symmetric spaces and obtain the surprising result that the eta invariant for such spaces is a differential invariant

group majorization the convex hulls of sets of matrices and the diagonal element singular value inequalities abstract a group induced preorder on matrix space permits the description of the convex hulls of sets of proper and improper real matrices with given singular values and also permits a new proof of the necessity of the known relations between the diagonal elements and singular values of a real matrix

on the completely positive and positive semidefinite preserving cones abstract the cone cp n q of completely positive linear transformations from m n c m n to m q is shown to be isometrically isomorphic to p nq the cone of nq by nq positive semidefinite matrices generalizations of scalar and matrix results to cp n q hp n q l m n m q where hp n q represents the hermitian preserving linear transformations are discussed relationships among the completely positives the set of positive semidefinite preservers p n and its dual m n are given left and right facial ideals of cp are characterized properties of the joint angular field of values of a finite sequence of hermitian matrices h 1 h m are studied leading to a characterization of p q p n

a one level feti method for the drift diffusion poisson system with discontinuities at an interface abstract a 3d feti method for the drift diffusion poisson system including discontinuities at a 2d interface is developed the motivation for this work is to provide a parallel numerical algorithm for a system of pdes that are the basic model equations for the simulation of semiconductor devices such as transistors and sensors moreover discontinuities or jumps in the potential and its normal derivative at a 2d surface are included for the simulation of nanowire sensors based on a homogenized model using the feti method these jump conditions can be included with the usual numerical properties and the original farhat roux feti method is extended to the drift diffusion poisson equations including discontinuities we show two numerical examples the first example verifies the correct implementation including the discontinuities on a 2d grid divided into eight subdomains the second example is 3d and shows the application of the algorithm to the simulation of nanowire sensors with high aspect ratios the poisson boltzmann equation and the drift diffusion poisson system with jump conditions are solved on a 3d grid with real world boundary conditions

the interior point revolution in optimization history recent developments and lasting consequences interior methods are a pervasive feature of the optimization landscape today but it was not always so although interior point techniques primarily in the form of barrier methods were widely used during the 1960s for problems with nonlinear constraints their use for the fundamental problem of linear programming was unthinkable because of the total dominance of the simplex method during the 1970s barrier methods were superseded nearly to the point of oblivion by newly emerging and seemingly more efficient alternatives such as augmented lagrangian and sequential quadratic programming methods by the early 1980s barrier methods were almost universally regarded as a closed chapter in the history of optimization this picture changed dramatically in 1984 when narendra karmarkar announced a fast polynomial time interior method for linear programming in 1985 a formal connection was established between his method and classical barrier methods since then interior methods have continued to transform both the theory and practice of constrained optimization we present a condensed unavoidably incomplete look at classical material and recent research about interior methods

finite element error analysis of elliptic pdes with random coefficients and its application to multilevel monte carlo methods we consider a finite element approximation of elliptic partial differential equations with random coefficients such equations arise for example in uncertainty quantification in subsurface flow modeling models for random coefficients frequently used in these applications such as log normal random fields with exponential covariance have only very limited spatial regularity and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter in our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter which then enables us to prove uniform bounds on the finite element error in standard bochner spaces these new bounds can then be used to perform a rigorous analysis of the multilevel monte carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness to conclude we give some numerical results that confirm the new bounds

topology of the cone of positive maps on qubit systems an alternative geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra has been presented the premise of the proof is the identification of positive maps with operators preserving the lorentz cone in four dimensions and it allows to decompose the positive maps with respect to those preserving the boundary of the cone in addition useful conditions implying complete positivity of a map of have been given together with a sufficient condition for complete positivity of an extremal schwarz map of lastly following the same geometrical approach a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented

homotopy invariance of η invariants abstract intersection homology and results related to the higher signature problem are applied to show that certain combinations of invariants of the signature operator are homotopy invariant in various circumstances

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

completely positive mappings and mean matrices abstract some functions f r r induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices moreover such a function f can define a linear mapping j d f 1 m n m n on matrices which is basic in the constructions of monotone metrics the present subject is to check the complete positivity of j d f 1 in the case of a few concrete functions f this problem has been motivated by applications in quantum information

optimal multilevel randomized quasi monte carlo method for the stochastic drift diffusion poisson system abstract in this paper an optimal multilevel randomized quasi monte carlo method to solve the stationary stochastic drift diffusion poisson system is developed we calculate the optimal values of the parameters of the numerical method such as the mesh sizes of the spatial discretization and the numbers of quasi points in order to minimize the overall computational cost for solving this system of stochastic partial differential equations this system has a number of applications in various fields wherever charged particles move in a random environment it is shown that the computational cost of the optimal multilevel randomized quasi monte carlo method which uses randomly shifted low discrepancy sequences is one order of magnitude smaller than that of the optimal multilevel monte carlo method and five orders of magnitude smaller than that of the standard monte carlo method the method developed here is applied to a realistic transport problem namely the calculation of random dopant effects in nanoscale field effect transistors

quasi monte carlo finite element methods for a class of elliptic partial differential equations with random coefficients in this paper quasi monte carlo qmc methods are applied to a class of elliptic partial differential equations pdes with random coefficients where the random coefficient is parametrized by a co

further analysis of multilevel monte carlo methods for elliptic pdes with random coefficients we consider the application of multilevel monte carlo methods to elliptic pdes with random coefficients we focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter and that only have limited spatial regularity we extend the finite element error analysis for this type of equation carried out in charrier et al siam j numer anal 2013 to more difficult problems posed on non smooth domains and with discontinuities in the coefficient for this wider class of model problem we prove convergence of the multilevel monte carlo algorithm for estimating any bounded linear functional and any continuously frechet differentiable non linear functional of the solution we further improve the performance of the multilevel estimator by introducing level dependent truncations of the karhunen loeve expansion of the random coefficient numerical results complete the paper

on birkhoff s theorem for doubly stochastic completely positive maps of matrix algebras abstract a study is made of the extreme points of the convex set of doubly stochastic completely positive maps of the matrix algebra m n if n 2 the extreme points are precisely the unitary maps but if n 3 there are nonunitary extreme points examples of which are exhibited a tilde operation is defined on the linear maps of m n and used to give an elementary derivation of a result of kummerer and maassen

multi level monte carlo finite element method for elliptic pdes with stochastic coefficients in monte carlo methods quadrupling the sample size halves the error in simulations of stochastic partial differential equations spdes the total work is the sample size times the solution cost of an instance of the partial differential equation a multi level monte carlo method is introduced which allows in certain cases to reduce the overall work to that of the discretization of one instance of the deterministic pde the model problem is an elliptic equation with stochastic coefficients multi level monte carlo errors and work estimates are given both for the mean of the solutions and for higher moments the overall complexity of computing mean fields as well as k point correlations of the random solution is proved to be of log linear complexity in the number of unknowns of a single multi level solve of the deterministic elliptic problem numerical examples complete the theoretical analysis