semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

on the regularity of matrices in min algebra abstract we consider the concept of regularity for square matrices with entries from a linearly ordered commutative group g the algebraic compositions being given by x y min x y x y x y for the case where g is cyclic we derive a necessary and sufficient condition for the regularity of a matrix which can be checked in o n 3 operations using standard algorithms

splitting in large dimension and infrared estimates ii moment inequalities this is the continuation of notes written for the nato asi conference in il ciocco september 96 consisting of the analysis of the links between estimating the splitting between the two first eigenvalues for the schrodinger operator h and the proof of infrared estimates for quantities attached to gaussian type measures these notes were mainly reporting on the old contributions of dyson frohlich glimm jaffe lieb simon and spencer in the 1970s in connection with more recent contributions of pastur khoruzhenko barbulyak and kondrat ev which treat in general more sophisticated models here we concentrate on the simplest model related to field theory and extend the results of barbulyak and kondrat ev by mixing ideas coming from pastur and khozurenko related to the use of bogolyubov s inequality with classical inequalities due to ginibre lebowitz sokal and others or in the case when the temperature t is zero by applying rather elementary estimates on schrodinger operators in order to find

semiclassical asymptotics of eigenvalues for schrodinger operators with magnetic fields abstract we consider schrodinger operators with magnetic fields on a two dimensional compact manifold or on r 2 the purpose is to study the semiclassical asymptotics of the eigenvalues by two different methods we obtain some facts on the harmonic oscillators under uniform magnetic fields and we also discuss them

simulation of multi dimensional random fields by karhunen loeve expansion abstract the karhunen loeve k l expansion method is a powerful tool for simulating stationary and nonstationary gaussian and non gaussian stochastic processes with explicitly known covariance functions since the k l expansion requires the solution of fredholm integral equation of the second kind it is generally not feasible to simulate multi dimensional random fields this is because even the numerical solution of the fredholm integral multi dimensional eigenvalue problem is difficult to obtain in order to address this problem this paper develops a consistent generalization of k l expansion for multi dimensional random field simulation the new method decomposes an n dimensional random field into a total of n stochastic processes each can be represented by using the traditional k l expansion thus the developed method is embedded into the well established framework of the k l expansion for simulating stochastic process and obviating the need for solving the multi dimensional integral eigenvalue problems four examples including random fields with different kinds of covariance functions are used to demonstrate the application of the proposed method

towards simultaneous reduction of both input and output spaces for interactive simulation based structural design abstract engineering design problems generally involve a high dimensional input space of design variables yielding an output space by means of costly high fidelity evaluations in order to decrease the overall cost reduced order models for the output space such as proper orthogonal decomposition pod and proper generalized decomposition pgd are an active area of research however little research has been conducted into alleviating the problems associated with a high dimensional input space in addition to higher dimensionality being an impediment to efficient design by itself complex shapes involve a high number of explicit implicit constraints restricting the design space geometric parameterization methods in traditional cad present difficulties in expressing these constraints leading to a high failure rate and the generation of inadmissible shapes in this paper we propose a simultaneous meta modeling protocol for both input and output spaces we perform a reparametrization of the input space using constrained shape interpolation by introducing the concept of an manifold of admissible meshed shapes the output space is reduced using constrained proper orthogonal decomposition by simultaneously using meta modeling for both spaces we facilitate interactive design space exploration for the purpose of design the proposed approach is applied to the industrial problem of designing a car engine intake port

trace formulas a golden thompson inequality and classical limit in boson fock space trace formulas for the heat semi groups of second quantization operators and their perturbations in the abstract boson fock space are given in terms of path functional integral representations as applications an inequality of golden thompson type and a classical limit are derived for the trace of the heat semi group of a perturbed second quantization operator the abstract results are applied to a model ofp type in quantum field theory

the vertex morphing method for node based shape optimization abstract the vertex morphing method a consistent surface control approach for shape optimization is presented this method introduces a control field and a map that relates it to the geometry field the optimization problem is solved in the control space and the map is defined such that both surface smoothness and mesh regularity criteria are satisfied simultaneously and synchronously in the geometry field the control field is discretized with the same discretization as the one of the geometry which leads to the richest design space possible for optimization of a discrete geometry the capabilities of the vertex morphing method are evaluated within an adjoint based cfd shape optimization workflow several engineering cases mainly from automotive industry are tested the method has shown significant success in performance efficiency and robustness even for problems with extreme geometrical complexities and millions of design parameters for applications in structural as well as fluid mechanics and their combination

spectral and scattering theory of spatially cut off p ϕ 2 hamiltonians we study spatially cut off p 2 hamiltonians we show the local finiteness of the pure point spectrum outside of thresholds the limiting absorption principle and asymptotic completeness of scattering for such hamiltonians our results imply the absence of singular continuous spectrum

a parameterized non intrusive reduced order model and error analysis for general time dependent nonlinear partial differential equations and its applications abstract a novel parameterized non intrusive reduced order model p nirom based on proper orthogonal decomposition pod has been developed this p nirom is a generic and efficient approach for model reduction of parameterized partial differential equations p pdes over existing parameterized reduced order models p rom most of them are based on the reduced basis method it is non intrusive and independent on partial differential equations and computational codes during the training process the smolyak sparse grid method is used to select a set of parameters over a specific parameterized space p r p for each selected parameter the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model more generally the snapshots and basis function sets for any parameters over p can be obtained using an interpolation method the p nirom can then be constructed by using our recently developed technique xiao et al 2015 41 42 where either the smolyak or radial basis function rbf methods are used to generate a set of hyper surfaces representing the underlying dynamical system over the reduced space the new p nirom technique has been applied to parameterized navier stokes equations and implemented with an unstructured mesh finite element model the capability of this p nirom has been illustrated numerically by two test cases flow past a cylinder and lock exchange case the prediction capabilities of the p nirom have been evaluated by varying the viscosity initial and boundary conditions the results show that this p nirom has captured the quasi totality of the details of the flow with cpu speedup of three orders of magnitude an error analysis for the p nirom has been carried out

parametric solutions involving geometry a step towards efficient shape optimization abstract optimization of manufacturing processes or structures involves the optimal choice of many parameters process parameters material parameters or geometrical parameters usual strategies proceed by defining a trial choice of those parameters and then solving the resulting model then an appropriate cost function is evaluated and its optimality checked while the optimum is not reached the process parameters should be updated by using an appropriate optimization procedure and then the model must be solved again for the updated process parameters thus a direct numerical solution is needed for each choice of the process parameters with the subsequent impact on the computing time in this work we focus on shape optimization that involves the appropriate choice of some parameters defining the problem geometry the main objective of this work is to describe an original approach for computing an off line parametric solution that is a solution able to include information for different parameter values and also allowing to compute readily the sensitivities the curse of dimensionality is circumvented by invoking the proper generalized decomposition pgd introduced in former works which is applied here to compute geometrically parametrized solutions

semi classical limit of the lowest eigenvalue of a schrodinger operator on a wiener space ii p ϕ 2 model on a finite volume abstract we determine the semi classical limit of the lowest eigenvalue of a p 2 hamiltonian on a finite volume interval

tunneling for spatially cut off p phi _2 hamiltonians we study the asymptotic behavior of low lying eigenvalues of spatially cut off p 2 hamiltonian in the semi classical limit we determine the semi classical limit of the lowest eigenvalue of the hamiltonian in terms of the hessian of the potential function of the corresponding classical equation moreover we prove that the gap of the lowest two eigenvalues goes to 0 exponentially fast in the semi classical limit when the potential function is double well type in fact we prove that the exponential decay rate is greater than or equal to the agmon distance between two zero points of the symmetric double well potential function also we study basic properties of the agmon distance and instanton

semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space abstract we study a semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space key results are semiboundedness theorem of the schrodinger operator laplace type asymptotic formula and ims localization formula we also make a remark on the semiclassical problem of a schrodinger operator on a path space over a riemannian manifold

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

on the spectrum of semi classical witten laplacians and schrodinger operators in large dimension we investigate the low lying spectrum of witten laplacians on forms of arbitrary degree in the semi classical limit and uniformly in the space dimension we show that under suitable assumptions implying that the phase function has a unique local minimum one obtains a number of clusters of discrete eigenvalues at the bottom of the spectrum moreover we are able to count the number of eigenvalues in each cluster we apply our results to certain sequences of schrodinger operators having strictly convex potentials and show that some well known results of semi classical analysis hold also uniformly in the dimension

shape optimization in ship hydrodynamics using computational fluid dynamics abstract a definite trend in computational applied mechanics is the development of integrated procedures for design optimization based on large scale numerical simulations simulation based design sbd in the present paper the fundamental elements of a sbd environment for shape optimization are presented and analyzed the focus is on complex engineering optimization problems which involve computationally highly expensive objective functions and nonlinear constraints advanced strategies adopted for reducing the overall computational effort are illustrated optimization algorithms for nonlinear programming problems are discussed as well as alternative techniques for shape variation and mesh manipulation necessary to automatically adapt the volume grid to the evolving shapes a new verification and validation v v methodology for assessing errors and uncertainties in simulation based optimization is also introduced based on the trends i e the differences between the numerically predicted improvement of the objective function and the actual improvement measured in a dedicated experimental campaign including consideration of numerical and experimental uncertainties two different sbd versions are then presented and demonstrated on a complex industrial problem namely the optimal shape redesign of a ship under real world geometrical and functional constraints whose evaluation during the optimization process involves repeated solutions of the reynolds averaged navier stokes equations finally an experimental campaign is carried out on the two optimized models to validate the computations and assess the success of the optimization process both the optimized models demonstrate improved characteristics beyond the numerical and experimental uncertainties confirming the validity of the sbd frameworks

leading large order asymptotics for φ4 2 perturbation theory we develop a rigorous semiclassical expansion to compute the radius of convergence of the borel transform for the pressure in 4 2 field theory this result gives a partial justification for the lipatov method of finding large order perturbation theory asymptotics in quantum field theory

semi classical limit of the bottom of spectrum of a schrodinger operator on a path space over a compact riemannian manifold abstract we determine the limit of the bottom of spectrum of schrodinger operators with variable coefficients on wiener spaces and path spaces over finite dimensional compact riemannian manifolds in the semi classical limit these are extensions of the results in s aida semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space j funct anal 203 2 2003 401 424 the problem on path spaces over riemannian manifolds is considered as a problem on wiener spaces by using ito s map however the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the riemannian curvature tensor on the underling manifold is not equal to 0 the difficulties are solved by using unitary transformations of the schrodinger operators by approximate ground state functions and estimates in the rough path analysis

classes of matrices associated with the optimal assignment problem abstract let e e ij be a matrix with integral elements and let x be an indeterminate defined over the rational field q we investigate matrices of the form x x e ij i 1 m j 1 n m n we may multiply the lines rows or columns of the matrix x by suitable integral powers of x in various ways and thereby transform x into a matrix y x f ij such that the f ij are nonnegative integers and each line of y contains at least one element x 0 1 we call y a normalized form of x and we denote by s x the class of all normalized forms associated with a given matrix x the classes s x have a fascinating combinatorial structure and the present paper is a natural outgrowth and extension of an earlier study we introduce new concepts such as an elementary transformation called an interchange we prove for example that two matrices in the same class are transformable into one another by interchanges our analysis of the class s x also yields new insights into the structure of the optimal assignments of the matrix e by way of the diagonal products of the matrix x

linear independence in bottleneck algebras abstract let r be a linearly ordered set a bottleneck algebra is the set r together with two operations a b max a b and a b min a b in this paper we suggest various definitions of linear independence of sets of n tuples over r study their characteristics and interrelations

design space dimensionality reduction in shape optimization by karhunen loeve expansion abstract the paper presents a methodology to reduce the dimension of design spaces in shape optimization problems while retaining a desired level of geometric variance the method is based on a generalized karhunen loeve expansion kle arbitrary shape modification spaces are assessed in terms of karhunen loeve modes eigenvectors and associated geometric variance eigenvalues the former are used as a basis in order to build a reduced dimensionality representation of the shape modification the method is demonstrated for the shape optimization of a high speed catamaran based on cfd simulations and aimed at the reduction of the wave component of calm water resistance kle is applied to three design spaces with large dimensionality 20 based on a free form deformation technique the space with the largest geometric variance is selected for dimensionality reduction and design optimization n dimensional design spaces are used with n 1 2 3 and 4 retaining up to the 95 of the geometric variance associated to the original space the correlation between the objective reduction achieved the dimension n and the geometric variance of the reduced dimensionality space is shown and found significant

the double well splitting of the low energy levels for the schrodinger operator of discrete φ4 models on tori the formula is obtained for the asymptotics of the splitting of the first two eigenvalues of the schrodinger operator that stands for the discrete multidimensional 4 model on tori the behavior of the main term is investigated in the limit of the large number of particles

an estimate of the gap of spectrum of schrodinger operators which generate hyperbounded semigroups abstract logarithmic sobolev inequalities with potential functions on loop spaces were proved by l gross 1991 j funct anal 102 265 313 s aida 1996 in proceedings of the fifth gregynog symposium stochastic analysis and applications pp 1 19 world scientific singapore and f z gong and z m ma 1998 j funct anal 599 623 the generators of the dirichlet forms are schrodinger operators on loop spaces and they generate hyperbounded semigroups recently f z gong m rockner and l m wu 2000 poincare inequality for weighted first order sobolev spaces on loop spaces submitted proved the existence of the gap of spectrum at the lowest eigenvalue of the schrodinger operators by using m hino s exponential decay estimates of the semigroup 2000 osaka j math 37 603 624 and the author s strong ergodicity property of the semi group s aida 1998 j funct anal 158 152 185 in this paper we will give a lower bound on the gap by using a weak poincare inequality which was introduced by m rockner and f y wang 2000 weak poincare inequalities and l 2 convergence rates of markov semigroups preprint also we will give estimates on the distribution function of ground states using the weak poincare inequality

survey of shape parameterization techniques for high fidelity multidisciplinary shape optimization a survey is provided of shape parameterization techniques formultidisciplinary optimization and some emerging ideas are highlighted the survey focuses on the suitability of available techniques for multidisciplinary applications of complex cone gurations using high e delity analysis tools such as computational e uid dynamics and computational structural mechanics the suitability criteria are based on the efe ciency effectiveness ease of implementation and availability of analytical sensitivities for geometry and grids a section on sensitivity analysis grid regeneration and grid deformation techniques is also provided