learning on dynamic statistical manifolds hyperbolic balance laws with uncertain random parameters and inputs are ubiquitous in science and engineering quantification of uncertainty in predictions derived from such laws and reduction of predictive uncertainty via data assimilation remain an open challenge that is due to nonlinearity of governing equations whose solutions are highly non gaussian and often discontinuous to ameliorate these issues in a computationally efficient way we use the method of distributions which here takes the form of a deterministic equation for spatio temporal evolution of the cumulative distribution function cdf of the random system state as a means of forward uncertainty propagation uncertainty reduction is achieved by recasting the standard loss function i e discrepancy between observations and model predictions in distributional terms this step exploits the equivalence between minimization of the square error discrepancy and the kullback leibler divergence the loss function is regularized by adding a lagrangian constraint enforcing fulfilment of the cdf equation minimization is performed sequentially progressively updating the parameters of the cdf equation as more measurements are assimilated

on regulation under sampling the paper deals with linear and nonlinear regulation under sampling it is shown that digital solutions exist under assumptions which are closely related to the existence of robust solutions to the continuous problem approximated solutions are computed starting from the continuous ones

higher order optimality conditions for set valued optimization via studniarski derivatives this paper deals with higher order optimality conditions of a set valued optimiza tion problem sop whose constraint condition is determined by a fixed set by virtue of studniarski derivatives of set valued maps and strict local efficiency higher order necessary and sufficient opti mality conditions for sop are obtained for the sufficient conditions the initial space must be finite dimensional higher order optimality conditions for sop are also investigated under the conditions that the initial and objective spaces are finite dimensional and the multifunction involved is mth order

optimality conditions for henig and globally proper efficient solutions with ordering cone has empty interior in this paper we consider the set valued vector optimization problems with constraint in locally convex spaces we present the necessary and sufficient conditions for henig efficient solution pair globally proper efficient solution pair and super efficient solution pair without the ordering cones having the nonempty interior

inferring solutions of differential equations using noisy multi fidelity data for more than two centuries solutions of differential equations have been obtained either analytically or numerically based on typically well behaved forcing and boundary conditions for well posed problems we are changing this paradigm in a fundamental way by establishing an interface between probabilistic machine learning and differential equations we develop data driven algorithms for general linear equations using gaussian process priors tailored to the corresponding integro differential operators the only observables are scarce noisy multi fidelity data for the forcing and solution that are not required to reside on the domain boundary the resulting predictive posterior distributions quantify uncertainty and naturally lead to adaptive solution refinement via active learning this general framework circumvents the tyranny of numerical discretization as well as the consistency and stability issues of time integration and is scalable to high dimensions

the lagrange multiplier rule in set valued optimization the known lagrange multiplier rule is extended to set valued constrained optimization problems using the contingent epiderivative as differentiability notion a necessary optimality condition for weak minimizers is derived which is also a sufficient condition under generalized convexity assumptions

positive proper efficient points and related cone results in vector optimization theory positive proper efficient points are defined as solutions of appropriate linear scalar optimization problems a geometric characterization of positive proper efficient points is given as well as conditions under which the set of positive proper efficient points is dense in the set of all efficient points it is shown that these results are applicable in the normed vector lattices c a b l p and l p for 1 leqq p leqq infty and that previous related results which required the ordering cone to have a compact or weak compact base are not applicable in many normed vector lattices including c a b l p and l p for 1 leqq p leqq infty

scalar multiplier rules in set valued optimization in this work we establish necessary and sufficient conditions for weak minimizers of a constrained set valued optimization problem these conditions are given by means of multiplier rules formulated in terms of contingent epiderivatives of scalar set valued maps we consider that the image spaces are finite dimensional and the set valued maps are convex and stable for these conditions our results provide a scalar version of analogous existing results in the literature under weaker existence conditions moreover we prove that the multiplier rules can be computed in terms of the directional derivative of associated maps of infima

information based objective functions for active data selection learning can be made more efficient if we can actively select particularly salient data points within a bayesian learning framework objective functions are discussed that measure the expected informativeness of candidate measurements three alternative specifications of what we want to gain information about lead to three different criteria for data selection all these criteria depend on the assumption that the hypothesis space is correct which may prove to be their main weakness

brief paper continuous time controller redesign for digital implementation a trajectory based approach given a continuous time nonlinear closed loop system we investigate sampled data feedback laws for which the trajectories of the sampled data closed loop system converge to the continuous time trajectories with a prescribed rate of convergence as the length of the sampling interval tends to zero we derive necessary and sufficient conditions for the existence of such sampled data feedback laws and in case of existence provide explicit redesign formulas and algorithms for these controllers

super efficiency in vector optimization we introduce a new concept of efficiency in vector optimization this concept super efficiency is shown to have many desirable properties in particular we show that in reasonable settings the super efficient points of a set are norm dense in the efficient frontier we also provide a chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex

discovering variable fractional orders of advection dispersion equations from field data using multi fidelity bayesian optimization abstract the fractional advection dispersion equation fade can describe accurately the solute transport in groundwater but its fractional order has to be determined a priori here we employ multi fidelity bayesian optimization to obtain the fractional order under various conditions and we obtain more accurate results compared to previously published data moreover the present method is very efficient as we use different levels of resolution to construct a stochastic surrogate model and quantify its uncertainty we consider two different problem set ups in the first set up we obtain variable fractional orders of one dimensional fade considering both synthetic and field data in the second set up we identify constant fractional orders of two dimensional fade using synthetic data we employ multi resolution simulations using two level and three level gaussian process regression models to construct the surrogates

probabilistic non linear principal component analysis with gaussian process latent variable models summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure in this paper we provide an overview of some existing techniques for discovering such embeddings we then introduce a novel probabilistic interpretation of principal component analysis pca that we term dual probabilistic pca dppca the dppca model has the additional advantage that the linear mappings from the embedded space can easily be non linearised through gaussian processes we refer to this model as a gaussian process latent variable model gp lvm through analysis of the gp lvm objective function we relate the model to popular spectral techniques such as kernel pca and multidimensional scaling we then review a practical algorithm for gp lvms in the context of large data sets and develop it to also handle discrete valued data and missing attributes we demonstrate the model on a range of real world and artificially generated data sets

super efficiency of vector optimization in banach spaces using variational analysis in terms of the clarke normal cone we consider super efficiency of vector optimization in banach spaces we establish some characterizations for super efficiency in particular dropping the assumption that the ordering cone has a bounded base we extend a result in borwein and zhuang j m borwein d zhuang super efficiency in vector optimization trans amer math soc 338 1993 105 122 to the nonconvex setting 2006 elsevier inc all rights reserved

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

weak clarke epiderivative in set valued optimization a new notion of weak clarke epiderivative for a set valued map is introduced using the concept of clarke tangent cone the existence characterization and properties of weak clarke epiderivative are then studied finally optimality criteria are established for a constrained set valued optimization problem in terms of weak clarke epiderivative

petrov galerkin and spectral collocation methods for distributed order differential equations distributed order fractional operators offer a rigorous tool for mathematical modelling of multiphysics phenomena where the differential orders are distributed over a range of values rather than being just a fixed integer fraction as in standard fractional odes pdes we develop two spectrally accurate schemes namely a petrov galerkin spectral method and a spectral collocation method for distributed order fractional differential equations these schemes are developed based on the fractional sturm liouville eigen problems fslps m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 in the petrov galerkin method we employ fractional nonpolynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind while we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind we define the underlying distributed sobolev space and the associated norms where we carry out the

digital redesign of infinite dimensional controllers based on numerical integration of new representation abstract continuous time infinite dimensional controllers that include laplace transforms of time functions with compact support are indispensable for the advanced control of delay systems however no study has yet been conducted on digital redesign for obtaining digital controllers that are used in sampled data control systems from predesigned continuous time infinite dimensional controllers we introduce a new representational form to describe the continuous time input output relation of linear time invariant systems called finite interval integral representation using numerical integration we approximate the continuous time input output relation of a predesigned infinite dimensional controller in the finite interval integral representation to obtain a digital controller

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

cone directed higher order generalized adjacent derivative and higher order mond weir duality in this paper cone directed higher order generalized adjacent derivative for a set valued map is defined by virtue of the cone directed higher order generalized adjacent derivative a higher order mond weir type dual problem for a constrained set valued optimiza tion problem is introduced and its weak duality strong duality and converse duality properties are established the results obtained improve the corresponding results in the literatures

conjugate duality in set valued vector optimization abstract in this paper conjugate duality results for convexlike set valued vector optimization problems are presented under closedness or boundedness hypotheses some properties of the value mapping of a set valued vector optimization problem are studied a conjugate duality result is also proved for a convex set valued vector optimization problem without the requirements of closedness and boundedness

about contingent epiderivatives abstract in this paper we analyze some questions concerning the contingent epiderivative when the ordering cone is not necessarily pointed we introduce the notion of family of contingent epiderivatives we also investigate the relationship between the contingent derivative and the generalized epiderivative furthermore we give existence theorems with respect to daniell cones as a particular case we study set valued maps that take values in r n

optimality conditions for proper efficient solutions of vector set valued optimization abstract based on the concept of an epiderivative for a set valued map introduced in j nanchang univ 25 2001 122 130 in this paper we present a few necessary and sufficient conditions for a henig efficient solution a globally proper efficient solution a positive properly efficient solution an f efficient solution and a strongly efficient solution respectively to a vector set valued optimization problem with constraints

machine learning of linear differential equations using gaussian processes this work leverages recent advances in probabilistic machine learning to discover governing equations expressed by parametric linear operators such equations involve but are not limited to ordinary and partial differential integro differential and fractional order operators here gaussian process priors are modified according to the particular form of such operators and are employed to infer parameters of the linear equations from scarce and possibly noisy observations such observations may come from experiments or black box computer simulations as demonstrated in several synthetic examples and a realistic application in functional genomics employ probabilistic machine learning to discover governing equations expressed by parametric linear operators proper placement of gaussian process priors allows one to efficiently infer model parameters via maximum likelihood estimation a general treatment of inverse problems governed by linear operators leading to model discovery from just a handful of noisy measurements

new coprimeness over multivariable polynomial matrices and its application to control delay systems abstract this paper presents a new notion of coprimeness over multivariable polynomial matrices where a single variable is given priority over the remaining variables from a characterization through a set of common zeros of the minors it is clarified that the presented coprimeness is equivalent to weakly zero coprimeness in the particular variable an application of the presented coprimeness to control systems with non commensurate delays and finite spectrum assignment is also presented because the presented coprimeness is stronger than minor coprimeness non commensurate delays are difficult to deal with in algebraic control theory the rational ratio condition between delays which can reduce non commensurate delays to commensurate delays proves to be both powerful and practical concept in algebraic control theory for delay systems

canonical realizations of systems with delays in this paper we consider the properties of the realizations of a class of nonrational transfer functions the transfer functions of this class can be realized as systems with delays we introduce

higher order optimality conditions for henig efficient solutions in set valued optimization in this paper higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set valued maps are introduced necessary and sufficient conditions for henig efficient solutions to a constrained set valued optimization problem are given by employing the higher order generalized epiderivatives

conjugate duality in vector optimization in this paper we develop a conjugate duality theory in vector optimiza tion conjugate duality was fully developed in scalar optimization by rockafellar 3 and provides a unified framework for the duality theory the author and sawaragi extended it to the case of multiobjective optimization by introducing some new concepts such as conjugate maps and subgradients for vector valued set valued maps 4 6 their results are based on the efficiency pareto maximality kawasaki refined their results and obtained a reciprocal duality by introducing the concept of supremum set in an extended euclidean space 1 2 his supremum is based on the weak efficiency weak pareto maximality of the closure of a set in an extended sense the author s newly defined the supremum of a set in the extended multi dimensional euclidean space on the basis of weak ehiciency through several discussions on supremum the definition satisfies some desirable fundamental properties it can be extended to the case of general partially ordered linear topological spaces in a straightforward manner based on this definition of supremum some useful concepts such as conjugate maps and subgradients are introduced for vector valued set valued maps in this paper these concepts enable us to develop the conjugate duality in vector optimization although the results obtained are quite similar to the earlier works by the author and sawaragi 4 or kawasaki 2 our new approach makes the proofs much easier and more understandable and provides a better version of conjugate duality

higher order weak epiderivatives and applications to duality and optimality conditions in this paper the notions of higher order weak contingent epiderivative and higher order weak adjacent epiderivative for a set valued map are defined by virtue of higher order weak adjacent contingent epiderivatives and henig efficiency we introduce a higher order mond weir type dual problem and a higher order wolfe type dual problem for a constrained set valued optimization problem sop and discuss the corresponding weak duality strong duality and converse duality properties we also establish higher order kuhn tucker type necessary and sufficient optimality conditions for sop

predicting the output from a complex computer code when fast approximations are available s we consider prediction and uncertainty analysis for complex computer codes which can be run at different levels of sophistication in particular we wish to improve efficiency by combining expensive runs of the most complex versions of the code with relatively cheap runs from one or more simpler approximations a bayesian approach is described in which prior beliefs about the codes are represented in terms of gaussian processes an example is presented using two versions of an oil reservoir simulator 1 c complex mathematical models implemented in large computer codes have been used to study real systems in many areas of scientific research sacks et al 1989 usually because physical experimentation is too costly and sometimes impossible as in the case of large environmental systems a computer experiment involves running the code with various input values for the purpose of learning something about the real system often a simulator can be run at different levels of complexity with versions ranging from the most sophisticated high level code to the most basic for example in 4 we consider two codes which simulate oil pressure at a well of a hydrocarbon reservoir both codes use finite element analysis in which the rocks comprising the reservoir are represented by small interacting grid blocks the flow of oil within the reservoir can be simulated by considering the interaction between the blocks the two codes differ in the resolution of the grid so that we have a very accurate slow version using many small blocks and a crude approximation using large blocks which runs much faster alternatively a mathematical model could be expanded to include more of the scientific laws underlying the physical processes simple fast versions of the code may well include the most important features and are useful for preliminary investigations in real time applications the number of runs from a high level simulator may be limited by expense then there is a need to trade off the complexity of the expensive code with the availability of the simpler approximations the purpose of the current paper is to explore ways in which runs from several levels of a code can be used to make inference about the output from the most complex code we may also have uncertainty about values for the input parameters which apply in any given application uncertainty analysis of computer codes describes how this uncertainty on the inputs affects our uncertainty about the output

higher order mond weir duality for set valued optimization in this paper we introduce a higher order mond weir dual for a set valued optimization problem by virtue of higher order contingent derivatives and discuss their weak duality strong duality and converse duality properties

neural net induced gaussian process regression for function approximation and pde solution abstract neural net induced gaussian process nngp regression inherits both the high expressivity of deep neural networks deep nns as well as the uncertainty quantification property of gaussian processes gps we generalize the current nngp to first include a larger number of hyperparameters and subsequently train the model by maximum likelihood estimation unlike previous works on nngp that targeted classification here we apply the generalized nngp to function approximation and to solving partial differential equations pdes specifically we develop an analytical iteration formula to compute the covariance function of gp induced by deep nn with an error function nonlinearity we compare the performance of the generalized nngp for function approximations and pde solutions with those of gps and fully connected nns we observe that for smooth functions the generalized nngp can yield the same order of accuracy with gp while both nngp and gp outperform deep nn for non smooth functions the generalized nngp is superior to gp and comparable or superior to deep nn

brief finite spectrum assignment for linear systems with non commensurate time delays this paper studies finite spectrum assignment for linear scalar systems with non commensurate time delays based on a practically important class of finite laplace transforms the finite spectrum assignability can be reduced to the solvability of a bezout equation over a multivariable polynomial ring with coefficients in the class of finite laplace transforms it is shown that in the non commensurate delay case spectral canonicity is not sufficient for the finite spectrum assignability

the lagrange multiplier rule for super efficiency in vector optimization in this paper we study constrained multiobjective optimization problems with objectives being closed graph multifunctions in banach spaces in terms of the coderivatives and clarke s normal cones we establish lagrange multiplier rules for super efficiency as necessary or sufficient optimality conditions of the above problems

solving partial differential equations by collocation using radial basis functions after a series of application papers have proven the approach to be numerically effective this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with possibly radial basis functions

controlling linear systems with nonlinear perturbations under sampled data output feedback digital redesign approach this article develops a digital redesign dr technique for sampled data observer based output feedback control of a continuous time linear system with nonlinear perturbation it is assumed that the nonlinear perturbation is a locally lipschitz function to deal with the discrete time modelling error in nonlinear systems as opposed to the previous approach the dr problem is configured as a stabilisation one for error dynamics between the closed loop system of nominal linear model under an analogue state feedback controller and that of the linear system with the nonlinear perturbation under a sampled data output feedback controller a constructive dr condition is formulated in the format of linear matrix inequalities the stability of the actual sampled data control system is guaranteed within the dr procedure the effectiveness of the proposed dr methodology is demonstrated through a numerical simulation

radial epiderivatives and asymptotic functions in nonconvex vector optimization the notions of lower and upper radial epiderivatives a radial notion different from that generally discussed for nonconvex vector valued functions are introduced many of their properties are established and some of the optimality conditions for a point to be an ideal pareto or weak pareto minimum involving these epiderivatives are also presented in particular a characterization for the ideal minima in terms of the lower radial epiderivative is proved such a result appears to be new in the literature under some mild assumptions on the given function it is proved that the asymptotic cone of its epigraph is the epigraph of its upper radial epiderivative moreover given a vector minimization problem we describe the asymptotic behavior of its solution set by introducing some cones of asymptotic directions of the function involved finally we define the lower and upper radial subdifferentials and express the optimality conditions by means of these subdifferentials certainly these optimality conditions subsume various necessary or sufficient conditions found in the literature for convex or nonconvex functions

from structured data to evolution linear partial differential equations abstract this paper is devoted to the derivation of computational methods for constructing partial differential equations from data following some recent works 7 14 15 20 we propose a methodology based on symbolic calculus 8 9 13 pseudospectral methods 2 3 and stochastic processes 6 in order to determine non constant coefficients of linear evolution partial differential equations pdes from a set of structured data constituted by solutions at given times and positions of an unknown linear pde

generalized constraint qualifications and optimality conditions for set valued optimization problems in this paper we discuss the connections of four generalized constraint qualifications for set valued vector optimization problems with constraints then some k t type necessary and sufficient optimality conditions are derived in terms of the contingent epiderivatives

on the existence of pareto efficient points we prove some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings these results are compared to the axiom of choice

lagrange multipliers for set valued optimization problems associated with coderivatives in this paper we investigate a vector optimization problem p where objective and constraints are given by set valued maps we show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of p as solutions of a scalar optimization problem sp with single valued objective and constraint functions then applying some classical or recent results in optimization theory to sp and using estimates of subdifferentials of marginal functions we obtain optimality conditions for p expressed in terms of lagrange or sequential lagrange multipliers associated with various coderivatives of the set valued data