the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

stochastic inverse heat conduction using a spectral approach an adjoint based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data process conditions and measurement noise the ill posed stochastic inverse problem is restated as a conditionally well posed l 2 optimization problem the gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field the l 2 optimization problem is solved using a conjugate gradient approach accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems

runge kutta methods applied to fully implicit differential algebraic equations of index 1 in this paper we study the order of runge kutta methods applied to differential algebraic equations of index one we derive general order conditions for the local order kl and give a convergence result which shows that the order kg of the global error satisfies kg kl 1 we also describe some numerical experiments which are in agreement with our results

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

statistical inverse problems discretization model reduction and inverse crimes the article discusses the discretization of linear inverse problems when an inverse problem is formulated in terms of infinite dimensional function spaces and then discretized for computational purposes a discretization error appears since inverse problems are typically ill posed neglecting this error may have serious consequences to the quality of the reconstruction the bayesian paradigm provides tools to estimate the statistics of the discretization error that is made part of the measurement and modelling errors of the estimation problem this approach also provides tools to reduce the dimensionality of inverse problems in a controlled manner the ideas are demonstrated with a computed example

reduced forms and reduced witt rings of higher level the present paper and its successor s deal with an extension of the socalled reduced theory of quadratic forms over a formally real field k 5 7 lo this theory deals with the reduced witt ring wred k w k nil w k and its natural embedding into c x z the ring of integer valued continuous functions on the space x x k of orderings of k recently it has been found that there is a far reaching generalization of the notion of ordering namely the orderings of higher level 11 the orderings of level s where s divides a given natural number n are related to sums of 2nth powers in the same way as the usual orderings are to sums of squares their intersection is the set of all sums of 2nth powers in k 11 in 6 orderings of levels were replaced by signatures of this level these are those characters x i s of order 2s whose kernels are additively closed where k k o it is exactly this change in point of view which leads to a natural definition of reduced witt rings of higher level let x x k denote the set of all signatures of k whose order divides 2n so that x k x k x being a closed subspace of the compact character group of x the set x k is also a compact topological space each element a e i induces a continuous function a x c in c x c defined by x x a where c is given the discrete topology by analogy with the natural representation of the ordinary reduced witt ring as a subring of 477 0021 8693 85 3 00

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

letter to the editor enumeration results for runge kutta methods for index 2 daes recursive formulas are presented that give the number of order conditions for single step runge kutta methods for index 2 daes

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

existence of solutions and optimal control for a model of tissue invasion by solid tumours abstract in this paper we study the distributed optimal control problem for the two dimensional mathematical model of cancer invasion existence of optimal state control and stability is proved and an optimality system is derived

on order reduction for runge kutta methods applies to differential algebraic systems and to stiff systems of odes in this short note the order reduction results of petzold siam j numer anal 23 1986 pp 837 852 for implicit runge kutta methods applied to index 1 differential algebraic systems are extended to include a larger class of methods the relationship between the order reduction results for differential algebraic systems and recent results for stiff systems of ordinary differential equations is explained

rosenbrook methods for differential algebraic equations this paper deals with the numerical solution of differential algebraic equations dae of index one it begins with the development of a general theory on the taylor expansion for the exact solutions of these problems which extends the well known theory of butcher for first order ordinary differential equations to dae s of index one as an application we obtain butcher type results for rosenbrock methods applied to dae s of index one we characterize numerical methods as applications of certain sets of trees we derive convergent embedded methods of order 4 3 which require 4 or 5 evaluations of the functions 1 evaluation of the jacobian and 1 lu factorization per step

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

runge kutta methods and differential algebraic systems for the solution of differential algebraic systems only those runge kutta formulae with nonsingular coefficient matrix a have been considered up to now to retain the algorithmic simplicity of dirk formulae while overcoming order reduction a is allowed here to be singular algebraic conditions on the formula are given that are necessary and sufficient for it to solve unambiguously constant coefficient linear differential algebraic systems of arbitrary index for a special class the regular formulae an expression is derived for the local discretization error this expression reduces to the one known when the runge kutta matrix a is nonsingular the paper concludes with some computational illustrations

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

order results for implicit runge kutta methods applied to differential algebraic systems in this paper we study the order stability and convergence properties of implicit runge kutta methods applied to a relatively simple class of nonlinear differential algebraic systems these methods often do not attain the same order of accuracy for differential algebraic systems as they do for purely differential systems we derive a set of order conditions which the method coefficients should satisfy in addition to the usual order conditions to ensure a given order of accuracy and we present results on the stability and convergence properties of these methods

application of fixed point collocation method for solving an optimal control problem of a parabolic hyperbolic free boundary problem modeling the growth of tumor with drug application abstract in this paper employing a fixed point collocation method we solve an optimal control problem for a model of tumor growth with drug application this model is a free boundary problem and consists of five time dependent partial differential equations including three different first order hyperbolic equations describing the evolution of cells and two second order parabolic equations describing the diffusion of nutrient and drug concentration in the mentioned optimal control problem the concentration of nutrient and drug is controlled using some control variables in order to destroy the tumor cells in this study applying the fixed point method we construct a sequence converging to the solution of the optimal control problem in each step of the fixed point iteration the problem changes to a linear one and the parabolic equations are solved using the collocation method the stability of the method is also proved some examples are considered to illustrate the efficiency of method

a scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties these advances raise the possibility of solving the corresponding stochastic inverse problem the problem of designing estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information a scalable parallel methodology for stochastic inverse design problems is developed in this article the representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology a novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems the stochastic inverse design problem is transformed to a deterministic optimization problem in a larger dimensional space that is subsequently solved using deterministic optimization algorithms the design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

the real spectrum of a noncommutative ring spaces of orderings were introduced by the second author in a series of w x papers 15 19 and various structure results were obtained generalizing results proved earlier for formally real fields by various people e becker w x l brocker r brown t craven a prestel and n schwartz in 6 26 w x also see 20 23 25 craven and tschimmel prove that formally real skew w x w x fields give rise to spaces of orderings in 8 also see 9 kalhoff extends w x this to ternary fields also called planar ternary rings in 10 11 also see w x 28 kleinstein and rosenberg and knebusch respectively show how spaces of orderings arise in the study of real commutative semi local rings orderings on commutative rings have been studied extensively since the introduction of the real spectrum by coste and roy in the early 1980s

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

stochastic collocation for optimal control problems with stochastic pde constraints we discuss the use of stochastic collocation for the solution of optimal control problems which are constrained by stochastic partial differential equations spde thereby the constraining spde depends on data which is not deterministic but random assuming a deter ministic control randomness within the states of the input data will propagate to the states of the system for the solution of spdes there has recently been an increasing effort in the development of efficient numerical schemes based upon the mathematical concept of generalized polynomial chaos modal based stochastic galerkin and nodal based stochastic collocation versions of this methodol ogy exist both of which rely on a certain level of smoothness of the solution in the random space to yield accelerated convergence rates in this paper we apply the stochastic collocation method to develop a gradient descent as well as a sequential quadratic program sqp for the minimization of objective functions constrained by an spde the stochastic function involves several higher order moments of the random states of the system as well as classical regularization of the control in particular we discuss several objective functions of tracking type numerical examples are presented to demonstrate the performance of our new stochastic collocation minimization approach

orderings and real places on commutative rings over fields there is a close relationship between orderings and real places which has been exploited in quadratic form theory 3 5 6 131 and in the study of the real holomorphy rings and sums of 2n powers 1 2 21 221 also orderings on an arbitrary commutative ring 4 141 have proved useful in real algebraic geometry and in quadratic form theory over rings more general than fields of course real places and valuations on fields have played an important role in this theory too at the same time there is a general theory of valuations on commutative rings which has been in existence for some time 9 10 15 23 24 in this paper k places k a formally real field in particular r places are considered on an arbitrary commutative ring r to obtain the good separation properties we want the places we consider are functions

valuations and higher level orders in commutative rings valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields see for example br in mw the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting while it seems desirable to extend these results further the techniques used for rings with many units will not work for general commutative rings at the same time there is a general theory of valuations in commutative rings see lm m and g which in ma was used to study orders and the reduced theory of quadratic forms over general commutative rings thus it seems natural to ask if the connections between valuations and higher level orders in fields exist in commutative rings in this paper we use valuation theory to study the space of orders and the reduced witt ring relative to a higher level preorder in a commutative ring as in ma we first localize our ring at a multiplicative set without changing the space of orders in order to make the valuation theory work better this is a standard idea from real algebraic geometry remarkably many of the notions methods and results for fields carry over to this new setting we define compatiblity between valuations and orders and preorders and the ring a t associated to a preorder t which turns out to be prufer ring as in the field case we define the relation of dependency on the set of valuations associated to a preorder and we use this to prove a decomposition theorem for the space of orders we can then apply this to show that under a certain finiteness condition the space of orders is equivalent to the space of orders of a preordered field