representations p adiques et equations differentielles in this paper we associate to every p adic representation v a p adic differential equation d rig v that is to say a module with a connection over the robba ring we do this via the theory of fontaine s k modules this construction enables us to relate the theory of k modules to p adic hodge theory we explain how to construct dcris v and dst v from d rig v which allows us to recognize semi stable or crystalline representations the connection is then unipotent or trivial on d rig v 1 t in general the connection has an infinite number of regular singularities but v is de rham if and only if those are apparent singularities a structure theorem for modules over the robba ring allows us to get rid of all singularities at once and to obtain a classical differential equation with a frobenius structure using this we construct a functor from the category of de rham representations to that of classical p adic differential equations with frobenius structure a recent theorem of y andre gives a complete description of the structure of the latter object this allows us to prove fontaine s p adic monodromy conjecture every de rham representation is potentially semi stable as an application we can extend to the case of arbitrary perfect residue fields some results of hyodo h1g h1st of perrin riou the semi stability of ordinary representations of colmez absolutely crystalline representations are of finite height and of bloch and kato if the weights of v are 2 then bloch kato s exponential expv is an isomorphism

analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

prequential randomness and probability this paper studies dawid s prequential framework from the point of view of the algorithmic theory of randomness our first main result is that two natural notions of randomness coincide one notion is the prequential version of the measure theoretic definition due to martin lof and the other is the prequential version of the game theoretic definition due to schnorr and levin this is another manifestation of the close relation between the two main paradigms of randomness the algorithmic theory of randomness can be stripped of its algorithmic aspect and still give meaningful results the measure theoretic paradigm then corresponds to kolmogorov s measure theoretic probability and the game theoretic paradigm corresponds to game theoretic probability our second main result is that measure theoretic probability coincides with game theoretic probability on all analytic in particular borel sets

an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

analysis of block parareal preconditioners for parabolic optimal control problems in this paper we describe block matrix algorithms for the iterative solution of a large scale linear quadratic optimal control problem involving a parabolic partial differential equation over a finite control horizon we consider an all at once discretization of the problem and formulate three iterative algorithms the first algorithm is based on preconditioning a symmetric positive definite reduced linear system involving only the unknown control variables however inner outer iterations are required the second algorithm modifies the first algorithm to avoid inner outer iterations by introducing an auxiliary variable it yields a symmetric indefinite system with a positive definite block preconditioner the third algorithm is the central focus of this paper it modifies the preconditioner in the second algorithm by a parallel in time preconditioner based on the parareal algorithm theoretical results show that the preconditioned algorithms have optimal convergence properties and parallel scalability numerical experiments confirm the theoretical results

stable parareal in time method for first and second order hyperbolic systems the parareal in time algorithm allows one to perform parallel simulations of time dependent problems this algorithm has been implemented on many types of time dependent problems with some success recent contributions have allowed one to extend the domain of application of the parareal in time algorithm so as to handle long time simulations of hamiltonian systems this improvement has managed to avoid the fatally large lack of accuracy of the plain parareal in time algorithm which does not conserve invariant quantities a somewhat similar difficulty occurs for problems where the solution lacks regularity either initially or during the evolution as is the case for hyperbolic systems of conservation laws in this paper we identify the reasons for instabilities of the parareal in time algorithm and propose a simple way to cure them we use the new method to solve a linear wave equation and a nonlinear burgers equation the results illustrate the stability of this variant of the parareal in time algorithm

a micro macro parareal algorithm application to singularly perturbed ordinary differential equations we introduce a micro macro parareal algorithm for the time parallel integration of multiscale in time systems the algorithm first computes a cheap but inaccurate solution using a coarse propagator simulating an approximate slow macroscopic model which is iteratively corrected using a fine scale propagator accurately simulating the full microscopic dynamics this correction is done in parallel over many subintervals thereby reducing the wall clock time needed to obtain the solution compared to the integration of the full microscopic model over the complete time interval we provide a numerical analysis of the algorithm for a prototypical example of a micro macro model namely singularly perturbed ordinary differential equations we show that the computed solution converges to the full microscopic solution when the parareal iterations proceed only if special care is taken during the coupling of the microscopic and macroscopic levels of description the error bound depends on the modeling error

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

analysis for parareal algorithms applied to hamiltonian differential equations long time integrations are an important issue in the numerical solution of hamiltonian systems they are time consuming and it is natural to consider the use of parallel architectures for reasons of efficiency in this context the parareal algorithm has been proposed by several authors the present work is a theoretical study of the parareal algorithm when it is applied to hamiltonian differential equations the idea of backward error analysis is employed to get insight into the long time behavior of numerical approximations one of the main results is that convergence of the parareal iterations restricts the length of the time window for nearly integrable systems its length is bounded by the square root of the inverse of the accuracy of the coarse integrator the theoretical bounds are confirmed by numerical experiments

prequential level of impossibility with some applications within dawid s prequential framework we introduce a quantitative measure of disagreement between a probabalistic theory and observations using this measure we reconsider howard s theorem on laplace s rule of succession and dawid s result on the prequential efficiency of the bayesian rule

kolmogorov s contributions to information theory and algorithmic complexity briefly information theory says a random object x p x has complexity entropy h xp x log p x with the attendant interpretation that h bits are sufficient to describe x on the average algorithmic complexity says an object x has a complexity k x equal to the length of the shortest binary program that describes x it is a beautiful fact that these ideas are much the same in fact it is roughly true that ek x h moreover if we let pu x pr u prints x be the probability that a given computer u prints x when given a random program it can be shown that log 1 pu x k x for all x thus establishing a vital link between the universal probability measure pu and the universal complexity k more on this later the relationship of these ideas to probability theory was summed up in kolmogorov s 1983 paper which was based on his 1970 talk in nice perhaps only

a logic of probability with application to the foundations of statistics a simple example shows that the classical theory of probability implies more than one can deduce via kolmogorov s calculus of probability developing dawid s ideas i propose a new calculus of probability which is free from this drawback this calculus naturally leads to a new interpretation of probability i argue that attempts to create a general empirical theory of probability should be abandoned and we should content ourselves with the logic of probability establishing relations between probabilistic theories and observations my approach to the logic of probability is based on a variant of ville s principle of the excluded gambling strategy

integral crystalline cohomology over very ramified valuation rings in this paper we intend to show how fontaine s comparison theory fol fo2 between crystalline and p adic etale cohomology can be extented to very ramified base rings so far it had been developed either for qp coefficients or zp coefficients if the base is unramified over the witt vectors see fa2 for example it states that for any p adic discrete valuation ring v there exists a ring b v with the following property if x denotes a smooth and proper v scheme the p adic etale cohomology of the generic fiber x xv k is related to the crystalline cohomology of x v0 vo defined below by an isomorphism

computability noncomputability and hyperbolic systems abstract in this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point these manifolds are the essential feature which characterizes a hyperbolic system having many applications in physical sciences and other fields we show that i locally these manifolds can be computed but ii globally they cannot since their degree of computational unsolvability lies on the second level of the borel hierarchy we also show that smale s horseshoe the first example of a hyperbolic invariant set which is neither an equilibrium point nor a periodic orbit is computable

the well calibrated bayesian abstract suppose that a forecaster sequentially assigns probabilities to events he is well calibrated if for example of those events to which he assigns a probability 30 percent the long run proportion that actually occurs turns out to be 30 percent we prove a theorem to the effect that a coherent bayesian expects to be well calibrated and consider its destructive implications for the theory of coherence

effective symbolic dynamics random points statistical behavior complexity and entropy we consider the dynamical behavior of martin lof random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure we use computable partitions to define a sort of effective symbolic model for the dynamics through this construction we prove that such points have typical statistical behavior the behavior which is typical in the birkhoff ergodic theorem and are recurrent we introduce and compare some notions of complexity for orbits in dynamical systems and prove i that the complexity of the orbits of random points equals the kolmogorov sinai entropy of the system ii that the supremum of the complexity of orbits equals the topological entropy

non stochastic infinite and finite sequences abstract combining outcomes of coin tossing and transducer algorithms it is possible to generate with probability close to 1 very pathological sequences for which computable probabilistic forecasting is impossible these sequences are not random with respect to any reasonable probability distribution a natural consequence from the definition of such sequences is that each simple measure of the set of all such sequences is equal to 0 it was kolmogorov s and levin s idea to estimate the probability of generating of such sequences in the combinations of probabilistic and algorithmic processes 8 14 21 we collect several results in this direction for infinite sequences and asymptotic results for finite sequences including estimation of space and time of losing randomness for time bounded forecasting systems a correction to 22

computability of probability measures and martin lof randomness over metric spaces in this paper we investigate algorithmic randomness on more general spaces than the cantor space namely computable metric spaces to do this we first develop a unified framework allowing computations with probability measures we show that any computable metric space with a computable probability measure is isomorphic to the cantor space in a computable and measure theoretic sense we show that any computable metric space admits a universal uniform randomness test without further assumption

most sequences are stochastic the central problem in machine learning and statistics is the problem of predicting future events xn 1 based on past observations x1x2 xn where n 1 2 the main goal is to find a method of prediction that minimizes the total loss suffered on a sequence x1x2 xn 1 for n 1 2 we say that a data sequence is stochastic if there exists a simply described prediction algorithm whose performance is close to the best possible one this optimal performance is defined in terms of vovk s predictive complexity which is a generalization of the notion of kolmogorov complexity predictive complexity gives a limit on the predictive performance of simply described prediction algorithms in this paper we argue that data sequences normally occurring in the real world are stochastic more formally we prove that levin s a priori semimeasure of nonstochastic sequences is small

ergodic theorems for individual random sequences in the framework of the kolmogorov s approach to the substantiation of the probability theory and information theory on the base of the theory of algorithms we try to formulate probabilistic laws i e statements of the form p a 1 where a is some formula in pointwise form if is random then a holds nevertheless not all proofs of such laws can be directly translated into the algorithmic form in 11 two examples have been distinguished birkhoff s 2 ergodic theorem and shannon mcmillan breiman theorem of information theory 1 in this paper an analysis of algorithmic effectiveness of these theorems is given we prove that birkhoff s ergodic theorem is indeed in some strong sense nonconstructive at the same time the claim to formulate probabilistic laws for algorithmically random sequences is not so restrictive we present the versions of these laws for individual random sequences

lindley s paradox 1982 lindley s paradox journal of the american statistical association vol 77 no 378 pp 325 334

abelian varieties with l adic galois representation of mumford s type nous etudions des varietes abeliennes sur des corps de nombres telles que l algebre de lie de l image d une representation l adique associee soit geometriquement isomorphe a c sl 2 3 on appelle de telles varietes abeliennes des varietes abeliennes avec representation l adique du type de mumford on montre que de telles varietes abeliennes ont potentiellement bonne reduction partout et on determine les polygones de newton et decompositions en facteurs simples de ces reductions

convergence analysis for three parareal solvers we analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem mathbf u a mathbf u g the coarse propagator mathcal g is fixed to the backward euler method and three time integrators are chosen for the mathcal f propagator the trapezoidal rule the third order diagonal implicit runge kutta rk dirk method and the fourth order gauss rk method it is well known that the parareal euler algorithm using the backward euler method for mathcal f and mathcal g converges rapidly but less is known when one uses for mathcal f the trapezoidal rule or the fourth order gauss rk method especially when the mesh ratio j delta t delta t is small we show that for a specified lambda max the maximal eigenvalue of a or its upper bound there exists some critical j rm cri such that the parareal solvers derived from these three choices of mathcal f converge as fast as parareal euler provided j geq j rm cri

a constructive version of birkhoff s ergodic theorem for martin lof random points we prove the effective version of email protected s ergodic theorem for martin lof random points and effectively open sets improving the results previously obtained in this direction in particular those of vyugin nandakumar and hoyrup rojas the proof consists of two steps first we prove a generalization of email protected s theorem which is a particular case of effective ergodic theorem a trajectory of a computable ergodic mapping that starts from a random point cannot remain inside an effectively open set of measure less than 1 second we show that the full statement of the effective ergodic theorem can be reduced to this special case both steps use the statement of classical ergodic theorem but not its usual classical proof therefore we get a new simple proof of the effective ergodic theorem with weaker assumptions than before this result was recently obtained independently by franklin greenberg miller and ng

a preconditioned numerical solver for stiff nonlinear reaction diffusion equations with fractional laplacians that avoids dense matrices the numerical solution of fractional partial differential equations poses significant computational challenges in regard to efficiency as a result of the spatial nonlocality of the fractional differential operators the dense coefficient matrices that arise from spatial discretisation of these operators mean that even one dimensional problems can be difficult to solve using standard methods on grids comprising thousands of nodes or more in this work we address this issue of efficiency for one dimensional nonlinear space fractional reaction diffusion equations with fractional laplacian operators we apply variable order variable stepsize backward differentiation formulas in a jacobian free newton krylov framework to advance the solution in time a key advantage of this approach is the elimination of any requirement to form the dense matrix representation of the fractional laplacian operator we show how a banded approximation to this matrix which can be formed and factorised efficiently can be used as part of an effective preconditioner that accelerates convergence of the krylov subspace iterative solver our approach also captures the full contribution from the nonlinear reaction term in the preconditioner which is crucial for problems that exhibit stiff reactions numerical examples are presented to illustrate the overall effectiveness of the solver

on the gaussianity of kolmogorov complexity of mixing sequences it has been proved that for all stationary and ergodic processes the average kolmogorov complexity of the first n observations converges almost surely to its shannon s conditional entropy this paper studies the convergence rate of this asymptotic result in particular we show that if the process satisfies certain mixing conditions then a central limit theorem will be respected furthermore we show that under slightly stronger mixing conditions one may obtain non asymptotic concentration bounds for the kolmogorov complexity

the dimensions of individual strings and sequences a constructive version of hausdorff dimension is developed using constructive supergales which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences this constructive dimension is used to assign every individual infinite binary sequence s a dimension which is a real number dim s in the interval 0 1 sequences that are random in the sense of martin lof have dimension 1 while sequences that are decidable 10 or 10 have dimension 0 it is shown that for every 20 computable real number in 0 1 there is a 20 sequence s such that dim s a discrete version of constructive dimension is also developed using termgales which are supergale like functions that bet on the terminations of finite binary strings as well as on their successive bits this discrete dimension is used to assign each individual string w a dimension which is a nonnegative real number dim w the dimension of a sequence is shown to be the limit inferior of the dimensions of its prefixes the kolmogorov complexity of a string is proven to be the product of its length and its dimension this gives a new characterization of algorithmic information and a new proof of mayordomo s recent theorem stating that the dimension of a sequence is the limit inferior of the average kolmogorov complexity of its first n bits every sequence that is random relative to any computable sequence of coin toss biases that converge to a real number in 0 1 is shown to have dimension h the binary entropy of

from exact sciences to life phenomena following schrodinger and turing on programs life and causality this text presents a survey and a conceptual analysis of a path which goes from programming to physics and biology schrodinger s early reflections on coding and the genome will be a starting point by his and turing s remarks a link is explicitly made between the notion of program and the analysis of causality and determination in physics in particular turing s work in computing and in morphogenesis his 1952 paper on continuous dynamics will be seen as part of a scientific path which goes from laplace s understanding of deterministic predictability to the developments of poincare s analysis of unpredictability in non linear systems at the core of turing s 1952 work the relevance of planetary resonance in poincare s three body theorem and its analogies and differences with logical circularities will then be discussed on these grounds some recent technical results will be mentioned relating algorithmic randomness a strong form of logical undecidability and physical deterministic unpredictability this will be a way to approach the issue of resonances and circularities in system biology where these notions have a deeply different nature in spite of some confusion which is often made finally three aspects of the author s and his collaborators recent work in system biology will be surveyed they concern an approach to biological structural stability as extended criticality the structure of time and of biological rhythms and the role of a proper biological observable organization this is described in terms of anti entropy a new notion inspired by a remark by schrodinger

random elements in effective topological spaces with measure following a suggestion of zvonkin and levin we generalize martin lof s definition of infinite random sequences over a finite alphabet via randomness tests to effective topological spaces with a measure we show that under weak computability conditions there is a universal randomness test we prove a theorem on randomness preserving functions which corrects and extends a result by schnorr and apply it to a number of examples in particular we show that a real number is random if and only if it has a random b ary representation for any b 2 we show that many computable continuously differentiable real functions preserve randomness especially all computable analytic functions which are not constant on any open subset of their domain preserve randomness finally we introduce a new randomness concept for subsets of natural numbers which we characterize in terms of random sequences surprisingly it turns out that there are infinite co r e random sets

on the empirical validity of the bayesian method the ideas in kolmogorov s programme for algorithmc substantiation of applications of probability make it possible to define a measure of disagreement between the probability distribution representing the attitude of a coherent individual towards a random experiment and the outcome of the experiment when there is agreement we say that the probability distribution is empirically valid we prove quantitatively that formulae of bayesian statistics transform empirically valid probability distributions into other empirically valid distributions

analysis of two parareal algorithms for time periodic problems the parareal algorithm which permits us to solve evolution problems in a time parallel fashion has created a lot of attention over the past decade the algorithm has its roots in the multiple shooting method for boundary value problems which in the parareal algorithm is applied to initial value problems with a particular coarse approximation of the jacobian matrix it is therefore of interest to formulate parareal type algorithms for time periodic problems which also couple the end of the time interval with the beginning and to analyze their performance in this context we present and analyze two parareal algorithms for time periodic problems one with a periodic coarse problem and one with a nonperiodic coarse problem an interesting advantage of the algorithm with the nonperiodic coarse problem is that no time periodic problems need to be solved during the iteration since on the time subdomains the problems are not time periodic either we prove for both linear and nonlinear problems convergence

a constructive borel cantelli lemma constructing orbits with required statistical properties in the general context of computable metric spaces and computable measures we prove a kind of constructive borel cantelli lemma given a sequence constructive in some way of sets a i with effectively summable measures there are computable points which are not contained in infinitely many a i as a consequence of this we obtain the existence of computable points which follow the typical statistical behavior of a dynamical system they satisfy the birkhoff theorem for a large class of systems having computable invariant measure and a certain logarithmic speed of convergence of birkhoff averages over lipschitz observables this is applied to uniformly hyperbolic systems piecewise expanding maps systems on the interval with an indifferent fixed point and it directly implies the existence of computable numbers which are normal with respect to any base

calibration based empirical probability probability forecasts for a sequence of uncertain events may be compared with the outcomes of those events by means of a natural criterion of empirical validity calibration it is shown that any two sequences of forecasts which both meet this criterion must be in asymptotic agreement these agreed values can then be considered as correct objective probability forecasts for the particular sequence of outcome results obtained however the objective forecasts vary with the extent of the information taken into account when they are formulated we thus obtain a general theory of empirical probability relative to an information base this theory does not require that such probabilities be interpreted in terms of repeated trials of the same event some implications of this theory are discussed

uniform test of algorithmic randomness over a general space the algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution these restrictions seem artificial some progress has been made to extend the theory to arbitrary bernoulli distributions by martin lof and to arbitrary distributions by levin we recall the main ideas and problems of levin s theory and report further progress in the same framework the issues are the following allow non compact spaces like the space of continuous functions underlying the brownian motion the uniform test deficiency of randomness dp x depending both on the outcome x and the measure p should be defined in a general and natural way see which of the old results survive existence of universal tests conservation of randomness expression of tests in terms of description complexity existence of a universal measure expression of mutual information as deficiency of independence the negative of the new randomness test is shown to be a generalization of complexity in continuous spaces we show that the addition theorem survives the paper s main contribution is introducing an appropriate framework for studying these questions and related ones like statistics for a general family of distributions

computational limits to nonparametric estimation for ergodic processes a new negative result for nonparametric distribution estimation of binary ergodic processes is shown the problem of estimation of distribution with any degree of accuracy is studied then it is shown that for any countable class of estimators there is a zero entropy binary ergodic process that is inconsistent with the class of estimators our result is different from other negative results for universal forecasting scheme of ergodic processes

algorithmic randomness and monotone complexity on product space we study algorithmic randomness and monotone complexity on product of the set of infinite binary sequences we explore the following problems monotone complexity on product space lambalgen s theorem for correlated probability classification of random sets by likelihood ratio tests decomposition of complexity and independence and bayesian statistics for individual random sequences formerly lambalgen s theorem for correlated probability is shown under a uniform computability assumption in h takahashi inform compt 2008 in this paper we show the theorem without the assumption

bayesianism an algorithmic analysis abstract the bayesian program in statistics starts from the assumption that an individual can always ascribe a definite probability to any event it will be demonstrated that this assumption is incompatible with the natural requirement that the individual s subjective probability distribution should be computable we shall construct a probabilistic algorithm producing with probability extremely close to 1 an infinite binary sequence which is not random with respect to any computable probability distribution we use dawid s notion of randomness computable calibration but the results hold for other widely known notions of randomness as well since the bayesian knows the algorithm he must believe that this sequence will be noncalibrable on the other hand it seems that the bayesian must believe that the sequence is random with respect to his own probability distribution we hope that the discussion of this apparent paradox will clarify the foundations of bayesian statistics we analyse also the time of computation and the place of losing randomness we show that we need only polynomial time and space to demonstrate non calibration effects on finite sequences

a finite difference scheme for option pricing in jump diffusion and exponential levy models we present a finite difference method for solving parabolic partial integro differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a levy process or more generally a time inhomogeneous jump diffusion process we discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the levy measure we propose an explicit implicit finite difference scheme which can be used to price european and barrier options in such models we study stability and convergence of the scheme proposed and under additional conditions provide estimates on the rate of convergence numerical tests are performed with smooth and nonsmooth initial conditions

a characterization of certain shimura curves in the moduli stack of abelian varieties let f x y be a semistable family of complex abelian varieties over a curve y of genus g y and smooth over the com

on calibration error of randomized forecasting algorithms it has been recently shown that calibration with an error less than d 0 is almost surely guaranteed with a randomized forecasting algorithm where forecasts are obtained by random rounding the deterministic forecasts up to d we show that this error cannot be improved for a vast majority of sequences we prove that using a probabilistic algorithm we can effectively generate with probability close to one a sequence resistant to any randomized rounding forecasting with an error much smaller than d we also reformulate this result by means of a probabilistic game

an asymptotic parallel in time method for highly oscillatory pdes we present a new time stepping algorithm for nonlinear pdes that exhibit scale separation in time of a highly oscillatory nature the algorithm combines the parareal method a parallel in time scheme introduced in j l lions y maday and g turinici c r acad sci paris ser i math 332 2001 pp 661 668 with techniques from the heterogeneous multiscale method cf w e and b engquist notices amer math soc 50 2003 pp 1062 1070 which make use of the slow asymptotic structure of the equations a j majda and p embid theoret comput fluid dyn 11 1998 pp 155 169 we present error bounds based on the analysis in m j gander and e hairer in domain decomposition methods in science and engineering xvii springer berlin 2008 pp 45 56 and g bal in domain decomposition methods in science and engineering springer berlin 2005 pp 425 432 that demonstrate convergence of the method a complexity analysis also demonstrates that the parallel speedup increases

an efficient parareal algorithm for a class of time dependent problems with fractional laplacian time dependent diffusion equations with fractional laplacian have received considerable attention in recent years for which numerical methods play an important role because a simple and analytic solution is often unavailable we analyze in this paper a parareal algorithm for this kind of problem which realizes parallel in time computation the algorithm is iterative and uses the 3rd order sdirk singly diagonally implicit runge kutta method with a small step size t as the f propagator and the implicit explicit euler method with a large step size t as the g propagator the two step sizes satisfy t t j with j 2 being an integer using the implicit explicit euler method as the g propagator potentially improves the parallel efficiency but complicates the convergence analysis by employing some technical analysis we provide a sharp estimate of the convergence rate which is independent of the mesh ratio j and the distribution of the eigenvalues of the coefficient matrix an extension of the results to problems with time periodic conditions is also given several numerical experiments are carried out to verify the theoretical results

computability on computable metric spaces in previous papers e g weihrauch 1987 type 2 theory of effectivity tte has been shown to be a very general and powerful computer oriented theory for studying effectivity in mathematics in this contribution computability on certain computable separable metric spaces is studied in detail by applying the framework of tte computationally admissible representations of metric spaces are introduced after showing that four different effective representations are computationally equivalent for extending computability to the set of continuous functions several effective namings which are related to definitions of continuity are introduced and compared the definitions are used to prove effective g extension theorems for continuous functions and an effective g characterization of the domains of strongly continous functions which generalize the known properties of real functions kreitz 1984

mathematical metaphysics of randomness abstract we consider various mathematical refinements of the notion of randomness of an infinite sequence and relations between them on the base of the game approach a new possible refinement of the notion of randomness is introduced unpredictability the case of finite sequences is also considered

efficient solution of two sided nonlinear space fractional diffusion equations using fast poisson preconditioners we develop a fast poisson preconditioner for the efficient numerical solution of a class of two sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines using the shifted gr unwald finite difference formulas to approximate the two sided i e the left and right riemann liouville fractional derivatives the resulting semi discrete nonlinear systems have dense jacobian matrices owing to the non local property of fractional derivatives we employ a modern initial value problem solver utilising backward differentiation formulas and jacobian free newton krylov methods to solve these systems for efficient performance of the jacobianfree newton krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver the key contribution of our work is to generalise the fast poisson preconditioner widely used for integer order diffusion equations so that it applies to the two sided space fractional diffusion equation a number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy