Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces $(\Omega,\sigma)$ and appropriate spaces of functions inside $L^2(\Omega,\sigma)$. The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups

Failure Probability Estimation and Detection of Failure Surfaces via Adaptive Sequential Decomposition of the Design Domain

We propose an algorithm for an optimal adaptive selection of points from the design domain of input random variables that are needed for an accurate estimation of failure probability and the determination of the boundary between safe and failure domains. The method is particularly useful when each evaluation of the performance function g(x) is very expensive and the function can be characterized as either highly nonlinear, noisy, or even discrete-state (e.g., binary). In such cases, only a limited number of calls is feasible, and gradients of g(x) cannot be used. The input design domain is progressively segmented by expanding and adaptively refining mesh-like lock-free geometrical structure. The proposed triangulation-based approach effectively combines the features of simulation and approximation methods. The algorithm performs two independent tasks: (i) the estimation of probabilities through an ingenious combination of deterministic cubature rules and the application of the divergence theorem and (ii) the sequential extension of the experimental design with new points. The sequential selection of points from the design domain for future evaluation of g(x) is carried out through a new learning function, which maximizes instantaneous information gain in terms of the probability classification that corresponds to the local region. The extension may be halted at any time, e.g., when sufficiently accurate estimations are obtained. Due to the use of the exact geometric representation in the input domain, the algorithm is most effective for problems of a low dimension, not exceeding eight. The method can handle random vectors with correlated non-Gaussian marginals. The estimation accuracy can be improved by employing a smooth surrogate model. Finally, we define new factors of global sensitivity to failure based on the entire failure surface weighted by the density of the input random vector.Comment: 42 pages, 24 figure

Rough Path Perspectives on the Itô-Stratonovich Dilemma

This thesis is comprised of six distinct research projects which share the theme of rough and stochastic integration theory. Chapter 1 deals with the problem of approximating an SDE X in R^d with one Y defined on a specified submanifold, so as to minimise quantities such as E[|Y_t − X_t|^2] for small t: this is seen to be best performed when using Itô instead of Stratonovich calculus. Chapter 2 develops the theory of not necessarily geometric 3 > p-rough paths on manifolds. Drawing on [FH14, É89, É90] we define controlled rough integration and RDEs both in the local and extrinsic framework, with the latter generalising [CDL15]. Finally, we lay out the theory of parallel transport and Cartan development, for which non-geometricity results in second-order conditions and corrections to the classical formulae. In Chapter 3 we treat the theory of geometric rough paths of arbitrary roughness in the framework of controlled paths of [Gub04], from an algebraic and combinatorial point of view, and avoiding the smooth approximation arguments used in [FV10b]. As an application, we show how our emphasis on functoriality allows for a simple transposition of the theory to the manifold setting. The goal of Chapter 4 is to treat the theory of branched rough paths on manifolds. Drawing on [HK15, Kel12], we show how to lift a controlled path to a rough path. The “transfer principle”, intended in the sense of Malliavin and Emery, refers to the expression of a connection-dependent “intrinsic differential” d_∇X that defines integration in a coordinate-invariant manner, which we derive by combining Kelly’s bracket corrections with certain higher-order Christoffel symbols. In reviewing branched rough paths, special attention is given to those that can be defined on Hoffman’s quasi-shuffle algebra [Hof00], for which some of the relations simplify. The final two chapters do not involve any differential geometry. Chapter 5 is a report on work in progress, the aim of which is to compute the Wiener chaos decomposition (and in particular the expectation) of the signature of certain multidimensional Gaussian processes such as 1/3 < H-fractional Brownian motion (fBm). This generalises the results of [BC07], arrived at through a piecewise-linear approximation argument which fails when 1/4 < H ≤ 1/2. Furthermore, our calculation restricts to that of [Bau04] in the case of Brownian motion, and can be applied to other semimartingales, such as the Brownian bridge. Our novel approach makes use of Malliavin calculus and the recent rough-Skorokhod conversion formula of [CL19]. Finally, in Chapter 6 we combine the topics of the previous two to define a branched rough path above multidimensional 1/4 < H-fBm, and compute its terms and correction terms. Our rough path is defined intrinsically and canonically in terms of the stochastic process, restricts to the Itô rough path when H = 1/2, has the property that its integrals of one-forms vanish in mean, and is not quasi-geometric when H ∈ (1/4, 1/3].Open Acces

Geometric Packings of Non-Spherical Shapes

The focus of this thesis lies on geometric packings of non-spherical shapes in three-dimensional Euclidean space. Since computing the optimal packing density is difficult, we investigate lower and upper bounds for the optimal value. For this, we consider two special kinds of geometric packings: translative packings and lattice packings. We study upper bounds for the optimal packing density of translative packings. These are packings in which just translations and no rotations of the solids are allowed. Cohn and Elkies determined a linear program for the computation of such upper bounds that is defined by infinitely many inequalities optimizing over an infinite dimensional set. We relax this problem to a semidefinite problem with finitely many constraints, since this kind of problem is efficiently solvable in general. In our computation we consider three-dimensional convex bodies with tetrahedral or icosahedral symmetry. To obtain a program that is not too large for current solvers, we use invariant theory of finite pseudo-reflection groups to simplify the constraints. Since we solve this program by using numerical computations, the solutions might be slightly infeasible. Therefore, we verify the obtained solutions to ensure that they can be made feasible for the Cohn-Elkies program. With this approach we find new upper bounds for three-dimensional superballs, which are unit balls for the l^3_p norm, for p ∈ (1, ∞) \ {2} . Furthermore, using our approach, we improve Zong’s recent upper bound for the translative packing density of tetrahedra from 0.3840 . . . to 0.3683... , which is very close to the best known lower bound of 0.3673... The last part of this thesis deals with lattice packings of superballs. Lattice packing sare translative packings in which the centers of the solids form a lattice. Thus, any lattice packing density is in particular a lower bound for the optimal translative packing density. Using a theorem of Minkowski, we compute locally optimal lattice packings for superballs. We obtain lattice packings for p ∈ [1, 8] whose density is at least as high as the density of the currently best lattice packings provided by Jiao, Stillinger, and Torquato. For p ∈ (1, 2)\[log 2 3, 1.6], we even improve these lattice packings. The upper bounds for p ∈ [3, 8], as well as the numerical results for the upper bounds for p ∈ [1, log 2 , 3], are remarkably close to the lower bounds we obtain by these lattice packings