Riemann integral | EPFL Graph Search

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration, or simulated using Monte Carlo Integration. Overview Let f be a non-negative real-valued function on the interval [a, b], and let S be the region of the plane under the graph of the function f and above the interval [a, b]. See the figure on the top right. This region can be expressed in set-builder notation as S = \left { (x, y),: ,a \leq x \leq b,,, 0 < y < f(x) \right }. We are interested in measuring the area of S. Once

Stochastic partial differential equations driven by Lévy white noises

Thomas Marie Jean-Baptiste Humeau

We study various aspects of stochastic partial differential equations driven by Lévy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a generalized random process or as an independently scattered random measure. After unifying these approaches and establishing appropriate stochastic integral representations, we show that a necessary and sufficient condition for a Lévy white noise to have values in the space of tempered Schwartz distributions, is that the underlying Lévy measure have a positive absolute moment. In the case of a linear stochastic partial differential equation with a general differential operator and driven by a symmetric pure jump Lévy white noise, we show that when the mild solution is locally Lebesgue integrable, then it is equal to the generalized solution, and that a random field representation exists for the generalized solution if and only if the fundamental solution of the operator has certain integrability properties. In that case, we show that the random field representation is equal to the mild solution. For this purpose, a new stochastic Fubini theorem is proved. These results are applied to the linear stochastic heat and wave equations driven by a symmetric alpha-stable noise. We then study the non-linear stochastic heat equation driven by a general type of Lévy white noise, possibly with heavy tails and non-summable small jumps. Our framework includes in particular the alpha-stable noise. In the case of the equation on the whole space, we show that the law of the solution that we construct does not depend on the space variable. Then we show in various domains D that the solution u to the stochastic heat equation is such that t -> u(t,·) has a càdlàg version in a fractional Sobolev space of order r < -d /2. Finally, we show that the partial functions have a continuous version under some optimal moment conditions. In the alpha-stable case, we show that for the choices of alpha for which this moment condition is not satisfied, the sample paths of the partial functions are unbounded on any non-empty open subset.

EPFL2017

A Functional Perspective on Information Measures

Amedeo Roberto Esposito

Since the birth of Information Theory, researchers have defined and exploited various information measures, as well as endowed them with operational meanings. Some were born as a "solution to a problem", like Shannon's Entropy and Mutual Information. Others were the fruit of generalisation and the mathematical genius of bright minds like Rényi, Csizsár and Sibson. These powerful objects allow us to manipulate probabilities intuitively and seem always to be somehow connected to concrete settings in communication, coding or estimation theory. A common theme is: take a problem in one of these areas, try to control (upper or lower-bound) the expected value of some function of interest (often, probabilities of error) and, with enough work, an information measure appears as a fundamental limit of the problem. The most striking example of this is in Shannon's seminal paper in 1948: his purpose was to characterise the smallest possible expected length of a uniquely decodable encoding that compresses the realisations of a random variable. As he brilliantly proved, the smallest expected length one can hope for is the Entropy of the random variable. In establishing this connection, another quantity needed to be implicitly controlled: the Kraft's sum of the code. Seemingly unrelated before, these three objects joined forces in harmony to provide a beautiful and fundamental result. But why are they related? The answer seems to be: duality. Duality is an abstract notion commonly used in linear algebra and functional analysis. It has been expanded and generalised over the years. Several incarnations have been discovered throughout mathematics. One particular instance of this involves vector spaces: given two vector spaces and a "duality pairing" one can jump from one space to the other (its dual) through Legendre-Fenchel-like transforms. In the most common settings in Information Theory, the two spaces and the pairing are, respectively: 1) the space of (probability)measures defined on X; 2) the space of bounded functions defined on X; 3) the Lebesgue integral of the function (the expected value of the function if the measure is a probability measure). Once these are set, Legendre-Fenchel-like transforms allow us to connect a) a functional acting on the space described in 1), b) a functional acting on the space described in 2) and the anchor point is c) the (expected) value described in 3).These three pieces (a), b) and c)) represent the actors of many of the results provided in Information Theory. Once they are found, one usually bounds the functional described in b) and obtains a bound connecting the expected value and the functional of measures (e.g., an information measure). Going back to Shannon's result, fixed a random variable (and thus, a probability measure) and selected the function to be the length of a code: the functional a) is the Shannon Entropy of the source; the functional b) is the Kraft sum of the code; the pairing c) is the expected length of the code. We explore this connection and this pattern throughout the thesis. We will see how it can be found in notable results like Coding Theorems for one-to-one codes, Campbell's Coding Theorem, Arikan's Guessing Theorem, Fano-like and Transportation-Cost Inequalities and so on. Moreover, unearthing the pattern allows us to generalise it to other information measures and apply the technique in a variety of fields, including Learning Theory, Estimation Theory and Hypothesis Testing.

EPFL2022

Generalized Flows Satisfying Spatial Boundary Conditions

Boris Buffoni

In a region D in R-2 or R-3, the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by partial derivative(t)v + (v . del(x))v = -del(xP), div(x)v = 0, where v(t, x) is the velocity of the particle located at x is an element of D at time t and p(t, x) is an element of R is the pressure. Solutions v and p to the Euler equation can be obtained by solving {del x {partial derivative t phi(t, x, a) + p(t, x) + (1/2)|del(x)phi(t, x, a)|(2)} = 0 at a = kappa(t, x), v(t, x) = del(x)phi(t, x, a) at a = kappa(t, x), partial derivative(t)kappa(t, x) | (v. del(x))kappa(t, x) = 0 div(x)v(t, x) = 0 where phi : R x D x R-l -> R and kappa : R x D -> R-l are additional unknown mappings (l >= 1 is prescribed). The third equation in the system says that kappa is an element of R-l is convected by the flow and the second one that phi can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition kappa(0, x) = x on D (and thus l = 2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52: 411-452, 1999) in his Eulerian-Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross. D and that carry each "particle" at time t = 0 at a prescribed location at time t = T > 0, that is, kappa(T, x) is prescribed in D for all x is an element of D. We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary partial derivative D of the bounded region D (a two-or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through partial derivative D of particles labelled by each value of. at each point of partial derivative D. One of the main novelties is the introduction of a prescribed "generalized" Bernoulli's function H : R-l -> R, namely, we add to (0.1) the requirement that partial derivative(t)phi(t, x, a) + p(t, x) + (1/2)|del(x)phi(t, x, a)|(2) = H(a) at a = kappa(t, x) with phi, p, kappa periodic in time of prescribed period T > 0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of "Lamb's surfaces" and "isotropic manifolds" in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier's formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional (kappa, v) -> integral(T)(0)integral(D) {1/2|v(t, x)|(2) + H(kappa(t, x))} dtdx defined for kappa and v that are T-periodic in t, such that partial derivative(t)kappa(t, x) + (v . del(x))kappa(t, x) = 0, div(x)v(t, x) = 0, and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize integral(]0, L[x]0,1[) {(1/2)|del psi|(2) + H(psi)}dx for psi is an element of W-1,W-2 (]0, L[x]0,1[) under appropriate boundary conditions, where. is the stream function. For a minimizer, corresponding functions phi and kappa are given in terms of the stream function psi.

Birkhäuser-Verlag, Springer2012