Push-forward Measure Research Articles

Data-driven uncertainty quantification for predictive flow and transport modeling using support vector machines

Specification of hydraulic conductivity as a model parameter in groundwater flow and transport equations is an essential step in predictive simulations. It is often infeasible in practice to characterize this model parameter at all points in space due to complex hydrogeological environments leading to significant parameter uncertainties. Quantifying these uncertainties requires the formulation and solution of an inverse problem using data corresponding to observable model responses. Several types of inverse problems may be formulated under various physical and statistical assumptions on the model parameters, model response, and the data. Solutions to most types of inverse problems require large numbers of model evaluations. In this study, we incorporate the use of surrogate models based on support vector machines to increase the number of samples used in approximating a solution to an inverse problem at a relatively low computational cost. To test the global capabilities of this type of surrogate model for quantifying uncertainties, we use a framework rooted in measure theory for constructing pullback and push-forward probability measures to study the data-to-parameter-to-prediction propagation of uncertainties under minimal statistical assumptions. Additionally, we demonstrate that it is possible to build a support vector machine using relatively low-dimensional representations of the hydraulic conductivity to propagate distributions. The numerical examples further demonstrate that we can make reliable probabilistic predictions of contaminant concentration at spatial locations.

Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems

We formulate, and present a numerical method for solving, an inverse problem for inferring parameters of a deterministic model from stochastic observational data on quantities of interest. The solution, given as a probability measure, is derived using a Bayesian updating approach for measurable maps that finds a posterior probability measure that when propagated through the deterministic model produces a push-forward measure that exactly matches the observed probability measure on the data. Our approach for finding such posterior measures, which we call consistent Bayesian inference or push-forward based inference, is simple and only requires the computation of the push-forward probability measure induced by the combination of a prior probability measure and the deterministic model. We establish existence and uniqueness of observation-consistent posteriors and present both stability and error analyses. We also discuss the relationships between consistent Bayesian inference, classical/statistical Bayesian inference, and a recently developed measure-theoretic approach for inference. Finally, analytical and numerical results are presented to highlight certain properties of the consistent Bayesian approach and the differences between this approach and the two aforementioned alternatives for inference.

The Order Properties and Karcher Barycenters of Probability Measures on the Open Convex Cone

We study the probability measures on the open convex cone of positive definite operators equipped with the Loewner ordering. We show that two crucial push-forward measures derived by the congruence transformation and power map preserve the stochastic order for probability measures. By the continuity of two push-forward measures with respect to the Wasserstein distance, we verify several interesting properties of the Karcher barycenter for probability measures with finite first moment such as the invariant properties and the inequality for unitarily invariant norms. Moreover, the characterization for the stochastic order of uniformly distributed probability measures has been shown.

The symplectic geometry of closed equilateral random walks in 3-space

A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with $n$ edges is the $(2n-3)$-dimensional Riemannian manifold of equilateral closed polygons in $\mathbb{R}^3$. We study closed random walks using the symplectic geometry of the $(2n-6)$-dimensional quotient of the manifold of polygons by the action of the rotation group $\operatorname {SO}(3)$. The basic objects of study are the moment maps on equilateral random polygon space given by the lengths of any $(n-3)$-tuple of nonintersecting diagonals. The Atiyah-Guillemin-Sternberg theorem shows that the image of such a moment map is a convex polytope in $(n-3)$-dimensional space, while the Duistermaat-Heckman theorem shows that the pushforward measure on this polytope is Lebesgue measure on $\mathbb{R}^{n-3}$. Together, these theorems allow us to define a measure-preserving set of "action-angle" coordinates on the space of closed equilateral polygons. The new coordinate system allows us to make explicit computations of exact expectations for total curvature and for some chord lengths of closed (and confined) equilateral random walks, to give statistical criteria for sampling algorithms on the space of polygons and to prove that the probability that a randomly chosen equilateral hexagon is unknotted is at least $\frac{1}{2}$. We then use our methods to construct a new Markov chain sampling algorithm for equilateral closed polygons, with a simple modification to sample (rooted) confined equilateral closed polygons. We prove rigorously that our algorithm converges geometrically to the standard measure on the space of closed random walks, give a theory of error estimators for Markov chain Monte Carlo integration using our method and analyze the performance of our method. Our methods also apply to open random walks in certain types of confinement, and in general to walks with arbitrary (fixed) edgelengths as well as equilateral walks.

Escape rates and singular limiting distributions for intermittent maps with holes

We study the escape dynamics in the presence of a hole of a standard family of intermittent maps of the unit interval with neutral fixed point at the origin (and finite absolutely continuous invariant measure). Provided that the hole (is a cylinder that) does not contain any neighborhood of the origin, the surviving volume is shown to decay at polynomial speed with time. The associated polynomial escape rate depends on the density of the initial distribution, more precisely, on its behavior in the vicinity of the origin. Moreover, the associated normalized push forward measures are proved to converge to the point mass supported at the origin, in sharp contrast to systems with exponential escape rate. Finally, a similar result is obtained for more general systems with subexponential escape rates; namely that the Cesaro limit of normalized push forward measures is generally singular, invariant and supported on the asymptotic survivor set.

The Filippov characteristic flow for the aggregation equation with mildly singular potentials

Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (2011) [17], a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being λ-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the λ-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms.

The interplay of classes of algorithmically random objects

We study algorithmically random closed subsets of $\cs$\!, algorithmically random continuous functions from $\cs$ to $\cs$\!, and algorithmically random Borel probability measures on $\cs$, especially the interplay between these three classes of objects. Our main tools are preservation of randomness and its converse, the no randomness ex nihilo principle, which say together that given an almost-everywhere defined computable map between an effectively compact probability space and an effective Polish space, a real is Martin-Lof random for the pushforward measure if and only if its preimage is random with respect to the measure on the domain. These tools allow us to prove new facts, some of which answer previously open questions, and reprove some known results more simply. Our main results are the following. First we answer an open question in \cite{Barmpalias:2008aa} by showing that $\X\subseteq\cs$\! is a random closed set if and only if it is the set of zeros of a random continuous function on $\cs$\!. As a corollary we obtain the result that the collection of random continuous functions on $\cs$ is not closed under composition. Next, we construct a computable measure $Q$ on the space of measures on $\cs$ such that $\X\subseteq\cs$ is a random closed set if and only if $\X$ is the support of a $Q$-random measure. We also establish a correspondence between random closed sets and the random measures studied in \cite{Culver:2014aa}. Lastly, we study the ranges of random continuous functions, showing that the Lebesgue measure of the range of a random continuous function is always contained in $(0,1)$.

Push-Forward Measures on Configuration Spaces: Integration by Parts and Log-Sobolev Inequality

We study a class of measures on the space \(\Gamma _{X}\) of locally finite configurations in \(X=\mathbb {R}^{d}\), obtained as images of “lattice” Gibbs measures on \(X^{\mathbb {Z} ^{d}}\) with respect to an embedding \(\mathbb {Z}^{d}\subset \mathbb {R}^{d}\). For these measures, we prove the integration by parts formula and log-Sobolev inequality.

Push forward measures and concentration phenomena

In this note we study how a concentration phenomenon can be transferred from one measure μ to a push‐forward measure ν. In the first part, we push forward μ by , where , and obtain a concentration inequality in terms of the medians of the given norms (with respect to μ) and the Banach‐Mazur distance between them. This approach is finer than simply bounding the concentration of the push forward measure in terms of the Banach‐Mazur distance between K and L. As a consequence we show that any normed probability space with exponential type concentration is far (even in an average sense) from subspaces of . The sharpness of this result is shown by considering the spaces.

Properties of convergence in Dirichlet structures

In univariate settings, we prove a strong reinforcement of the energy image density criterion for local Dirichlet forms admitting square field operators. This criterion enables us to redemonstrate some classical results of Dirichlet forms theory (Ancona, 1976 [2]). Besides, when X=(X1,…,Xp) belongs to the D domain of the Dirichlet form, and when its square field operator matrix Γ[X,tX] is almost surely definite, we prove that LX is a Rajchman measure. This is the first result in full generality in the direction of Bouleau–Hirsch conjecture. Moreover, in multivariate settings, we study the particular case of Sobolev spaces: we prove that a convergence for the Sobolev norm W1,p(Rd,Rp) toward a non-degenerate limit, entails convergence of push-forward measures in the total variation topology. Our proofs are based on a new kind of integration by parts which is of independent interest.

Recurrence rates for observations of flows

AbstractWe study Poincaré recurrence for flows and observations of flows. For Anosov flow, we prove that the recurrence rates are linked to the local dimension of the invariant measure. More generally, we give for the recurrence rates for the observations an upper bound depending on the push-forward measure. When the flow is metrically isomorphic to a suspension flow for which the dynamic on the base is rapidly mixing, we prove the existence of a lower bound for the recurrence rates for the observations. We apply these results to the geodesic flow and we compute the recurrence rates for a particular observation of the geodesic flow, i.e. the projection on the manifold.

Ergodic actions of mapping class groups on moduli spaces of representations of non-orientable surfaces

Let M be a non-orientable surface with Euler characteristic χ(M) ≤ −2. We consider the moduli space of flat SU(2)-connections, or equivalently the space of conjugacy classes of representations $$\mathfrak{X} (M) = {\rm Hom} (\pi_1 (M), {\rm SU} (2)) / {\rm SU} (2).$$There is a natural action of the mapping class group of M on \({\mathfrak{X} (M)}\). We show here that this action is ergodic with respect to a natural measure. This measure is defined using the push-forward measure associated to a map defined by the presentation of the surface group. This result is an extension of earlier results of Goldman for orientable surfaces (see [8]).

The Equivariant Cohomology Theory of Twisted Generalized Complex Manifolds

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kahler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the \({\overline{\partial} \partial}\)-lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized Kahler condition, we prove the Kirwan injectivity in this setting. We then consider the Hamiltonian action of a torus on an H-twisted generalized Calabi-Yau manifold and extend to this case the Duistermaat-Heckman theorem for the push-forward measure.

Brown measures of sets of commuting operators in a type II 1 factor

Using the spectral subspaces obtained in [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II 1-factor, preprint, 2005], Brown's results (cf. [L.G. Brown, Lidskii's theorem in the type II case, in: H. Araki, E. Effros (Eds.), Geometric Methods in Operator Algebras, Kyoto, 1983, in: Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., 1986, pp. 1–35]) on the Brown measure of an operator in a type II 1 factor ( M , τ ) are generalized to finite sets of commuting operators in M . It is shown that whenever T 1 , … , T n ∈ M are mutually commuting operators, there exists one and only one compactly supported Borel probability measure μ T 1 , … , T n on B ( C n ) such that for all α 1 , … , α n ∈ C , τ ( log | α 1 T 1 + ⋯ + α n T n − 1 | ) = ∫ C n log | α 1 z 1 + ⋯ + α n z n − 1 | d μ T 1 , … , T n ( z 1 , … , z n ) . Moreover, for every polynomial q in n commuting variables, μ q ( T 1 , … , T n ) is the push-forward measure of μ T 1 , … , T n via the map q : C n → C . In addition it is shown that, as in [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II 1-factor, preprint, 2005], for every Borel set B ⊆ C n there is a maximal closed T 1 - , … , T n -invariant subspace K affiliated with M , such that μ T 1 | K , … , T n | K is concentrated on B. Moreover, τ ( P K ) = μ T 1 , … , T n ( B ) . This generalizes the main result from [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II 1-factor, preprint, 2005] to n-tuples of commuting operators in M .

Heat kernels, symplectic geometry, moduli spaces, and finite groups

In this note we apply heat kernels to derive some localization formula in sympletcic geometry, to study moduli spaces of flat connections on a Riemann surface, to obtain the push-forward measures for certain maps between Lie groups and to solve equations in finite groups.

Logarithmic convexity of push-forward measures

Let M be a compact symplectic manifold with a Hamiltonian action o f a compact torus T, and moment map q5 : M --~ t*. The push-forward by q5 of the symplectic measure on M equals Lebesgue measure on q~(M) multiplied by an "interpolating function" f . These interpolating functions are important because if M is a smooth projective variety, and the symplectic form on M represents the first Chern class o f a T-equivariant ample line bundle L -~ M, then the interpolating function f describes the asymptotic behavior of the multiplicities o f the representation of T on H~ | ([He], [GS2]). One striking property of the moment map is that its image qS(M) is a convex polytope in t* ([A], [GS1]). The purpose of this paper is to show that if the symplectic form is K~ihler, then the interpolating function also possesses a certain convexity property. Recall that a continuous function h on a convex set C is called convex if

Logarithmic convexity of push-forward measures

Logarithmic convexity of push-forward measures