Multi-scale uncertainty quantification in geostatistical seismic inversion

Geostatistical seismic inversion is commonly used to infer the spatial distribution of the subsurface petro-elastic properties by perturbing the model parameter space through iterative stochastic sequential simulations/co-simulations. The spatial uncertainty of the inferred petro-elastic properties is represented with the updated a posteriori variance from an ensemble of the simulated realizations. Within this setting, the large-scale geological (metaparameters) used to generate the petro-elastic realizations, such as the spatial correlation model and the global a priori distribution of the properties of interest, are assumed to be known and stationary for the entire inversion domain. This assumption leads to underestimation of the uncertainty associated with the inverted models. We propose a practical framework to quantify uncertainty of the large-scale geological parameters in seismic inversion. The framework couples geostatistical seismic inversion with a stochastic adaptive sampling and Bayesian inference of the metaparameters to provide a more accurate and realistic prediction of uncertainty not restricted by heavy assumptions on large-scale geological parameters. The proposed framework is illustrated with both synthetic and real case studies. The results show the ability retrieve more reliable acoustic impedance models with a more adequate uncertainty spread when compared with conventional geostatistical seismic inversion techniques. The proposed approach separately account for geological uncertainty at large-scale (metaparameters) and local scale (trace-by-trace inversion)

Existence and regularity of solutions to evolutionary problems in perfect plasticity

In this work we develop a rigorous mathematical analysis of variational problems describing the quasistatic evolutionary problems in plasticity. The common feature of the problems under consideration is the variational energy formulation, where the mathematical difficulties arise due to the presence of a term with a linear growth in the symmetric part of the gradient of the unknown vector-valued functions in a volumetric case, or on the Hessian of the unknown scalar function in two-dimensional problems for plates. In the applications these functions represent displacement fields of a body..

pp-adic Zeros of Systems of Quadratic Forms

This survey describes work on the number of variables required to ensure that a system of r quadratic forms over the p-adics has a non-trivial common zero

Quasidifferentiable Functions: Necessary Conditions and Descent Directions

In this paper the author studies the necessary conditions for an extremum when either the function to be optimized or the function describing the set on which optimization must be carried out is nondifferentiable. The author's main concern is with quasidifferentiable functions but smooth and convex cases are also discussed

Directional Differentiability of a Continual Maximum Function of Quasidifferentiable Functions

Much recent work in optimization theory has been concerned with the problems caused by nondifferentiability. Some of these problems have now been at least partially overcome by the definition of a new class of nondifferentiable functions called quasidifferentiable functions, and the extension of classical differential calculus to deal with this class of functions. This has led to increased theoretical research in the properties of quasidifferentiable functions and their behavior under different conditions. In this paper, the problem of the directional differentiability of a maximum function over a continual set of quasidifferentiable functions is discussed. It is shown that, in general, the operation of taking the "continual" maximum (or minimum) leads to a function which is itself not necessarily quasidifferentiable

A Directional Implicit Theorem for Quasidifferentiable Functions

The implicit and inverse function theorems of classical differential calculus represent an essential element in the structure of the calculus. In this paper the authors consider problems related to deriving analogous theorems in quasidifferential calculus

Tight concentration of star saturation number in random graphs

For given graphs $F$ and $G$, the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$ is called the $F$-saturation number and denoted $\mathrm{sat}(G, F)$. For the star $F=K_{1,r}$, the asymptotics of $\mathrm{sat}(G(n,p),F)$ is known. We prove a sharper result: whp $\mathrm{sat}(G(n,p), K_{1,r})$ is concentrated in a set of 2 consecutive points