Geophysical inverse problems typically involve a trade off between data misfit and some prior. Pareto curves trace the optimal trade off between these two competing aims. These curves are commonly used in problems with two-norm priors where they are plotted on a log-log scale and are known as L-curves. For other priors, such as the sparsity-promoting one norm, Pareto curves remain relatively un...

We develop and analyze a variant of Nesterov’s accelerated gradient descent (AGD) for minimization of smooth non-convex functions. We prove that one of two cases occurs: either our AGD variant converges quickly, as if the function was convex, or we produce a certificate that the function is “guilty” of being non-convex. This non-convexity certificate allows us to exploit negative curvature and ...

A secondary structure is a planar, labeled graph on the vertex set {1; : : : ; n} having two kind of edges: the segments [i; i+1], for 16 i6 n− 1 and arcs in the upper half-plane connecting some vertices i; j, i6 j, where j− i ¿ l, for some :xed integer l. Any two arcs must be totally disjoint. We enumerate secondary structures with respect to their size n, rank l and order k (number of arcs), ...

This paper studies an extended Bessel function of the form [Formula: see text] Representation formulations for [Formula: see text] are derived in terms of the parameters a, b, and p. An important consequence is the derivation of an [Formula: see text]-order differential equation satisfied by the function [Formula: see text]. Interesting functional inequalities are established, particularly for ...

We generalize a well-known result of L. Caffarelli on Lipschitz estimates for optimal transportation T between uniformly log-concave probability measures. Let T : R → R be an optimal transportation pushing forward μ = e dx to ν = e dx. Assume that 1) the second differential quotient of V can be estimated from above by a power function, 2) modulus of convexity of W can be estimated from below by...

‎In this paper‎, ‎some properties of pre-monotone operators are proved‎. ‎It is shown that in a reflexive Banach space‎, ‎a full domain multivalued $sigma$-monotone operator with sequentially norm$times$weak$^*$ closed graph is norm$times$weak$^*$ upper semicontinuous‎. ‎The notion of $sigma$-convexity is introduced and the‎ ‎relations between the $sigma$-monotonicity and $sigma$-convexity is i...

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this paper, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a convex combination of the potential functions when the underlying manifold is the sphere and the cost function is the distance squared. The proof uses the n...

Geophysical inverse problems typically involve a tradeoff between data misfit and some prior model. Pareto curves trace the optimal trade-off between these two competing aims. These curves are used commonly in problems with two-norm priors in which they are plotted on a log-log scale and are known as L-curves. For other priors, such as the sparsity-promoting one-norm prior, Pareto curves remain...

in this paper, we study the convexity of the integral operator

We consider the problem of optimizing an approximately convex function over a bounded convex set in Rn using only function evaluations. The problem is reduced to sampling from an approximately log-concave distribution using the Hit-and-Run method, which is shown to have the same O∗ complexity as sampling from log-concave distributions. In addition to extend the analysis for log-concave distribu...