We propose a family of Markov chain Monte Carlo methods whose performance is unaffected by affine tranformations of space. These algorithms are easy to construct and require little or no additional computational overhead. They should be particularly useful for sampling badly scaled distributions. Computational tests show that the affine invariant methods can be significantly faster than standard MCMC methods on highly skewed distributions.

/pdf/ensemble-samplers-with-affine-invariance-4t8yddl1e2.pdf

Ensemble samplers with affine invariance

Since its popularization in the late 1970s, Sequential Quadratic Programming (SQP) has arguably become the most successful method for solving nonlinearly constrained optimization problems. As with most optimization methods, SQP is not a single algorithm, but rather a conceptual method from which numerous specific algorithms have evolved. Backed by a solid theoretical and computational foundation, both commercial and public-domain SQP algorithms have been developed and used to solve a remarkably large set of important practical problems. Recently large-scale versions have been devised and tested with promising results.

Sequential Quadratic Programming

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations

Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle

The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.

Dynamics and stability of thin liquid films

Nonlinear asymptotic stability of viscous shock profiles for conservation laws

Multi-grid Monte Carlo (II). Two-dimensional XY model

We prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions. The proof uses the "integrated equation" and an effective equation for the motion of the front itself. We derive energy estimates that balance terms from the integrated equation with terms from the front motion equation. In this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions. These multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts (Lu) and in shock waves with phase changes. The proof here is a rigorous version of arguments that are used to derive effective equations (such as the Kuramoto-Sivashinsky equation) to describe the behavior of fronts. The one dimensional stability for scalar conservation laws was proven by II ' in and Oleinik (IO) using the "integrated equation" (see below) and a max- imum principle. Another proof, based on weighted norms and spectral theory for the linearized problem, was given by Sattinger (S). The multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by Kawashima and Mat- sumura (KM) and Goodman (Go) and completed by Liu (Li). These proofs use L2 energy estimates for the integrated equation. We consider equations of the form (!) ut + f(u)x + g(u)y=:uxx + uyy, where f(u) is a strictly convex function of u :

/pdf/stability-of-viscous-scalar-shock-fronts-in-several-3gp7nhk2m9.pdf

Stability of viscous scalar shock fronts in several dimensions

We describe and analyze some Monte Carlo methods for manifolds in Euclidean space defined by equality and inequality constraints. First, we give an MCMC sampler for probability distributions defined by un-normalized densities on such manifolds. The sampler uses a specific orthogonal projection to the surface that requires only information about the tangent space to the manifold, obtainable from first derivatives of the constraint functions, hence avoiding the need for curvature information or second derivatives. Second, we use the sampler to develop a multi-stage algorithm to compute integrals over such manifolds. We provide single-run error estimates that avoid the need for multiple independent runs. Computational experiments on various test problems show that the algorithms and error estimates work in practice. The method is applied to compute the entropies of different sticky hard sphere systems. These predict the temperature or interaction energy at which loops of hard sticky spheres become preferable to chains.

Monte Carlo on Manifolds: Sampling Densities and Integrating Functions

We derive a quadratically convergent algorithm for minimizing a nonlinear function subject to nonlinear equality constraints. We show, following Kaufman [4], how to compute efficiently the derivative of a basis of the subspace tangent to the feasible surface. The derivation minimizes the use of Lagrange multipliers, producing multiplier estimates as a by-product of other calculations. An extension of Kantorovich's theorem shows that the algorithm maintains quadratic convergence even if the basis of the tangent space changes abruptly from iteration to iteration. The algorithm and its quadratic convergence are known but the drivation is new, simple, and suggests several new modifications of the algorithm.

Newton's method for constrained optimization

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