We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

On robust estimation of the location parameter

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging

IEEE Transactions on Pattern Analysis and Machine Intelligence

/pdf/a-multiscale-finite-volume-method-for-maxwell-s-equations-at-3nfehxo2pd.pdf

A multiscale finite volume method for Maxwell's equations at low frequencies

In this work we establish the relation between optimal control and training deep Convolution Neural Networks (CNNs). We show that the forward propagation in CNNs can be interpreted as a time-dependent nonlinear differential equation and learning as controlling the parameters of the differential equation such that the network approximates the data-label relation for given training data. Using this continuous interpretation we derive two new methods to scale CNNs with respect to two different dimensions. The first class of multiscale methods connects low-resolution and high-resolution data through prolongation and restriction of CNN parameters. We demonstrate that this enables classifying high-resolution images using CNNs trained with low-resolution images and vice versa and warm-starting the learning process. The second class of multiscale methods connects shallow and deep networks and leads to new training strategies that gradually increase the depths of the CNN while re-using parameters for initializations.

Learning Across Scales - Multiscale Methods for Convolution Neural Networks.

Estimating parameters of partial differential equations (PDEs) from noisy and indirect measurements often requires solving ill-posed inverse problems. These so-called parameter estimation or inverse medium problems arise in a variety of applications such as geophysical and medical imaging and nondestructive testing. Their solution is computationally intense since the underlying PDEs need to be solved numerous times until the reconstruction of the parameters is sufficiently accurate. Typically, the computational demand grows significantly when more measurements are available, which poses severe challenges to inversion algorithms as measurement devices become more powerful. In this paper we present jInv, a flexible framework and open source software that provides parallel algorithms for solving parameter estimation problems with many measurements. Being written in the expressive programming language Julia, jInv is portable, easy to understand and extend, cross-platform tested, and well documented. It provid...

jInv--a Flexible Julia Package for PDE Parameter Estimation

Mean field games (MFG) and mean field control (MFC) are critical classes of multi-agent models for efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse-of-dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using an Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that were beyond reach with existing numerical methods.

A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems

We present an efficient solver for diffeomorphic image registration problems in the framework of large deformation diffeomorphic metric mapping (LDDMM). We use an optimal control formulation, in which the velocity field of a hyperbolic PDE needs to be found such that the distance between the final state of the system (the transformed/transported template image) and the observation (the reference image) is minimized. Our solver supports both stationary and nonstationary (i.e., transient or time-dependent) velocity fields. As transformation models, we consider both the transport equation (assuming intensities are preserved during the deformation) and the continuity equation (assuming masses are preserved). We consider the reduced form of the optimal control problem and solve the resulting unconstrained optimization problem using a discretize-then-optimize approach. A key contribution is the elimination of the PDE constraint using a Lagrangian hyperbolic PDE solver. Lagrangian methods rely on the concept of ...

/pdf/a-lagrangian-gauss-newton-krylov-solver-for-mass-and-5c9li4k1yr.pdf

Lars Ruthotto

Papers

A multiscale finite volume method for Maxwell's equations at low frequencies

Learning Across Scales - Multiscale Methods for Convolution Neural Networks.

jInv--a Flexible Julia Package for PDE Parameter Estimation

A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems

A lagrangian gauss-newton-krylov solver for mass- and intensity-preserving diffeomorphic image registration.