Multilevel model

Summary

Multilevel models (also known as hierarchical linear models, linear mixed-effect model, mixed models, nested data models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parameters that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models (in particular, linear regression), although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available. Multilevel models are particularly appropriate for research designs where data for participants are organized at more than one level (i.e., nested data). The units of analysis are usually individuals (at a lower level) who are nested within contextual/a

Official source

https://en.wikipedia.org/wiki/Multilevel_model

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Multilevel tensor approximation of PDEs with random data

Jonas Ballani, Daniel Kressner, Michael Dieter Peters

In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximation, we make use of a multilevel framework in which we consider the differences of the solution on two consecutive finite element levels at the collocation points. We then address the approximation of these high-dimensional differences by adaptive low-rank tensor techniques. This allows to equilibrate the error on all levels by exploiting regularity and additional low-rank structure of the solution. We arrive at an explicit representation in a low-rank tensor format of the approximate solution on the entire parameter domain, which can be used for, e.g., the direct and cheap computation of statistics. Numerical results are provided in order to illustrate the approach.

Springer2017

MATHICSE Technical Report : Multilevel tensor approximation of PDEs with random data

Jonas Ballani, Daniel Kressner, Michael Dieter Peters

In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hier-archy of finite element discretizations for the spatial approximation, we make use of a multilevel framework in which we consider the differences of the solution on two consecutive finite element levels in the collocation points. We then address the approximation ofthese high-dimensional differences by adaptive low-rank tensor techniques. This allows to equilibrate the error on all levels by exploiting analytic and algebraic properties of thesolution at the same time. We arrive at an explicit representation in a low-rank tensor format of the approximate solution on the entire parameter domain, which can be used for, e.g., the direct and cheap computation of statistics. Numerical results are provided in order to illustrate the approach.

MATHICSE2016

A multi-level model of human actions involving full body motion is proposed. Actions are considered as the combination of action primitives at different levels: motions of the center of mass, end effectors, and final postures of the virtual skeleton. The model is especially designed for fast action recognition from the input of the human motion at the joint level

1997