# Difference between revisions of "Applied/ACMS/absF18"

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=== Nan Chen (University of Wisconsin-Madison) === | === Nan Chen (University of Wisconsin-Madison) === | ||

− | '' | + | ''A simple stochastic model for El Nino with westerly wind bursts and the prediction of super El Nino events'' |

Atmospheric wind bursts in the tropics play a key role in the dynamics of the El Nino Southern Oscillation (ENSO). A simple modeling framework is proposed that summarizes this relationship and captures major features of the observational record while remaining physically consistent and amenable to detailed analysis. Within this simple framework, wind burst activity evolves according to a stochastic two-state Markov switching–diffusion process that depends on the strength of the western Pacific warm pool, and is coupled to simple ocean–atmosphere processes that are otherwise deterministic, stable, and linear. A simple model with this parameterization and no additional nonlinearities reproduces a realistic ENSO cycle with intermittent El Nino and La Nina events of varying intensity and strength as well as realistic buildup and shutdown of wind burst activity in the western Pacific. The wind burst activity has a direct causal effect on the ENSO variability: in particular, it intermittently triggers regular El Nino or La Nina events, super El Nino events, or no events at all, which enables the model to capture observed ENSO statistics such as the probability density function and power spectrum of eastern Pacific sea surface temperatures. The present framework is then applied to understand the mechanism of different super El Ninos. In particular, the framework is used to simulate and analyze the two famous super El Nino events in 1997-1998 and 2014-2016, with the conclusion that the delayed super El Nino events in 2014-2016 are not necessarily unusual in the tropical Pacific despite not appearing in the recent observational record and could reoccur in the future. | Atmospheric wind bursts in the tropics play a key role in the dynamics of the El Nino Southern Oscillation (ENSO). A simple modeling framework is proposed that summarizes this relationship and captures major features of the observational record while remaining physically consistent and amenable to detailed analysis. Within this simple framework, wind burst activity evolves according to a stochastic two-state Markov switching–diffusion process that depends on the strength of the western Pacific warm pool, and is coupled to simple ocean–atmosphere processes that are otherwise deterministic, stable, and linear. A simple model with this parameterization and no additional nonlinearities reproduces a realistic ENSO cycle with intermittent El Nino and La Nina events of varying intensity and strength as well as realistic buildup and shutdown of wind burst activity in the western Pacific. The wind burst activity has a direct causal effect on the ENSO variability: in particular, it intermittently triggers regular El Nino or La Nina events, super El Nino events, or no events at all, which enables the model to capture observed ENSO statistics such as the probability density function and power spectrum of eastern Pacific sea surface temperatures. The present framework is then applied to understand the mechanism of different super El Ninos. In particular, the framework is used to simulate and analyze the two famous super El Nino events in 1997-1998 and 2014-2016, with the conclusion that the delayed super El Nino events in 2014-2016 are not necessarily unusual in the tropical Pacific despite not appearing in the recent observational record and could reoccur in the future. | ||

+ | |||

+ | === Sulian Thual (Fudan University) === | ||

+ | ''A Stochastic Skeleton Model for the Madden-Julian Oscillation and El Nino-Southern Oscillation'' | ||

+ | |||

+ | A broad range of random atmospheric disturbances in the tropics may be considered as possible | ||

+ | triggers to the El Niño Southern Oscillation (ENSO), such as for example westerly wind bursts, easterly wind bursts, as well as the convective envelope of the Madden-Julian Oscillation (MJO). Here a simple dynamical stochastic model for the tropical ocean-atmosphere is proposed that captures those processes as well as their multiscale interactions. Realistic features include for the first time altogether the MJO wavenumber-frequency power spectra, eastward propagation, structure and confinement to the warm pool region and similarly for atmospheric Kelvin and Rossby equatorial waves, in addition to the ENSO intermittency, power spectrum and non-Gaussian statistics of sea surface temperatures, among others. | ||

+ | |||

+ | Importantly, intraseasonal atmospheric disturbances such as the MJO are here solved dynamically which renders more explicit their upscale contribution to the interannual flow as well as their modulation in return. First, the background red noise spectrum of atmospheric disturbances rather than their individual characteristics is shown to be most important for the triggering of the ENSO. Second, the onset, strength and demise of El Niño events is linked to the increase and eastward expansion of atmospheric disturbances eastward of the warm pool region. The present framework serves as a prototype for general circulation models that solve similar dynamical interactions on several spatial and temporal scales. | ||

+ | |||

+ | === Matthew Thorpe (Cambridge University) === | ||

+ | ''Continuum Limits of Semi-Supervised Learning on Graphs'' | ||

+ | |||

+ | Given a data set $\{x_i\}_{i=1}^n$ with labels $\{y_i\}_{i=1}^N$ on the first $N$ data points the goal of semi-supervised is to infer labels on the remaining $\{x_i\}_{i=N+1}^n$ data points. In this talk we use a random geometric graph model with connection radius $r(n)$. The framework is to consider objective functions which reward the regularity of the estimator function and impose or reward the agreement with the training data, more specifically we will consider discrete p-Laplacian and fractional Laplacian regularization. | ||

+ | |||

+ | The talk concerns the asymptotic behaviour in the limit where the number of unlabelled points increases while the number of training points remains fixed. The results are to uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. I will give almost optimal ranges on the scaling of $r(n)$ for asymptotic consistency to hold. Furthermore, I will setup the Bayesian interpretation of this problem. | ||

+ | |||

+ | This is joint work with Matt Dunlop (Caltech), Dejan Slepcev (CMU) and Andrew Stuart (Caltech). | ||

+ | |||

+ | === Fei Lu (Johns Hopkins University) === | ||

+ | ''Data-informed stochastic model reduction for complex dynamical systems'' | ||

+ | |||

+ | The need to develop reduced nonlinear statistical-dynamical models from time series of partial observations of complex systems arises in many applications such as geophysics, biology and engineering. The challenges come mainly from memory effects due to the nonlinear interactions between resolved and unresolved scales, and from the difficulty in inference from discrete data. | ||

+ | |||

+ | To address these challenges, we introduce a discrete-time stochastic parametrization framework, in which we infer nonlinear autoregression moving average (NARMA) type models to take the memory effects into account. We show by examples that the NARMA type stochastic reduced models that can capture the key statistical and dynamical properties, and therefore can improve the performance of ensemble prediction in data assimilation. The examples include the Lorenz 96 system (which is a simplified model of the global atmosphere) and the Kuramoto-Sivashinsky equation of spatiotemporally chaotic dynamics. Applications of this inference approach to model reduction for stochastic Burgers equations will be discussed. | ||

Line 23: | Line 47: | ||

We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the ”Quantum Equilibrium-Disequilibrium” model. Our model gives rise to non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, ”classical” GBM scale-invariant dynamics with an exponential asset growth and without defaults are formally recovered from our model. However, we argue that this is only a formal mathematical limit, and in reality the GBM limit is non- analytic due to non-linear effects that produce both defaults and divergence of perturbation theory in a small market friction parameter. | We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the ”Quantum Equilibrium-Disequilibrium” model. Our model gives rise to non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, ”classical” GBM scale-invariant dynamics with an exponential asset growth and without defaults are formally recovered from our model. However, we argue that this is only a formal mathematical limit, and in reality the GBM limit is non- analytic due to non-linear effects that produce both defaults and divergence of perturbation theory in a small market friction parameter. | ||

+ | |||

+ | |||

+ | === Karl Rohe (UW-Madison, statistics) === | ||

+ | |||

+ | ''Making Spectral Graph Theory work in practice. Making the practice work in theory'' | ||

+ | |||

+ | After introducing Cheeger's Inequality and spectral clustering, this talk has two parts. The first part will (1) show how spectral clustering gives "bad results" in many applied settings and (2) illustrate a "hack" that makes it work very well. Most of the talk will be spent on the second part, which will provide a simple theory to provide a deeper understanding of where the bad results come from and why the hack works so well. | ||

+ | |||

+ | There are four pieces to this simple theory. First, sparse and stochastic graphs create a lot of small trees that are connected to the core of the graph by only one edge. Second, graph conductance is sensitive to these noisy "dangling sets." Third, by Cheeger's inequality and an inequality from Ky Fan, spectral clustering inherits this sensitivity. These three pieces explain why spectral clustering gives bad results in practice. The fourth piece uses Cheeger's inequality to show how the hack creates a new form of graph conductance that we call CoreCut. Simple inspection of CoreCut reveals why it is less sensitive to small cuts in the graph. In addition to this statistical benefit, these results also demonstrate why the hack also improves the computational speed of spectral clustering. | ||

+ | |||

+ | |||

+ | === Yimin Zhong (Univ. of California - Irvine) === | ||

+ | |||

+ | ''Instability of an inverse problem for the stationary radiative transport near the diffusion limit'' | ||

+ | |||

+ | In this work, we study the instability of an inverse problem of radiative transport equation with angularly averaged measurement near the diffusion limit, i.e. the normalized mean free path (the Knudsen number) $0 < \eps \ll 1$. It is well-known that there is a transition of stability from H\"{o}lder type to logarithmic type with $\eps\to 0$, the theory of this transition of stability is still an open problem. In this study, we show the transition of stability by establishing the balance of two different regimes depending on the relative sizes of $\eps$ and the perturbation in measurements. When $\eps$ is sufficiently small, we obtain exponential instability, which stands for the diffusive regime, and otherwise we obtain H\"{o}lder instability instead, which stands for the transport regime. | ||

+ | |||

+ | |||

+ | === Alfredo N Wetzel (UW-Madison) === | ||

+ | |||

+ | ''Discontinuous Fronts as Exact Solutions to Precipitating Quasi-Geostrophy'' | ||

+ | |||

+ | Atmospheric fronts may be idealized as boundaries between two air masses with different temperature, density, moisture, etc. In this presentation, we discuss exact discontinuous solutions of a simplified model for moist mid-latitude synoptic atmospheric flows, the precipitating quasi-geostrophic (PQG) equations. These simple discontinuous solutions correspond to propagating moist fronts that require both rainfall and a phase change of water at the front interface to exist. The fronts propagate at speeds related to the rainfall velocity, temperature/wind jump magnitudes, and front geometry. Moreover, the relative simplicity of the model and front geometry gives rise to readily accessible conditions for the front’s existence. As an initial assessment of the realism of these fronts, we use rough estimates of relevant physical parameters to show that cold, warm, and stationary fronts are sensibly captured by the model. Simple exact solutions have previously been presented in the context of the well-known Margules’ front slope formula for dry fronts but have not been generalized to include propagation and moisture. | ||

+ | |||

+ | |||

+ | === Lukas Einkemmer (University of Tübingen and University of Innsbruck) === | ||

+ | |||

+ | ''A low-rank projector-splitting integrator for kinetic plasma simulation: from structure preservation to fluid limits'' | ||

+ | |||

+ | Many problems encountered in plasma physics require a kinetic description. The associated partial differential equations are posed in an up to six-dimensional phase space. Thus, a direct discretization of this phase space, often called the Eulerian approach, is extremely expensive from a computational point of view. | ||

+ | |||

+ | In this talk we propose a dynamical low-rank approximation for the Vlasov--Poisson and Vlasov--Maxwell equations. This approximation is derived by constraining the dynamics to a manifold of low-rank functions via a tangent space projection and by splitting this projection into the subprojections from which it is built. This reduces a time step for the six- (or four-) dimensional Vlasov equation to solving two systems of three- (or two-) dimensional advection equations. By a hierarchical dynamical low-rank approximation, a time step for the Vlasov equation can be further reduced to a set of six (or four) systems of one-dimensional advection equations. | ||

+ | |||

+ | The resulting systems of advection equations can then be solved by semi-Lagrangian or spectral methods. We highlight the favorable behavior of the proposed numerical method by performing numerical simulation for a number of problems. These simulation show that that the proposed algorithm is able to drastically reduce the required computational effort. Unfortunately, much of the physical structure inherent in the continuous problem is destroyed in the process. We will discuss an approach to restore some of the physical structure to the low-rank approximation, which is important even for relatively short times. In addition, we will discuss the underlying fluid limit. |

## Latest revision as of 14:23, 6 December 2018

## Contents

- 1 ACMS Abstracts: Fall 2018
- 1.1 Ting Zhou (Northeastern University)
- 1.2 Daniel Sanz-Alonso (University of Chicago)
- 1.3 Nan Chen (University of Wisconsin-Madison)
- 1.4 Sulian Thual (Fudan University)
- 1.5 Matthew Thorpe (Cambridge University)
- 1.6 Fei Lu (Johns Hopkins University)
- 1.7 Matthew Dixon (Illinois Institute of Technology)
- 1.8 Karl Rohe (UW-Madison, statistics)
- 1.9 Yimin Zhong (Univ. of California - Irvine)
- 1.10 Alfredo N Wetzel (UW-Madison)
- 1.11 Lukas Einkemmer (University of Tübingen and University of Innsbruck)

# ACMS Abstracts: Fall 2018

### Ting Zhou (Northeastern University)

*Nonparaxial near-nondiffracting accelerating optical beams*

We show that new families of accelerating and almost nondiffracting beams (solutions) for Maxwell’s equations can be constructed. These are complex geometrical optics (CGO) solutions to Maxwell’s equations with nonlinear limiting Carleman weights. They have the form of wave packets that propagate along circular trajectories while almost preserving a transverse intensity profile. We also show similar waves constructed using the approach combining CGO solutions and the Kelvin transform.

### Daniel Sanz-Alonso (University of Chicago)

*Discrete and Continuous Learning in Information and Geophysical Sciences*

The formulation of Bayesian inverse problems in function space has led to new theoretical and computational developments, providing improved understanding on regularization techniques and suggesting new scalable algorithms. The approach has found numerous applications throughout the geophysical and medical sciences, where interest often lies in recovering an unknown field defined on a physical domain. Learning problems in the information sciences, in contrast, typically seek to recover functions defined on discrete point clouds. My talk will have two parts. In the first one, I will prove that in certain large data limit, discrete learning problems converge to a continuous one, thus allowing to transfer scalable Markov chain Monte Carlo methodology developed in the geophysical sciences to novel applications in the information sciences. In the second part I will introduce a fully Bayesian, data-driven methodology to discretize complex forward models with the specific goal of solving inverse problems. This methodology has the potential of producing cheap surrogates that still allow for satisfactory input reconstruction.

### Nan Chen (University of Wisconsin-Madison)

*A simple stochastic model for El Nino with westerly wind bursts and the prediction of super El Nino events*

Atmospheric wind bursts in the tropics play a key role in the dynamics of the El Nino Southern Oscillation (ENSO). A simple modeling framework is proposed that summarizes this relationship and captures major features of the observational record while remaining physically consistent and amenable to detailed analysis. Within this simple framework, wind burst activity evolves according to a stochastic two-state Markov switching–diffusion process that depends on the strength of the western Pacific warm pool, and is coupled to simple ocean–atmosphere processes that are otherwise deterministic, stable, and linear. A simple model with this parameterization and no additional nonlinearities reproduces a realistic ENSO cycle with intermittent El Nino and La Nina events of varying intensity and strength as well as realistic buildup and shutdown of wind burst activity in the western Pacific. The wind burst activity has a direct causal effect on the ENSO variability: in particular, it intermittently triggers regular El Nino or La Nina events, super El Nino events, or no events at all, which enables the model to capture observed ENSO statistics such as the probability density function and power spectrum of eastern Pacific sea surface temperatures. The present framework is then applied to understand the mechanism of different super El Ninos. In particular, the framework is used to simulate and analyze the two famous super El Nino events in 1997-1998 and 2014-2016, with the conclusion that the delayed super El Nino events in 2014-2016 are not necessarily unusual in the tropical Pacific despite not appearing in the recent observational record and could reoccur in the future.

### Sulian Thual (Fudan University)

*A Stochastic Skeleton Model for the Madden-Julian Oscillation and El Nino-Southern Oscillation*

A broad range of random atmospheric disturbances in the tropics may be considered as possible triggers to the El Niño Southern Oscillation (ENSO), such as for example westerly wind bursts, easterly wind bursts, as well as the convective envelope of the Madden-Julian Oscillation (MJO). Here a simple dynamical stochastic model for the tropical ocean-atmosphere is proposed that captures those processes as well as their multiscale interactions. Realistic features include for the first time altogether the MJO wavenumber-frequency power spectra, eastward propagation, structure and confinement to the warm pool region and similarly for atmospheric Kelvin and Rossby equatorial waves, in addition to the ENSO intermittency, power spectrum and non-Gaussian statistics of sea surface temperatures, among others.

Importantly, intraseasonal atmospheric disturbances such as the MJO are here solved dynamically which renders more explicit their upscale contribution to the interannual flow as well as their modulation in return. First, the background red noise spectrum of atmospheric disturbances rather than their individual characteristics is shown to be most important for the triggering of the ENSO. Second, the onset, strength and demise of El Niño events is linked to the increase and eastward expansion of atmospheric disturbances eastward of the warm pool region. The present framework serves as a prototype for general circulation models that solve similar dynamical interactions on several spatial and temporal scales.

### Matthew Thorpe (Cambridge University)

*Continuum Limits of Semi-Supervised Learning on Graphs*

Given a data set $\{x_i\}_{i=1}^n$ with labels $\{y_i\}_{i=1}^N$ on the first $N$ data points the goal of semi-supervised is to infer labels on the remaining $\{x_i\}_{i=N+1}^n$ data points. In this talk we use a random geometric graph model with connection radius $r(n)$. The framework is to consider objective functions which reward the regularity of the estimator function and impose or reward the agreement with the training data, more specifically we will consider discrete p-Laplacian and fractional Laplacian regularization.

The talk concerns the asymptotic behaviour in the limit where the number of unlabelled points increases while the number of training points remains fixed. The results are to uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. I will give almost optimal ranges on the scaling of $r(n)$ for asymptotic consistency to hold. Furthermore, I will setup the Bayesian interpretation of this problem.

This is joint work with Matt Dunlop (Caltech), Dejan Slepcev (CMU) and Andrew Stuart (Caltech).

### Fei Lu (Johns Hopkins University)

*Data-informed stochastic model reduction for complex dynamical systems*

The need to develop reduced nonlinear statistical-dynamical models from time series of partial observations of complex systems arises in many applications such as geophysics, biology and engineering. The challenges come mainly from memory effects due to the nonlinear interactions between resolved and unresolved scales, and from the difficulty in inference from discrete data.

To address these challenges, we introduce a discrete-time stochastic parametrization framework, in which we infer nonlinear autoregression moving average (NARMA) type models to take the memory effects into account. We show by examples that the NARMA type stochastic reduced models that can capture the key statistical and dynamical properties, and therefore can improve the performance of ensemble prediction in data assimilation. The examples include the Lorenz 96 system (which is a simplified model of the global atmosphere) and the Kuramoto-Sivashinsky equation of spatiotemporally chaotic dynamics. Applications of this inference approach to model reduction for stochastic Burgers equations will be discussed.

### Matthew Dixon (Illinois Institute of Technology)

*"Quantum Equilibrium-Disequilibrium”: Asset Price Dynamics, Symmetry Breaking and Defaults as Dissipative Instantons*

We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the ”Quantum Equilibrium-Disequilibrium” model. Our model gives rise to non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, ”classical” GBM scale-invariant dynamics with an exponential asset growth and without defaults are formally recovered from our model. However, we argue that this is only a formal mathematical limit, and in reality the GBM limit is non- analytic due to non-linear effects that produce both defaults and divergence of perturbation theory in a small market friction parameter.

### Karl Rohe (UW-Madison, statistics)

*Making Spectral Graph Theory work in practice. Making the practice work in theory*

After introducing Cheeger's Inequality and spectral clustering, this talk has two parts. The first part will (1) show how spectral clustering gives "bad results" in many applied settings and (2) illustrate a "hack" that makes it work very well. Most of the talk will be spent on the second part, which will provide a simple theory to provide a deeper understanding of where the bad results come from and why the hack works so well.

There are four pieces to this simple theory. First, sparse and stochastic graphs create a lot of small trees that are connected to the core of the graph by only one edge. Second, graph conductance is sensitive to these noisy "dangling sets." Third, by Cheeger's inequality and an inequality from Ky Fan, spectral clustering inherits this sensitivity. These three pieces explain why spectral clustering gives bad results in practice. The fourth piece uses Cheeger's inequality to show how the hack creates a new form of graph conductance that we call CoreCut. Simple inspection of CoreCut reveals why it is less sensitive to small cuts in the graph. In addition to this statistical benefit, these results also demonstrate why the hack also improves the computational speed of spectral clustering.

### Yimin Zhong (Univ. of California - Irvine)

*Instability of an inverse problem for the stationary radiative transport near the diffusion limit*

In this work, we study the instability of an inverse problem of radiative transport equation with angularly averaged measurement near the diffusion limit, i.e. the normalized mean free path (the Knudsen number) $0 < \eps \ll 1$. It is well-known that there is a transition of stability from H\"{o}lder type to logarithmic type with $\eps\to 0$, the theory of this transition of stability is still an open problem. In this study, we show the transition of stability by establishing the balance of two different regimes depending on the relative sizes of $\eps$ and the perturbation in measurements. When $\eps$ is sufficiently small, we obtain exponential instability, which stands for the diffusive regime, and otherwise we obtain H\"{o}lder instability instead, which stands for the transport regime.

### Alfredo N Wetzel (UW-Madison)

*Discontinuous Fronts as Exact Solutions to Precipitating Quasi-Geostrophy*

Atmospheric fronts may be idealized as boundaries between two air masses with different temperature, density, moisture, etc. In this presentation, we discuss exact discontinuous solutions of a simplified model for moist mid-latitude synoptic atmospheric flows, the precipitating quasi-geostrophic (PQG) equations. These simple discontinuous solutions correspond to propagating moist fronts that require both rainfall and a phase change of water at the front interface to exist. The fronts propagate at speeds related to the rainfall velocity, temperature/wind jump magnitudes, and front geometry. Moreover, the relative simplicity of the model and front geometry gives rise to readily accessible conditions for the front’s existence. As an initial assessment of the realism of these fronts, we use rough estimates of relevant physical parameters to show that cold, warm, and stationary fronts are sensibly captured by the model. Simple exact solutions have previously been presented in the context of the well-known Margules’ front slope formula for dry fronts but have not been generalized to include propagation and moisture.

### Lukas Einkemmer (University of Tübingen and University of Innsbruck)

*A low-rank projector-splitting integrator for kinetic plasma simulation: from structure preservation to fluid limits*

Many problems encountered in plasma physics require a kinetic description. The associated partial differential equations are posed in an up to six-dimensional phase space. Thus, a direct discretization of this phase space, often called the Eulerian approach, is extremely expensive from a computational point of view.

In this talk we propose a dynamical low-rank approximation for the Vlasov--Poisson and Vlasov--Maxwell equations. This approximation is derived by constraining the dynamics to a manifold of low-rank functions via a tangent space projection and by splitting this projection into the subprojections from which it is built. This reduces a time step for the six- (or four-) dimensional Vlasov equation to solving two systems of three- (or two-) dimensional advection equations. By a hierarchical dynamical low-rank approximation, a time step for the Vlasov equation can be further reduced to a set of six (or four) systems of one-dimensional advection equations.

The resulting systems of advection equations can then be solved by semi-Lagrangian or spectral methods. We highlight the favorable behavior of the proposed numerical method by performing numerical simulation for a number of problems. These simulation show that that the proposed algorithm is able to drastically reduce the required computational effort. Unfortunately, much of the physical structure inherent in the continuous problem is destroyed in the process. We will discuss an approach to restore some of the physical structure to the low-rank approximation, which is important even for relatively short times. In addition, we will discuss the underlying fluid limit.