Flemming Holtorf

We present LowRankArithmetic.jl and LowRankIntegrators.jl. The conjunction of both packages forms the backbone of a computational infrastructure that enables simple and non-intrusive use of dynamical low rank approximation for on-the-fly compression of large matrix-valued data streams or the approximate solution of otherwise intractable matrix-valued ODEs. We showcase the utility of these packages for the quantification of uncertainty in scientific models.

Many scientific computing problems boil down to solving large matrix-valued ordinary differential equations (ODE); prominent examples for that are the propagation of uncertainties through (partial-)differential equation models or on-the-fly compression of large-scale simulation or experimental data. While the naive integration of such matrix-valued ODEs often remains prohibitively expensive, it is in many cases found that their solution admit an accurate low-rank approximation. Exploiting such a low rank structure generally holds the potential for substantial computational resource savings (time and memory) over naive integration approaches, often recovering the tractability of integration. 

Dynamical low rank approximation (DLRA), a concept also known under the names Dirac-Fraenkel time-varying variational principle or dynamically orthogonal schemes, seeks to exploit the low-rank structure of the solution of matrix-valued ODEs by performing the integration within the manifold of fixed (low-)rank matrices.  However, while theoretically elegant, the effective use of DLRA in practice is often cumbersome due to the need for custom implementations of integration routines that take advantage of the assumed low-rank structure. To alleviate this limitation, we present the packages LowRankArithmetic.jl and LowRankIntegrators.jl. The conjunction of both packages forms the backbone of a computational infrastructure that enables simple and non-intrusive use of DLRA. To that end, LowRankArithmetic.jl facilitates the propagation of low rank matrix representations through finite compositions of a rich set of algebraic operations, alleviating the need for custom implementations. Based on this key functionality, LowRankIntegrators.jl implements state-of-the-art integration routines for DLRA that automatically take advantage of low rank structure; the user needs to supply nothing more than the the right-hand-side of a matrix-valued ODE of interest. 

In this talk, we briefly review the conceptual idea behind DLRA, outline the primitives underpinning LowRankArthmetic.jl and LowRankIntegrators.jl, and showcase their utility for the propagation of uncertainties through scientific models ranging from stochastic PDEs to the chemical master equation.

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Dynamical Low Rank Approximation in Julia

We present MarkovBounds.jl -- a meta-package to SumOfSquares.jl which enables the computation of guaranteed bounds on the optimal value of a large class of stochastic optimal control problems via a high-level, practitioner-friendly interface.

The optimal control of stochastic processes is arguably one of the most fundamental questions in the context of decision-making under uncertainty. When the controlled process is a jump-diffusion process characterized by polynomial data (drift- and diffusion coefficient, jumps, etc.), it is well known that polynomial optimization and the machinery of the moment-sum-of-squares (SOS) hierarchy provides a systematic way to construct informative (and often tight) convex relaxations for the associated optimal control problems. While the JuMP ecosystem offers with SumOfSquares.jl in principle everything that is required to study stochastic optimal control problems from this perspective, it remains a cumbersome and error-prone process to translate a concrete stochastic optimal control problem into its SOS relaxation. Moreover, this translation process requires expert knowledge, rendering it inaccessible to a large audience. MarkovBounds.jl is intended to close this gap by providing a high-level interface which allows the user to define stochastic optimal control problems in symbolic form using for example DynamicPolynomials.jl or Symbolics.jl, and automates subsequent translation to associated SOS relaxations.  Finite and (discounted) infinite horizon problems are supported as well as several common objective function types. Furthermore, MarkovBounds.jl supports the combination of standard SOS relaxations with discretization approaches to tighten the relaxations. 

In this talk, we will briefly review the conceptual ideas behind constructing SOS relaxations for stochastic optimal control problems and showcase the use of MarkovBounds.jl for the optimal control of populations in a predator-prey system, expression of protein in a stochastic bio circuit and the bounding of rare event probabilities.

Flemming Holtorf

Talks:

Dynamical Low Rank Approximation in Julia

Stochastic Optimal Control with MarkovBounds.jl

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