DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:**Denis Ridzal, Sandia National Laboratories

**Title: ***
Optimization-based models and algorithms for the feature-preserving solution of transport equations
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**Abstract:**
A fundamental mathematical challenge in the numerical modeling and solution of transport equations is to simultaneously achieve high spatial accuracy and preserve essential physical solution features, such as positivity, monotonicity, and mass or energy balance. To reconcile the notion of accuracy with the preservation of physical features we employ global nonlinear optimization models at various stages of the discretization and solution process. Specifically, the optimization problems are designed to minimize the distance to a suitable target quantity, responsible for the solution accuracy, subject to a system of equalities and inequalities that maintain physical constraints.

This approach is significantly more flexible than traditional discretizations of transport equations, which rely on some form of flux limiting, and offers valuable computational and theoretical advantages. First, by design it is guaranteed to compute the most accurate solution representation that simultaneously satisfies physical constraints. Second, it is independent of the underlying spatial discretization scheme and the mesh representation, and can be thought of as a mathematically rigorous "post-processing" of the solution.

As the first example, motivated by Semi-Lagrangian and Arbitrary Lagrangian-Eulerian methods for transport, we use optimization ideas to perform conservative and monotone incremental remap, i.e., to transfer a scalar conserved quantity between a sequence of meshes subject to local solution bounds. Two formulations of remap are compared and contrasted -- the flux-variable flux-target (FVFT) formulation and the mass-variable mass-target (MVMT) formulation -- yielding optimization algorithms with distinct performance characteristics. We highlight the MVMT formulation, a singly linearly constrained quadratic program with simple solution bounds, for which we design an efficient and easily parallelizable optimization algorithm.

As the second example, motivated by Eulerian finite element methods, we design a monotone and mass-conserving scheme for finite element transport. The scheme shares the optimization formulation and the algorithm with the above MVMT formulation. It provably maintains linear relationships between simulated quantities in multi-variable systems, such as those arising in models for atmospheric transport. As a side benefit, our framework also enables robust failure recovery on future high-performance computing architectures.

Bio:
Dr. Denis Ridzal is a Principal Member of Technical Staff at Sandia National Laboratories. Dr. Ridzal received a Ph.D. in Computational and Applied Mathematics from Rice University in 2006. His research includes the development and analysis of algorithms for large-scale optimization, with emphasis on the efficient treatment of simulation constraints, iterative linear solvers and preconditioners; optimization-based models and algorithms for transport equations and optimization-based physics coupling; and the development and implementation of high-order and compatible PDE discretizations. Dr. Ridzal is a lead developer of Sandia's Trilinos package Rapid Optimization Library (ROL) for matrix-free large-scale optimization and the Trilinos package Intrepid for compatible PDE discretizations.

**Time:** Friday, April 3, 2015, 1:30-2:30 p.m.

**Place:** Exploratory Hall, Room 4106

Department of Mathematical Sciences

George Mason University

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