Advancing numerical methods for viscosity solutions and applications (11w5086)

Arriving in Banff, Alberta Sunday, February 13 and departing Friday February 18, 2011

Organizers

(Università di Roma "La Sapienza")

(Università Roma Tre)

(University of British Columbia)

(University of California, Irvine)

Objectives

We aim to gather together viscosity solution experts from applied mathematics, analysis and numerical methods as well as scientists and engineers researching related applications. The recent explosion of approaches to solving these problems gives us confidence that the opportunity to exchange and integrate ideas from these various fields offered by a BIRS five day workshop will lead to significant advances in numerical methods for tackling these problems, and consequently to broad impact in the many related application fields.

*Relevance*

The theory of viscosity solutions for nonlinear PDEs was born in the 1980s and is still an active and important field of research in mathematical analysis. It has provided a sound framework for a number of differential models of great importance in applications. Such problems are characterized by common mathematical features including:

- Solutions with low regularity. Viscosity solutions are often continuous, but not differentiable, and singularities may encode information important to applications; for example, switching curves in control problems. Moreover, in some applications to differential games and image processing even continuity may not be present.

- Strong nonlinearities and possible degeneracies of differential operators.

- Solutions defined over high dimensional spaces; for example, the Hamilton-Jacobi-Bellman equations for optimal flight trajectories evolve on an underlying space of at least six dimensions, while stochastic control problems from mathematical finance may have twenty dimensions.

After some initial difficulty, development of effective numerical techniques has demonstrated rapid growth in recent years, and a number of methods have been proposed for problems of this class. However, the features listed above imply serious difficulties at a numerical level -- for both accuracy and complexity -- and although current approximation schemes are well suited to solve academic models, many real-life problems are still out of the reach of present day computational methods. Further development of techniques for efficient computation of viscosity solutions, as well as integration of existing techniques, could therefore have a huge impact in application areas.

*Timeliness*

We have seen significant advances in our ability to numerically approximate viscosity solutions in recent years. Many strategies have been proposed in order to improve accuracy and reduce complexity, and the performance of computers has dramatically increased; consequently, the set of tractable problems is growing continuously. In this context, the opportunity to gather together experts on the numerical, analytical, and applied components of viscosity solution theory would allow us:

- To identify the common elements in these techniques and thereby transfer knowledge between fields in order to develop more general methods.

- To merge approaches acting on different stages of the problem -- e.g. high-order discretizations, fast solvers, state-space reduction and so on -- to further increase the range of problems that can be treated.

- To compare the needs of application scientists and engineers, the supporting analytical theory, and the numerical schemes in order to better understand the issues and focus the various lines of research.

*Importance*

The theory of viscosity solutions appears in a variety of applications, all of which would benefit from an improvement of the related computational methods. We mention here some of the main fields of application which will be considered in the workshop.

- Level Set Methods: level set methods are currently considered a key tool for tracking the motion of interfaces in low dimensional spaces for models in combustion, fluid dynamics, phase transitions, material science, and so on. Flexible, high-order numerical schemes have been developed and have proven very effective. Beside further advances in this direction, recent work has focused on the reduction of computational complexity by means of fast solvers.

- Optimal control: two factors are conspiring to shift dynamic programming techniques from infeasible to feasible in fields such as control engineering and mathematical finance. The first is the continuing rapid increase in the computation power of hardware. The second is the development of scalable approximate dynamic programming (ADP) algorithms. While there have been many advances in ADP in the discrete domain, its relationship to viscosity solution theory and numerical methods in the continuous domain is a key topic of exploration for workshop participants.

- Image Processing: digital images and movies can be regarded as functions defined in two to four dimensions. A large class of image transforms, including various filtering and segmentation, can be interpreted as short time (viscosity) solutions of nonlinear PDEs; a key feature of these formulations is that the theory and algorithms can be developed in a dimensionally independent manner. Applications appear in medical image processing (including volumetric data), digital photography, and computer vision.