### 07.09.14 - 12.09.14, Seminar 14371

# Adjoint Methods in Computational Science, Engineering, and Finance

### The following text appeared on our web pages prior to the seminar, and was included as part of the invitation.

## Motivation

The focus of this Dagstuhl Seminar will be on fundamental theoretical issues in discrete and continuous adjoint methods with applications in Computational Science, Engineering, and Finance (CSEF).

With the ever increasing performance of HPC systems the resolution at which problems in CSEF formulated as (systems of) partial differential equations (PDE) can be solved numerically, is rising steadily. Adjoint methods enable the highly desirable transition from numerical simulation to deterministic (derivative-based nonlinear) optimization as they provide sensitivity information of target functionals with respect to a potentially very large number of parameters with a computational complexity that is independent of this number. Thus many (inverse) problems can be dealt with numerically the solution of which would be beyond modern computational capabilities otherwise. Typical examples include PDE-constrained optimization and optimal control, uncertainty quantification and error analysis / correction, and large-scale parameter estimation / data assimilation.

The development of adjoint numerical methods yields a large number of theoretical, algorithmic, and practical (implementation) challenges most of them to be addressed by state of the art Computer Science and Applied Mathematics methodology including parallel high-performance computing, domain-specific program analysis and compiler construction, combinatorial scientific computing, numerical linear algebra / analysis, and functional analysis.

One aim of this seminar is to tackle these challenges by setting the stage for accelerated development and deployment of such methods based on in-depth discussions between computer scientists, mathematicians, and practitioners from various application areas. We plan to initiate a sequence of related events alternating in between Schloss Dagstuhl and the Mathematisches Forschungsinstitut Oberwolfach, thus, emphasizing the obvious synergies between computer science and mathematics in the given context.

For this first meeting we would like to focus on some fundamental theoretical issues arising in the context of "continuous vs. discrete adjoints" in addition to introductory lectures and round table discussions / working groups on both approaches. Motivation will be provided by presentations of various applications of adjoint methods in CSEF.