On Efficient Solutions to the Continuous Sensitivity Equation Using Automatic Differentiation

Abstract

Abstract. Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic differentiation (AD) can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation. The approximate solution of the PDE uses a spatial discretization (mesh) that often depends on the shape parameters. Therefore, the straightforward application of AD introduces derivatives of the mesh. There are two drawbacks to this approach. First, extra computational effort (especially memory) is used in these calculations due to mesh sensitivities. Second, this mesh sensitivity information needs to be computed in order to obtain accurate results. In this work, we provide a methodology that avoids mesh sensitivities (and their drawbacks) by defining a modified PDE on a fixed domain (i.e., independent of the shape parameter) such that AD provides the desired approximation of the CSE. Using two examples, we demonstrate significant improvement in the computational effort, both in terms of floating point operations and memory requirements. We explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code. These changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation.

Sensitivity equations for a size-structured population model

“…Hence, Result 2.1 guarantees that there exists a unique bounded nonnegative solution to (17). Thus, this solution of (17) must be the same as B used in equations (11) and (12) in representing the unique nonnegative solution to (1).…”

“…However, to the best of our knowledge, thus far there is no literature on sensitivity equations and the related analysis for size-structured population models. Sensitivity analysis of dynamical systems has drawn the attention of numerous researchers [1,6,9,10,11,13,14,15,16,17,20,24,25,27,28,35,38,40] for many years because the resulting sensitivity functions can be used in many areas such as optimization and design [16,26,27,34,38], computation of standard errors [9,10,19,21,36], and information theory [12] related quantities (e.g., the Fisher information matrix) as well as control theory, parameter estimation and inverse problems [5,8,9,10,11,40,41]. One of our motivations for investigating sensitivity for size-structured population models derives from our efforts reported in [7], where a shrimp biomass production system and a…”

Reduced-order models for parameter dependent geometries based on shape sensitivity analysis

“…[27][28][29]). However, the CSE method offers a number of advantages over discrete sensitivity algorithms that have been extensively discussed in the literature [30,20]. Among them, we adopt the CSE method here for the two following reasons : First, in the case of shape parameters, the CSE method avoids the delicate issue of evaluating mesh sensitivities.…”

“…Each column of U represents a single POD spatial vector / j and they are ordered such that k i P k iþ1 . Due to the assumptions on the FE snapshots, the matrix B ¼ Y T MY 2 R mÂm is symmetric, positive definite so that the spectral theorem states that (30) exists and the eigenfunctions form a complete real orthonormal set associated to positive eigenvalues. From the physical point of view, k k measures the kinetic energy of the data captured by mode k on average over the time interval T :…”

Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space

“…Applications of sensitivity methods for finite dimensional vector parameters are widespread and can be found in many applications including biology [24] and physiology [22], mechanics [1,41], and control theory [61]. More recently, investigators' attention has turned to more complex formulations for sensitivity of infinite dimensional systems with function space parameters in problems involving shape sensitivities or sensitivities with discontinuous coefficients (for interesting and well motivated pioneering results in this area, we recommend that the reader see [25,26,30,31,32,33] and [54] and the references therein). However, sensitivity for the systems we investigate here of the forṁ x(t) = F (t, x(t), P ),…”