E6: Functional Interpolation

“Functional Interpolation” is a section of the open access peer-reviewed journal Mathematics, that publishes advanced studies on the theory of functional interpolation and its applications in engineering, physics, and science.

Functional interpolation is a mathematical framework that derives functionals containing a free function and, no matter what the free function is, always satisfies a set of user-defined linear constraints. These functionals reduce the solution search space of constrained problems to just the subspace of functions that analytically satisfy the given constraints. Constraints on univariate or multivariate functions can be specified by point constraints, derivative constraints, integral constraints, infinite constraints, component constraints, or linear combinations of them. Functional interpolation for inequality constraints is a special subject of high interest.

The primary aim of “Functional Interpolation” is the publication and dissemination of relevant mathematical works on this new and fast-growing subject in mathematics. Topics of interest include but are not limited to differential equations, calculus of variations, continuation methods, optimization problems, inverse problems, stiff problems, hybrid systems, and any other subject in which “Functional Interpolation” can play a key role. Applications of functional interpolation that improve the state-of-the-art solution methods in any field or problem in engineering, physics, or science are of high interest. Direct comparisons against state-of-the-art solution methods and the use of functional interpolation within existing solution approaches to make them more accurate and/or faster are also of high interest to this journal section. All papers are peer-reviewed.

Functional Interpolation FAQ

DEFINITION. Functional interpolation (FI) is the body of mathematics associated with deriving functions that satisfy a set of constraints.

Q: What kind of constraints can FI include?
A: FI includes any nonlinear constraints in any dimensional space. The Theory of Functional Connections (TFC) is a subset of FI that includes linear constraints in any real dimensional space. These constraints include point constraints, derivative constraints, integral constraints, infinite constraints, vector component constraints, or linear combinations of them. Inequality constraints were partially integrated into TFC.

Q: What kind of domains can FI be applied in?
A: FI can be applied to any domain. Currently, TFC can only be applied to real-valued n-dimensional rectangular domains. The extension to generic domains via domain mapping and patching is currently under investigation, while the extension to complex domains has never been attempted.

Q: Can FI be used for optimization?
A: Yes. FI can be used to reduce the search space to the set of functions that satisfy the constraints. Benefits of doing so may include improved and faster convergence as well as robustness to initialization.

Q: What are examples of mathematical problems that can be solved using FI?
A: Currently, FI can be applied to constrained optimization problems subject to linear constraints. This reduces the function solution space to the subspace of functions, fully satisfying the constraints. Examples of applications include (but are not limited to) differential equations, calculus of variations, continuation methods, optimization problems, constrained regression, inverse problems, stiff problems, and hybrid systems.

Q: Distinctions with respect to Lagrange multipliers technique.
A: The technique of Lagrange multipliers is used for finding the local maxima and minima of a function subject to equality constraints. Constraints can be any nonlinear function or differential function. This is carried out by introducing additional unknowns (Lagrange multipliers) that must be computed to obtain the solution. Computing Lagrange multipliers is sometimes easy, sometimes very difficult, and sometimes impossible. FI searches for a solution of constrained problems by analytically embedding the constraints in a functional without adding any additional unknowns.