C GasparAdvances in Numerical Algorithms, Graz, Fast interpolation techniques and meshless methods Csaba Gáspár Széchenyi István University, Department of Mathematics Győr, Hungary

C GasparAdvances in Numerical Algorithms, Graz, surface approximation completion of incomplete data sets construction of meshfree methods Motivations

C GasparAdvances in Numerical Algorithms, Graz, The scattered data interpolation problem in Vectorial problems: some additional conditions are required

C GasparAdvances in Numerical Algorithms, Graz, Method of radial basis functions where the basis functions depend on ||x|| only. The coefficients are a priori unknown and can be determined by solving the interpolation equations: The form of the interpolation function: and the orthogonality conditions:

C GasparAdvances in Numerical Algorithms, Graz, Some popular RBFs Some popular RBFs Method of Multiquadrics (Hardy): Thin Plate Splines (Duchon): (biharmonic fundamental solution) Traditional DRM (Partridge, Brebbia): Gaussian functions: Compactly supported RBFs Compactly supported RBFs (Wendland): Globally supported RBFs

C GasparAdvances in Numerical Algorithms, Graz, Why do we like the RBFs so much? Main advantages: excellent approximation properties easy to implement But what is the price to pay? We have to face that: RBFs generally lead to a system with large, dense and ill-conditioned matrix memory requirement: O(N 2 ) computational cost: O(N 3 )

C GasparAdvances in Numerical Algorithms, Graz, domain decomposition preconditioning fast multipole evaluation methods compactly supported RBFs multi-level techniques direct multi-elliptic interpolationdirect multi-elliptic interpolation Remedies

C GasparAdvances in Numerical Algorithms, Graz, Methods based on iterated elliptic operators Observation: Thin plate splines are fundamental solutions of the biharmonic operator Idea: Let us define the interpolation function in such a form: Second-order operators: unusable! Fourth-order operators: biharmonic operator; bi-Helmholtz-operator; Laplace-Helmholtz- operator Higher order operators: triharmonic operator; tetraharmonic operator; iterated Helmholtz- operator, etc.

C GasparAdvances in Numerical Algorithms, Graz, ”Harmonic interpolation”: a counterexample Numerical features: ill-posed problem poor interpolation surface logarithmic singularitieslogarithmic singularities at the interpolation points at the interpolation points

C GasparAdvances in Numerical Algorithms, Graz, Biharmonic interpolation Equivalent formulations: Let us introduce the closed subspace: Classical formulation: For a given function, find a function such that Direct problem: For a given function, find a function such that Variational problem: For a given function, find a function such that Theorem: The direct and the variational problems are equivalent and have a unique solution in W.

C GasparAdvances in Numerical Algorithms, Graz, Representation of the biharmonic interpolation Theorem: u is uniquely represented in the form: where w is an everywhere biharmonic function, and  is the biharmonic fundamental solution:...but the coefficients can be calculated from: without solving any system of equations! without solving any system of equations! This is an RBF-like interpolation...

C GasparAdvances in Numerical Algorithms, Graz, Approximation Separation distance: Theorem: The following estimations hold:

C GasparAdvances in Numerical Algorithms, Graz, Bi-Helmholtz interpolation Classical formulation: For a given function, find a function such that Direct problem: For a given function, find a function such that Variational problem: For a given function, find a function such that The direct and the variational problems are equivalent, and have a unique solution in W. The representation and approximation theorems are still valid. Now the fundamental solution is rapidly decreasing ("almost compactly supported"). Representation formula: Coefficients can be computed directly:

C GasparAdvances in Numerical Algorithms, Graz, Further generalizations The use of even higher order differential operators The use of different operators simultaneously In each case, the representation automatically produces an RBF-like interpolation based on the fundamental solution of the applied operator. Triharmonic operator: Mixed Laplace-Helmholtz operators:

C GasparAdvances in Numerical Algorithms, Graz,

C GasparAdvances in Numerical Algorithms, Graz, Numerical techniques Main features: No algebraic interpolation equation No algebraic interpolation equation has to be solved. auxiliary differential equation Instead, the problem is converted to an auxiliary differential equation. The domain is not predefined The boundary conditions are not strictly prescribed Numerical solution techniques: Traditional FDM or FEM ? FFT ? MGR ? quadtree-based multigrid techniques

C GasparAdvances in Numerical Algorithms, Graz, Quadtree cell systems Non-uniform cell subdivision generated by a finite set of points Schemes based on finite volumes Multigrid solution technique Computational costComputational cost can be reduced to O(N log N) Demo application

C GasparAdvances in Numerical Algorithms, Graz, Biharmonic interpolation on a QT cell system: an example

C GasparAdvances in Numerical Algorithms, Graz, Bi-Helmholtz interpolation: an example c = 10 c = 50

C GasparAdvances in Numerical Algorithms, Graz, Construction of meshless methods. General approaches Original problem:+ boundary conditions Method 1. The solution u is directly interpolated: Method 2. First, the function f is interpolated:

C GasparAdvances in Numerical Algorithms, Graz, Construction of meshless methods by direct multi-elliptic interpolation Model problem: + boundary conditions Method 1. The solution u is directly interpolated by tetraharmonic interpolation: Method 2. First, the function f is interpolated by biharmonic interpolation. Then, using the same QT cell system, the model problem is Then, using the same QT cell system, the model problem is solved, without boundary conditions. solved, without boundary conditions....A bit uncomfortable... This procedure mimics the computation of Newtonian potential Reduces the domain problem to a boundary problem

C GasparAdvances in Numerical Algorithms, Graz, How to solve the boundary problem? Traditional tool: the Boundary Element Method Direct Multi-Elliptic Boundary Interpolation Drawbacks Drawbacks: large, dense and ill-conditioned matrices Advantages Advantages: reduction of dimension reduction of mesh

C GasparAdvances in Numerical Algorithms, Graz, Boundary interpolation 1. The basic idea Model problem: Laplace equation with Dirichlet boundary condition The interpolation points are located at the boundary only, and... mixed Laplace-Helmholtz operator The use of the mixed Laplace-Helmholtz operator: 'Harmonic' interpolation: Biharmonic interpolation: 'Quasi-harmonic' interpolation: Fundamental solution:

C GasparAdvances in Numerical Algorithms, Graz, Boundary interpolation 2. Application to the Laplace equation Key issue: the proper choice of the scaling constant c The original problem: The associated boundary interpolation problem: Theorem 1: Theorem 2: With more regular data:

C GasparAdvances in Numerical Algorithms, Graz, Dirichlet boundary condition Examples

C GasparAdvances in Numerical Algorithms, Graz, Neumann boundary condition The naive approach The original problem: The associated boundary interpolation problem: The method fails... u approximates v 0 but fails to approximate u* !

C GasparAdvances in Numerical Algorithms, Graz, Neumann boundary condition An improved approach Idea: Locate additional Dirichlet points along Neumann boundary in outward normal direction; Define Dirichlet data to enforce Neumann conditions or: This procedure mimics the method of fundamental solutions

C GasparAdvances in Numerical Algorithms, Graz, Neumann boundary condition Examples

C GasparAdvances in Numerical Algorithms, Graz, Boundary interpolation 3. Application to the biharmonic equation The original problem: The associated boundary interpolation problem: Theorem 3: Further applications: 2D Stokes flow

C GasparAdvances in Numerical Algorithms, Graz, Vectorial interpolation problems Componentwise interpolation works well but often additional conditions are required: Use the potential/stream function approach RBF-method fails

C GasparAdvances in Numerical Algorithms, Graz, Tri-Helmholtz-interpolation for vectorial problems Classical formulation: For a given function find a function such that Variational formulation: Define For a given function find a function such that Theorem: The variational problem has a unique solution in W. The vectorial interpolation function: (grad  )  is divergence-free.

C GasparAdvances in Numerical Algorithms, Graz, Representation and RBF-like solution Representation formula: If c > 0 then  is represented in the form: where  is the fundamental solution of the operator : The theory assures existence and uniqueness also for the RBF-like interpolation The theory assures existence and uniqueness also for the RBF-like interpolation: Componentwise TPS ExactVectorial tri-Helmholtz

C GasparAdvances in Numerical Algorithms, Graz, Summary and conclusions Interpolation based on iterated elliptic equations Interpolation based on iterated elliptic equations Boundary interpolation by singularly perturbed equations Boundary interpolation by singularly perturbed equations Special RBF- and RBF-like methods are automatically Special RBF- and RBF-like methods are automatically generated generated No explicit use of RBFs is needed No explicit use of RBFs is needed No large and dense algebraic equations are to be solved No large and dense algebraic equations are to be solved Quadtree-based multigrid methods are used Quadtree-based multigrid methods are used

C GasparAdvances in Numerical Algorithms, Graz, Thank you for your attention!