The Solution of Multidimensional Real Helmholtz Equations on Sparse Grids

Divergence-Free Wavelets on the Hypercube: General Boundary Conditions

“…Due to the anisotropic nature of the basis functions, the stiffness matrix is, however, generally far from being sparse. Nevertheless, for (optimized) sparse grid index sets, the application of this matrix to a vector can be computed in linear complexity using the so-called unidirectional approach [1]. This approach relies on the availability of locally supported single scale bases and corresponding refinement relations for the univariate wavelets that are used as building blocks of the divergence-free wavelets.…”

“…By deleting the first column of 1 2 T − +1 M ,0 T , and by realizing that by span + ∪ + +1 = span + +1 , and + +1 being independent, the first row of 1 2 T − +1 M ,0 T , without its first column, and that of 1 2 T − +1 M ,1 are zero, we get the expressions of + and + +1 in terms of + +1 . Remark 5.3 Biorthogonality shows that in the situation of Proposition 5.2, we also have…”

“…λ 2 e 1 −ψ (1) λ 1 ⊗ + ψ (2) λ 2 e 2 : (λ 1 , λ 2 ) ∈ • ∇ (1) × • ∇ (2) , |λ 1 |=|λ 2 |= ∪ + ψ (1) λ 1 ⊗ θ (2) λ 2 e 1 − ψ (1) (2) λ 2 e 1 − θ (1) …”

Principal manifold learning by sparse grids

“…Furthermore there are so-called energy-norm based sparse grids which only need O(m) degrees of freedom but result in O(h r−1 ) accuracy with respect to the energy norm. This approach completely eliminates the dependence of the dimension d in the complexities at least for the m-asymptotics, the order constants however still depend exponentially on d. The sparse grid method has successfully been applied to problems from quantum mechanics [30], to stochastic differential equations [57], to highdimensional integration problems from physics and finance [12,32,55] and to the solution of moderately higher-dimensional partial differential equations, mainly of elliptic type [6,7,16]. For a survey, see [18].…”

A multilevel algorithm for the solution of second order elliptic differential equations on sparse grids

“…First, the sparse grid method was applied to Poisson's equation and to the Helmholtz equation (see Balder and Zenger [1] and Bungartz [2]). Different kinds of solvers, like conjugate gradients and Gauss-Seidel iteration, were used for the solution of the discrete equation system.…”