In this post we illustrate the normal spline interpolation approach with a few simple examples.
Let there is the following information of a smooth function φ(x,y),(x,y)∈R2:
φ(0,0)=0,∂φ∂x(0,0)=1,∂φ∂y(0,0)=1,
and it is necessary to reconstruct φ using this data.
Assume this function is an element of Hilbert space H2.5ε(R2), thus it can be treated as a continuously differentiable function φ∈C1(R2), and construct a normal spline σ(2.5)1 approximating φ:
σ(2.5)1=argmin{‖φ‖2H2.5ε:(1),(2),(3),∀φ∈H2.5ε(R2)}.
This spline can be presented as
This spline can be presented as
σ(2.5)1=μ1h1+μ′1h′1+μ′2h′2,
here
here
h1(η1,η2,ε)=exp(−ε√η21+η22)(1+ε√η21+η22),h′1(η1,η2,ε)=ε2exp(−ε√η21+η22)(η1+η2),h′2(η1,η2,ε)=h′1(η1,η2,ε), (η1,η2)∈R2,
and coefficients (μ1,μ′1,μ′2) are defined from the system (see the previous post):
[10002ε20002ε2][μ1μ′1μ′2]=[011].
Eventually
σ(2.5)1(x,y,ε)=exp(−ε√x2+y2)(x+y),(x,y)∈R2.
Fig.1 Spline σ(2.5)1,ε=1
Fig.2 Spline σ(2.5)1,ε=0.1
Now let function φ(x,y),(x,y)∈R2 is a twice continuously differentiable function which satisfies constraints:
φ(0,0)=0,∂φ∂x(0,0)+∂φ∂y(0,0)=2.
where
Fig.1 Spline σ(2.5)1,ε=1
Fig.2 Spline σ(2.5)1,ε=0.1
Now let function φ(x,y),(x,y)∈R2 is a twice continuously differentiable function which satisfies constraints:
φ(0,0)=0,∂φ∂x(0,0)+∂φ∂y(0,0)=2.
We approximate it by constructing a normal spline σ(3.5)1 in H3.5ε(R2):
σ(3.5)1=argmin{‖φ‖2H3.5ε:(11),(12),∀φ∈H3.5ε(R2)},σ(3.5)1=μ1h1+μ′1h′1,where
h1(η1,η2,ε)=exp(−ε√η21+η22)(3+3ε√η21+η22+ε2(η21+η22)),h′1(η1,η2,ε)=ε2exp(−ε√η21+η22)(1+ε√η21+η22)(η1+η2),
and coefficients (μ1,μ′1) are defined from the system:
[3002ε2][μ1μ′1]=[02].
Therefore
σ(3.5)1(x,y,ε)=exp(−ε√x2+y2)(1+ε√x2+y2)(x+y),(x,y)∈R2.
As the last example consider a problem of reconstructing a continuously differentiable function φ(x),x∈R, which satisfies constraint
dφdx(0)=1,
and it is closest to function z(x)=2x,x∈R. We approximate it by constructing a normal spline σ(2)1 in H2ε(R):
σ(2)1=argmin{‖φ−z‖2H2ε:(19),∀φ∈H2ε(R)},σ(2)1=z+μ′1h′1=2x+μ′1h′1,
Performing calculations analogous to previous ones, we'll receive:
As the last example consider a problem of reconstructing a continuously differentiable function φ(x),x∈R, which satisfies constraint
dφdx(0)=1,
and it is closest to function z(x)=2x,x∈R. We approximate it by constructing a normal spline σ(2)1 in H2ε(R):
σ(2)1=argmin{‖φ−z‖2H2ε:(19),∀φ∈H2ε(R)},σ(2)1=z+μ′1h′1=2x+μ′1h′1,
Performing calculations analogous to previous ones, we'll receive:
σ(2)1(x,ε)=2x−xexp(−ε|x|),x∈R.
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