RBF-based partition of unity method for elliptic PDEs: Adaptivity and stability issues via VSKs

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RBF-based partition of unity method

for elliptic PDEs:

Adaptivity and stability issues via VSKs

Niccol´o Tonicello

RBF-based partition of unity method for elliptic PDEs:

Adaptivity and stability issues via VSKs

Niccol´o Tonicello

19 Dicembre 2017

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for elliptic PDEs:

Niccol´o Tonicello

global vs local

Global:

1 Spectral Methods (orthogonal polynomials-based methods)

2 Meshless Methods (RBF-based methods) Local:

1 Finite Element Method

2 Finite Volume Method

3 Finite Differences Method

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for elliptic PDEs:

Niccol´o Tonicello

Global approach

Global

I(x) = R(x) =

N

X

k=1

c_kΦ(x , x_k) Ac = f with A_{i ,k} = Φ(x_i, x_k)

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for elliptic PDEs:

Niccol´o Tonicello

Partition of Unity

d

[

j =1

Ωj = Ω

W_j s.t.

d

X

j =1

W_j(x ) = 1 ∀x ∈ Ω

I(x) =

d

X

j =1

Wj(x )Rj(x ) with Rj(x ) =

Nj

X

k=1

c_k^jΦ(x , x_k^j) A_jc_j = f_j

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for elliptic PDEs:

Niccol´o Tonicello

Collocation method

Collocation

L(I(x_i)) = g₁(x_i) x_i ∈ Ω I(x_i) = g₂(x_i) x_i ∈ ∂Ω

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for elliptic PDEs:

Niccol´o Tonicello

Poisson’s equation

In our case L = −∆.

Local Matrix:

L_j = ( ¯W_j^∆A_j + 2 ¯W_j^∇A^∇_j + ¯W_jA^∆_j )A⁻¹_j Where:

(A^∆_j )_{i ,k} = ∆Φ(x_i^j, x_k^j) (A^∇_j )_{i ,k} = ∇Φ(x_i^j, x_k^j) (Aj)_{i ,k} = Φ(x_i^j, x_k^j)

( ¯W_j^∆)_k,k = ∆W_j(x_k^j) ( ¯W_j^∇)_k,k = ∇W_j(x_k^j)

( ¯Wj)k,k = Wj(x_k^j)

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for elliptic PDEs:

Niccol´o Tonicello

Poisson’s equation

Global Matrix

L_{i ,k} =

d

X

j =1

(L_j)_ζ_{i ,j}_ζ_k,j

Finally you solve the global linear system:

LI = f with

I = (I(x₁), ..., I(x_N)) f = (f₁, ..., f_N) with fi = g1(xi) for xi ∈ ˙Ω and fi = g2(xi) for xi ∈ ∂Ω

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for elliptic PDEs:

Niccol´o Tonicello

Stability Issues

1 Tikhonov regularization

2 Variably Scaled Kernel

3 Hybrid Variably Scaled Kernel

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for elliptic PDEs:

Niccol´o Tonicello

Tikhonov regularization

Original LS system

minI (||LI − f||²₂)

Tikhonov regularization

minI (||LI − f||²₂+ ||ΓI||²₂) Where usually Γ =p(γ)I and γ = 10⁻¹⁰∼ 10⁻¹⁵

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for elliptic PDEs:

Niccol´o Tonicello

Separation Distance & fill distance

Separation distance:

qX_N = 1 2min

i 6=k ||x_i − x_k||₂ Fill Distance:

hX_N = sup

x∈Ω

( min

x∈XN

||x − x_k||₂)

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for elliptic PDEs:

Niccol´o Tonicello

Convergence Estimate for interpolation

Suppose Ω ⊆ R^M is bounded and satisfies an interior cone condition. Suppose that Φ ∈ C^2k(Ω × Ω) is symmetric and strictly positive definite and let f ∈ N_Φ(Ω), where N_Φ is the native space of Φ. Then, there exist positive constants h₀ and C , independent of x, f , and Φ, such that:

|f (x) − R(x)| ≤ Ch^k_X

N

pC_Φ(x)||f ||_N_Φ_(Ω) Provided h_N ≤ h₀ and f ∈ N_Φ(Ω), where

C_Φ(x) = max

|β|=2k( max

x,z∈Ω∩B(w ,C2h_N(|D₂^βΦ(w, z)|))

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for elliptic PDEs:

Niccol´o Tonicello

Variably Scaled Kernel

Ψj : x → (x, ψj(x)) Increase of distances:

||Ψ_j(x) − Ψ_j(y)||²₂ ≥ ||x − y||²₂ VSK:

K((x, ψ(x)), (y, ψ(y))) ∀x, y ∈ R^M Local interpolant:

R_ψ_j(x) =

Nj

X

k=1

c_k^jK((x, ψ(x)), (x^j_k, ψ(x^j_k))) x ∈ Ω_j

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for elliptic PDEs:

Niccol´o Tonicello

Variably Scaled Kernel

Discrete local operator:

L¯_ψ_j = ( ¯W_j^∆A_ψ_j + 2 ¯W_j^∇A^∇_ψ_j + ¯WjA^∆_ψ_j)A⁻¹_ψ

j

Where simply A^∆_ψ

j, A^∇_ψ

j, A_ψ_j are little modifications of the previous definitions arising from the fact that:

Φ = Φ((||x − x^j_k||²+ (ψ_j(x) − ψ_j(x^j_k))²)¹²)

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for elliptic PDEs:

Niccol´o Tonicello

HVSK

Algorithm 1 HVSK

1: for j = 1 to d do

2: Computing A_j

3: Checking σm

4: if σ_m < (1e − 16)/ε⁴ then

5: VSK

6: else

7: STANDARD

8: end if

9: end for

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for elliptic PDEs:

Niccol´o Tonicello

Residual Adaptive Subsampling Scheme

We first start from a set

X_N = {x⁽¹⁾_i , i = 1, ..., N⁽¹⁾}

where we compute the solution and a test set of interior points:

Y_N_˜(1) = {y⁽¹⁾_i , i = 1, ..., ˜N⁽¹⁾} then we compute the residual on the test set:

r_i⁽¹⁾= f (y⁽¹⁾_i ) − I_N(1)(y⁽¹⁾_i )

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for elliptic PDEs:

Niccol´o Tonicello

Residual Adaptive Subsampling Scheme

then we define the following two sets of points:

S_T(1) 1

= {y⁽¹⁾_i : r_i⁽¹⁾> τ1, i = 1, ..., T₁⁽¹⁾}

ST₂⁽¹⁾ = {¯x⁽¹⁾_i : r_i⁽¹⁾< τ₂, i = 1, ..., T₂⁽¹⁾} Finally at the next step we will consider a new set X :

X_N^(k+1) = X_N^(k+1)

b

[X_N^(k+1)

c

where

X_N^(k+1)

c = (X_N^(k)

c

[S

T₁^(k))\S

T₂^(k)

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for elliptic PDEs:

Niccol´o Tonicello

Results

RMSE

RMSE :=

v u u t 1 s

s

X

i =1

|f (˜xi) − I(˜xi)|²

ψj Map

ψj =

Nj

X

i =1

|p_i^j(x, x^j_i)|

with

p_i^j(x, x^j_i) = 1

π arctan(h^j_i(x₁− x_{i 1}^j )e^−5(x²^−x^{i 2}^j⁾)

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for elliptic PDEs:

Niccol´o Tonicello

Map ψ

_j

Nc data set hX_N qXN

81 original 1.03e-1 1.07e-2 mapped 2.93e-1 3.68e-2 289 original 5.72e-2 2.07e-3 mapped 6.34e-2 6.73e-3 1089 original 3.27e-2 1.12e-3 mapped 5.79e-2 4.34e-3 4225 original 1.68e-2 1.74e-4 mapped 4.85e-2 6.79e-4

Table:Separation and fill distances of the original data set compared with the ones mapped via VSKs (Halton points)

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for elliptic PDEs:

Niccol´o Tonicello

RMSE vs ε

Figure:RMSEs obtained by varying ε for the Gaussian C^∞kernel.

From left to right, top to bottom, we consider Nc = 81, 289, 1089 and 4225 Halton data.

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for elliptic PDEs:

Niccol´o Tonicello

RASS

Figure: An illustrative example of RASS procedure

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for elliptic PDEs:

Niccol´o Tonicello

RASS

Figure:The iterations versus the MR and RMSE for Halton data. In the left frame we use the HVSK approach, while in the right one the standard bases

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for elliptic PDEs:

Niccol´o Tonicello

Summary

1 Global + Local

1 Classical PU collocation method → stability issues

1 Tikhonov regularization

2 VSK → HVSK + RASS

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for elliptic PDEs:

Niccol´o Tonicello

Work in progress

1 Different PDEs (parabolic)

2 Parallel computing implementation