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Preconditioning RBF collocation method by adapted finite difference matrix

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Abstract

In this paper, we shall consider an adapted finite difference preconditioning technique for radial basis function (RBF) method. This technique, for the system of equations that arises from solving Helmholtz equation with Dirichlet boundary conditions by RBF collocation method, is considered. We prove that the eigenvalues of preconditioned matrices are bounded for one and two-dimensional cases. We also show that RBF interpolating matrix can be well-conditioned by using appropriate number of polynomial bases in augmented term of RBF approximation. Numerical experiments show the efficiency and robustness of the preconditioning procedure, particularly when there are non-zero boundary conditions.

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Acknowledgments

We thank the respected reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

    F. Ghoreishi & Changiz Goli Keshavarzi

Authors
  1. F. Ghoreishi
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  2. Changiz Goli Keshavarzi
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Correspondence to F. Ghoreishi.

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Communicated by: Robert Schaback

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Appendix

Appendix

(**********The Mathematica code for Example 2. *********)

Clear[“Global*”];

n = Input[“Please enter n”];

b1 = 0;b2 = y;b3 = 0;

f = Input[“Please enter the RHS of your equation”];

Do[x[j] = N[Cos[j ∗Pi/n]],{j,0,n}]

(******** The Finite difference matrix ********)

h[n] = 1;

h[− 1] = 1;

Do[h[k] = x[k + 1] − x[k],{k,0,n − 1}]

equations1 = Table[2/(h[j] + h[j − 1]) ∗ (−u1[k, j + 1]/h[j] + (1/h[j] + 1/h[j − 1])

∗u1[k, j] −u1[k, j − 1]/h[j − 1]) + (b1/.{x → x[k],y → x[j]}) ∗ ((h[k − 1]

∗u1[k + 1,j] + (h[k] − h[k − 1]) ∗u1[k, j] − h[k] ∗u1[k − 1,j])/(2 ∗ h[k] ∗ h[k − 1]))

+(b2/.{x → x[k],y → x[j]}) ∗ ((h[j] ∗u1[k, j + 1] + (h[j] − h[j − 1])

∗u1[k, j] − h[j − 1]∗u1[k, j − 1])/(2 ∗ h[j] ∗ h[j − 1])) + (b3/.{x → x[k],y → x[j]}) ∗u1[k, j]

−(f/.{x → x[k],y → x[j]}),{j,0,n},{k,0,n}];

FinitUnknowns = Table[u1[0,j],{j,0,n}];

Do[FinitUnknowns = Join[FinitUnknowns,Table[u1[i, j],{j,0,n}]],{i,1,n}];FiniteDifferenceMatrix = (Normal[CoefficientArrays[Flatten[equations1],FinitUnknowns]][[2]]);

FiniteDifferenceMatrix = PadRight[FiniteDifferenceMatrix,{(n + 1)∧2,2 ∗ (n + 1)∧2}];

AugMat = PadLeft[IdentityMatrix[(n + 1)∧2],{(n + 1)∧2,2 ∗ (n + 1)∧2}];

FiniteDifferenceMatrix = Join[FiniteDifferenceMatrix,AugMat];

MM = PadRight[IdentityMatrix[(n + 1)∧2],{(n + 1)∧2,2 ∗ (n + 1) ∧2}];

Do[

a = Position[FinitUnknowns,u1[0,j]][[1]][[1]];

FiniteDifferenceMatrix = ReplacePart[FiniteDifferenceMatrix,a →MM[[a]]]

,{j,0,n}]

Do[

a = Position[FinitUnknowns,u1[n, j]][[1]][[1]];

FiniteDifferenceMatrix = ReplacePart[FiniteDifferenceMatrix,a →MM[[a]]],{j,0,n}]Do[a = Position[FinitUnknowns,u1[j,0]][[1]][[1]];

FiniteDifferenceMatrix = ReplacePart[FiniteDifferenceMatrix,a →MM[[a]]]

,{j,0,n}]

Do[

a = Position[FinitUnknowns,u1[j, n]][[1]][[1]];

FiniteDifferenceMatrix = ReplacePart[FiniteDifferenceMatrix,a →MM[[a]]]

,{j,0,n}]

(******* The RBF Collocation Matrix *******)

m = n;

c[i_,j_] = 1 + KroneckerDelta[i, j];

L[u_]:=D[u,{y,2}] + b1 ∗ D[u, y] + b2 ∗ D[u, x] + b3 ∗ u

φ[r_,i_,b_]:=Sqrt[r∧2 + (c[i, b])∧2]

norm[{x_,y_}]:=Sqrt[x∧2 + y∧2]

Do[Do[t = i + j ∗ (n + 1);X[t] = {x[j],x[i]},{j,0,n}],{i,0,n}]

(********** Lagrange interpolation ***********)

Do[pp[j] = Product[x − x[i],{i,0,j − 1}] ∗Product[x − x[i],{i, j + 1,n}],{j,0,n}]

Do[qq[j] = Product[y − x[i],{i,0,j − 1}] ∗Product[y − x[i],{i, j + 1,n}],{j,0,n}]

Do[p[i, x] = pp[i]/(pp[i]/.x → x[i]),{i,0,n}];

Do[p[i, y] = qq[i]/(qq[i]/.y → x[i]),{i,0,n}];

(************* Nodal points *************)

Do[Do[t = i + j ∗ (m + 1);p[t] = p[i, x] ∗ p[j, y],{j,0,m}],{i,0,m}];

PolynomialPart = Sum[𝜃[k] ∗ p[k],{k,0,(m + 1)∧2 − 1}];

Y = {x, y};

approximation = Sum[λ[k] ∗ φ[norm[Y − X[k]],k, b],{k,0,(n + 1)∧2 − 1}] + PolynomialPart;

For[i = 0,i ≤ (n + 1)∧2 − 1,i++,

If[((X[i][[1]]≠ 1)&&(X[i][[1]]≠ − 1)&&(X[i][[2]]≠ 1)&&(X[i][[2]]≠ − 1)),

r[i] = ((L[approximation])/.{x → X[i][[1]],y → X[i][[2]],b → i}) − (f/.{x → X[i][[1]],y → X[i][[2]]}),

r[i] = (approximation/.{x → X[i][[1]],y → X[i][[2]],b → i})]]

RBFUnknowns = Join[Table[λ[j],{j,0,(n + 1)∧2 − 1}],Table[𝜃[j],{j,0,(n + 1)∧2 − 1}]];

equations2 = Table[r[i],{i,0,(n + 1)∧2 − 1}];

RBFCollocationMatrix = Normal[CoefficientArrays[equations2,RBFUnknowns]][[2]];

Ihat = PadLeft [IdentityMatrix[2], {(n + 1)2,(n + 1)2}] ;

Ibast = PadRight [IdentityMatrix [(n + 1)2 − 2], {(n + 1)2,(n + 1)2}] ;

cons1 = Transpose[Join[Ibast,Ihat]];

RBFCollocationMatrix = Join[RBFCollocationMatrix,cons1];

RHS1 = Normal[CoefficientArrays[equations2,RBFUnknowns]][[1]];

RHS2 = ConstantArray [0,(n + 1)2] ;

RHS = Join[RHS1,RHS2];

solution1 = Chop[Solve[RBFCollocationMatrix.RBFUnknowns −RHS == 0]];

approximation = approximation/.solution1[[1]];

RBFUnknowns = Join[Table[λ[j],{j,0,(n + 1)∧2 − 1}],Table[𝜃[j],{j,0,(n + 1)∧2 − 1}]];

RBFPsmat = RBFCollocationMatrix;eigenvalues1 = Eigenvalues[RBFPsmat];real1 = Re[eigenvalues1];imaginary1 = Im[eigenvalues1];

eigenvalues1 = Table [{real1[[i]],imaginary1[[i]]}, {i,1,2(n + 1)2}] ;

m1 = Graphics[{EdgeForm[{Red,Thickness[.4]}],FaceForm[Lighter[White,1]],Disk[]}];

p1 = ListPlot[eigenvalues1,Axes →False,Frame →True,PlotMarkers →{{m1,0.029}},FrameLabel →

{“ Real ”,“Imaginary ”,,“”},GridLines →Automatic,GridLinesStyle →

Directive[RGBColor[0.7,1,0.7],Dashed],BaseStyle- >{FontSize → 6,FontFamily →Times},PlotRange →{{− 70,70},{− 80,80}},

PlotLabel →"WithoutPrecondition"]

(************ RBF interpolation matrix************)

approximation = Sum[λ[k] ∗ φ[norm[Y − X[k]],k, b],{k,0,(n + 1)∧2 − 1}] + PolynomialPart;

Do[r[i] = ((approximation)/.{x → X[i][[1]],y → X[i][[2]],b → i}),{i,0,(n + 1)∧2 − 1}]

equations3 = Table[r[i],{i,0,(n + 1)∧2 − 1}];

RBFInterpolationMatrix = Normal[CoefficientArrays[equations3,RBFUnknowns]][[2]];

cons1 = Transpose[Join[Ibast,Ihat]];

RBFInterpolationMatrix = Join[RBFInterpolationMatrix,cons1];

(**** Computing the inverse of interpolation matrix by using algorithm 1. ****)

A1 = RBFInterpolationMatrix;

AA = ToeplitzMatrix[A1[[1]]];

ZeroMatrix = Table [0, {i,0,2(n + 1)2 − 1}, {j,0,2(n + 1)2 − 1}];

Do[B[i] = Transpose[(A1 −AA)][[i]];

B[i] = Transpose[ReplacePart[ZeroMatrix,i- >B[i]]], {i,1,2(n + 1)2}] ;

AI[1] = Chop[Inverse[AA]];

Do [AI[i + 1] = Chop[AI[i] − (1/(N[1 + Tr[B[i].AI[i]],30]) ∗AI[i].B[i].AI[i])], {i,1,2(n + 1)2}]

InvRBFInterpolationMatrix = AI [2(n + 1)2 + 1];

(************** Collocation and interpolation with boundary conditions ***********)

matrbf = RBFPsmat.InvRBFInterpolationMatrix;

MM = PadRight[IdentityMatrix[(n + 1)∧2],{(n + 1)∧2,2 ∗ (n + 1)∧2}];

Do[

a = Position[FinitUnknowns,u1[0,j]][[1]][[1]];

matrbf = ReplacePart[matrbf,a →MM[[a]]],

{j,0,n}];

Do[

a = Position[FinitUnknowns,u1[n, j]][[1]][[1]];

matrbf = ReplacePart[matrbf,a →MM[[a]]],

{j,0,n}];

Do[

a = Position[FinitUnknowns,u1[j,0]][[1]][[1]];

matrbf = ReplacePart[matrbf,a →MM[[a]]]

,{j,0,n}];

Do[

a = Position[FinitUnknowns,u1[j, n]][[1]][[1]];

matrbf = ReplacePart[matrbf,a →MM[[a]]]

,{j,0,n}];

PreconditionedMatrix = Inverse[FiniteDifferenceMatrix].matrbf;

eigenvalues2 = Eigenvalues[PreconditionedMatrix];

real2 = Re[eigenvalues2];

Imaginary2 = Im[eigenvalues2];

eigenvalues2 = Table [{real2[[i]],Imaginary2[[i]]}, {i,1,2(n + 1)2}] ;

m1 = Graphics[{EdgeForm[{Blue,Thickness[0.4]}],FaceForm[Lighter[White,1]],Disk[]}];

p2 = ListPlot[eigenvalues2,Axes →False,Frame →True,PlotMarkers →{{m1,0.029}},FrameLabel →

{“ Real ”,“Imaginary ”,,“”},GridLines →Automatic,GridLinesStyle →

Directive[RGBColor[0.7,1,0.7],Dashed],BaseStyle- >{FontSize → 6,FontFamily →

Times},PlotRange →{{− 2.4,1.2},{−.1,.1}},PlotLabel →"WithPrecondition"]

spectrum = Eigenvalues[PreconditionedMatrix.Transpose[PreconditionedMatrix]];

Print[“The condition number of coefficient matrix after preconditioning = ”,Sqrt[Max[Abs[spectrum]]/Min[Abs[spectrum]]]]

solution2 = −Inverse[PreconditionedMatrix].Inverse[FiniteDifferenceMatrix].RHS;

ExactSolutionution = Input[“Please enter the exact solutionution For computing the uniform norm of the error”];

error = Table[Abs[solution2[[s + 1]] − (ExactSolutionution/.{x → X[s][[1]],y → X[s][[2]]})],{s,0,(n + 1)∧2 − 1}];

UniformError = Max[error];

Print[“The uniform norm of the error for given n after preconditioning = ”,UniformError]

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Ghoreishi, F., Keshavarzi, C.G. Preconditioning RBF collocation method by adapted finite difference matrix. Adv Comput Math 45, 3293–3325 (2019). https://doi.org/10.1007/s10444-019-09738-1

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