Energy-stable global radial basis function methods on summation-by-parts form

Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop and construct energy-stable radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite Element communities.

1. Introduction.We investigate energy stability of global radial basis function (RBF) methods for time-dependent partial differential equations (PDEs).Unlike finite differences (FD) or finite element (FE) methods, RBF schemes are mesh-free, making them flexible with respect to the geometry of the computational domain since the only used geometrical property is the pairwise distance between two centers.Further, they are suitable for problems with scattered data like in climate [10,28] or stock market [5,33] simulations.Finally, for smooth solutions, one can reach spectral convergence [9,11].In addition, they have recently become more and more popular for solving time-dependent problems in quantum mechanics, fluid dynamics, etc. [6,24,25,40].One distinguishes between global RBF methods (Kansa's methods) [26] and local RBF methods, such as the RBF generated finite difference (RBF-FD) [39] and RBF partition of unity (RBF-PUM) [42] method.See the monograph [12] and references therein.Even though their efficiency and good performance have been demonstrated for various problems, only a few stability results are known for advection-dominated problems.For example, an eigenvalue analysis was performed for a linear advection equation in [34], and it was found that RBF discretizations often produced eigenvalues lending to an exponential increase of the  L 2 norm when boundary conditions were introduced.To illustrate this, consider the following example (also found in [18,Section 6.1]): (1.1) with x ∈ [−1, 1], t > 0, and where periodic boundary conditions are applied.In this example, a bump is traveling to the right, leaving the domain and coming back to the left.In Figure 1, we plot the numerical solution and its energy up to t = 10 using a global RBF method with a Gaussian kernel and N = 20 points.An increase of the size of the bump and of the L 2 energy can be seen.For longer times, the computation breaks down.The discrete setting does not reflect the continuous one with zero energy growth and demonstrates the stability problems.
To overcome those, it was shown in [18,19] that a weak formulation could result in a stable method.Recently, L 2 estimates were obtained using an oversampling technique [41].Both these efforts use special techniques, and the question we address in this paper is how to stabilize RBF methods in a general way.Classical summation-by-parts (SBP) operators were introduced during the 1970s in the context of FD schemes and they allow for a systematic development of energy-stable semidiscretizations of well-posed initial-boundary-value problems (IBVPs) [7,38].The SBP property is a discrete analog to integration by parts, and proofs from the continuous setting carry over directly to the discrete framework [31] if proper boundary procedures are added [38].First based on polynomial approximations, the SBP theory has recently been extended to general function spaces developing so-called FSBP operators in [20].Here, we investigate stability of global RBF methods through the lens of the FSBP theory.We demonstrate that many existing RBF discretizations do not satisfy the FSBP property, which opens up for instabilities in these methods.Based on these findings, we show how RBF discretizations can be modified to obtain an SBP property.This then allows for a systematic development of energy-stable RBF methods.We give a couple of concrete examples including the most frequently used RBFs, where L 2 estimates are derived using an oversampling technique.For simplicity, we focus on the univariate setting for developing an SBP theory in the context of global RBF methods.That said, RBF methods and SBP operators can easily be extended to the multivariate setting, which is also demonstrated in our numerical tests.The rest of this work is organized as follows.In section 2, we provide some preliminaries on energy-stability of IBVPs and global RBF methods.Next, the concept of FSBP operators is shortly revisited in section 3. We adapt the FSBP theory to RBF function spaces in section 4. Here, it is also demonstrated that many existing RBF methods do not satisfy the SBP property and how to construct RBF operators in SBP form (RBFSBP).In section 5, we give a couple of concrete examples of RBFSBP operators resulting in energy-stable methods.Finally, we provide numerical tests in section 6 and concluding thoughts in section 7.

Preliminaries.
We now provide a few preliminaries on IBVPs and RBF methods.
2.1.Well-posedness and Energy Stability.Following [23,31,38], we consider (2.1) where u is the solution and L is a differential operator with smooth coefficients.Further, B 0 and B 1 are operators defining the boundary conditions, F is a forcing function, f is the initial data, and g x L , g where C, α c are constants independent of f .Moreover, the IBVP (2.1) is strongly wellposed, if it is well-posed and holds, where the function C(t) is bounded for finite t and independent of F, g x L , g x R , and f .
Switching to the discrete framework, our numerical approximation u h of (2.1) should be constructed in such a way that similar estimates to (2.4) and (2.5) are obtained.We denote our grid quantity (a measure of the grid size) by h.In the context of RBF methods, h denotes the maximum distance between two neighboring points.We henceforth denote by • h a discrete version of the L 2 -norm and • b represents a discrete boundary norm.Then, we define stability of the numerical solution as follows.
Definition 2.2.Let F = g x L = g x R = 0 and f h be an adequate projection of the initial data f which vanishes at the boundaries.The approximation u h is stable if holds for all sufficiently small h, where C and α d are constants independent of f h .The approximated solution u h is called strongly energy stable if it is stable and holds for all sufficiently small h.The function C(t) is bounded for finite t and independent of F, g x L , g x R , and f h .

Discretization.
To discretize the IBVP (2.1), we apply the method of lines.The space discretization is done using a global RBF method resulting in a system of ordinary differential equations (ODEs): Here, u denotes the vector of coefficients and L represents the spatial operator.We used the explicit strong stability preserving (SSP) Runge-Kutta (RK) method of third-order with three stages (SSPRK(3,3)) [36] for all subsequent numerical tests.
2.2.1.Radial Basis Function Interpolation.RBFs are powerful tools for interpolation and approximation [43,8,12].In the context of the present work, we are especially interested in RBF interpolants.Let u : R ⊃ Ω → R be a scalar valued function and X K = {x 1 , . . ., x K } a set of interpolation points, referred to as centers.The RBF interpolant of u is (2.9) Here, ϕ : R + 0 → R is the RBF (also called kernel) and {p l } m l=1 is a basis for the space of polynomials up to degree m − 1, denoted by P m−1 .Furthermore, the RBF interpolant (2.9) is uniquely determined by the conditions Incorporating polynomial terms of degree up to m−1 in the RBF interpolant (2.9) is important for several reasons: (i) The RBF interpolant (2.9) becomes exact for polynomials of degree up to m − 1, i. e., u h = u for u ∈ P m−1 .(ii) For some (conditionally positive) kernels ϕ, the RBF interpolant (2.9)only exists uniquely when polynomials up to a certain degree are incorporated.In addition, we will show that (i) is needed for the RBF method to be conservative [18,20].The property (ii) is explained in more detail in Appendix A as well as in [8,Chapter 7] and [15,Chapter 3.1].For simplicity and clarity, we will focus on the choices of RBFs listed in Table 1.More types of RBFs and their properties can be found in the monographs [43,8,12].Note that the set of all RBF interpolants (2.9) forms a K-dimensional linear space, denoted by R m (X K ).This space is spanned by the cardinal functions which are uniquely determined by the cardinal property and condition (2.11).They also provide us with the following (nodal) representation of the RBF interpolant: (2.16) 2.2.2.Radial Basis Function Methods.We outline the standard global RBF method for the IBVP (2.1).The domain Ω on which we solve (2.1) is discretized using two point sets: • The nodal point set (centers) where N ≥ K.By selecting Y N = X K , we get a collocation method, and with N > K, a method using oversampling.The numerical solution u is defined by the values of u h at Y N and the operator L(u) by using the spatial derivative of the RBF interpolant u h , also at Y N .The RBF discretization can be summarized in the following three steps: 1. Determine the RBF interpolant u h ∈ R m (X K ).
2. Define L(u) in the semidiscrete equation by inserting (2.16) into the continuous spatial operator.This yields 3. Use a classical time integration scheme to evolve (2.8).
Global RBF methods come with several free parameters.These include the center and evaluation points X K and Y N , the kernel ϕ, the degree m − 1 of the polynomial term included in the RBF interpolant (2.9).The kernel ϕ might come with additional free parameters such as the shape parameter ε.Finally, we note that also the basis of the RBF approximation space R m (X K ), that one uses for numerically computing the RBF approximation u h and its derivatives, can influence how well-conditioned the RBF method is in practice.Discussions of appropriate choices for these parameters are filling whole books [43,8,13,12] and are avoided here.In this work, we have a different point in mind and focus on the basic stability conditions of RBF methods.

Summation-by-parts
Operators on General Function Spaces.SBP operators were developed to mimic the behavior of integration by parts in the continuous setting and provide a systematic way to build energy-stable semi-discrete approximations.First, constructed for an underlying polynomial approximation in space, the theory was recently extended to general function spaces in [20].For completeness, we shortly review the extended framework of FSBP operators and repeat their basic properties.We consider the FSBP concept on the interval [x L , x R ] where the boundary points are included in the evaluation points Y N .Using this framework, we give the following definition originally found in [20]: (ii) P is a symmetric positive definite matrix, and Here, f (y) = [f (y 1 ), . . ., f (y N )] T and f ′ (y) = [f ′ (y 1 ), . . ., f ′ (y N )] T respectively denote the vector of the function values of f and its derivative f ′ at the evaluation points y 1 , . . ., y N .Further, D denotes the differentiation matrix and P is a matrix defining a discrete norm.
In order to produce an energy estimate, P must be positive definite and symmetric.In this manuscript and in [20], we focus for stability reasons on diagonal norm FSBP operators [29,14,35].The matrix Q is nearly skew-symmetric and can be seen as the stiffness matrix in context of FE.With these operators, integration-by-parts is mimicked discretely as: (3.1) for all f, g ∈ F.
3.1.Properties of FSBP Operators.In [20], the authors proved that the FSBP-SAT semi-discretization of the linear advection equation yields an energy stable semi-discretization.The so-called SAT term imposes the boundary condition weakly.Moreover, the underyling function space F should contain constants in order to ensure conservations.
In context of RBF methods, constants have to be included in the RBF interpolants (2.9), also for the reasons discussed above.We will extend the previous investigation to the linear advection-diffusion equation. (3.2) where a > ß is a constant and κ > 0 can depend on x and t.The problem (3.2) is strongly well-posed, as can be seen by the energy rate To translate this estimate to the discrete setting, we discretize (3.2).The most straightforward FSBP-SAT discretization reads (3.4) We can prove the following result using the FSBP Definition 3.1.
Proof.We use the energy method together with the FSBP property to obtain with Du 2 K = (Du) T P KDu.This is similar to the continuous estimate (3.3).Note that P and K have to be diagonal to ensure that P K defines a norm.
Clearly, the FSBP operators automatically reproduce the results from the continuous setting, similar to the classical SBP operators based on polynomial approximations [38].Note that no details are assumed on the specific function space, grid or the underlying methods.The only factors of importance is that the FSBP property is fulfilled and that well posed boundary condition are used.In what follows, we will adapt the FSBP theory to radial basis functions.
4. SBP operators for RBFs.First, we adapt the FSBP theory in subsection 2.2 to the RBF framework.Next, we investigate classical RBF methods concerning the FSBP property, and demonstrate that standard global RBF schemes does not fulfill this property.Finally, we describe how RBFSBP operators can be constructed that lead to stability.

RBF-based SBP operators.
The function space F ⊂ C 1 for RBF methods is defined by the description in Subsection 2.2.Consider a set of K points, The set of all RBF interpolants (2.9) forms a K-dimensional approximation space, which we denote by R m (X K ).Let {c k } K k=1 be a basis in R m (X K ).Further, we have the grid points which include the boundaries.They are used to define the RBFSBP operators.
is a symmetric positive definite matrix, and In the classical RBF discretizations, the exactness of the derivatives of the cardinal functions is the only condition which is imposed.However, to construct energy stable RBF methods, the existence of an adequate norm is as important as the condition on the derivative matrix.Hence it is often necessary to use a higher number of grid points than centers to ensure the existence of a positive quadrature formula to guarantee the conditions in Definition 4.1.The norm matrix P in Definition 4.1 has only been assumed to be symmetric positive definite.However, as mentioned above for the remainder of this work, we restrict ourselves to diagonal norm matrices where ω i is the associated quadrature weight because Diagonal-norm operators are i) required for certain splitting techniques [14,30,32], and variable coefficients, see for example (3.6).ii) better suited to conserve nonquadratic quantities for nonlinear stability [27], iii) easier to extend to, for instance, curvilinear coordinates [4,35,37].
Remark 4.2.In Definition 4.1, we have two sets of points, the interpolation points X K and the grid points Y N .The derivative matrix is constructed with respect to the exactness of the cardinal functions c k related to the interpolation points X K .However, all operators are constructed with respect to the grid points Y N , i.e.D, P, Q ∈ R N ×N .This is in particular essential when ensuring the existence of suitable norm matrix P .This means that the size of the SBP operator is determine by the quadrature formula.So, the number of grid points and their placing highly effects the size of the operators and so the efficiency of the underlying method itself.In the future, this will be investigated in more detail.

Existing
Collocation RBF Methods and the FSBP Property.In the classical collocation RBF approach, the centers intersect with the grid points, i.e.X K = Y N .It was shown in [20] that a diagonal-norm F-exact SBP operator exists on the grid with the coefficient matrix G and vector of moments m given by In this case, L > N and the linear system in (4.1) is overdetermined and has no solution.This is demonstrated in Table 2, which reports on the residual and smallest element of the least squares solution (solution with minimal ℓ 2 -error) of (4.1) for different cases.In all of our considered tests, the residuals were always larger than zero indicating that the operator is not in SBP form.Similar results are obtained for non-diagonal norm matrices P , which is outlined in Appendix B.

Existence and Construction of RBFSBP Operators.
Translating the main result from [20], we need quadrature formulas to ensure the exact integration of (R m (X K )R m (X K )) ′ .For RBF spaces, we use least-squares formulas, which can be used on almost arbitrary sets of grid points Y N and to any degree of exactness.The least squares ansatz always leads to a positive and (R m (X K )R m (X K )) ′ -exact quadrature formula as long a sufficiently large number of data points Y N is used.
Remark 4.3.Existing results on positivity and exactness of least squares quadrature formulas usually assume that the function space contains constants [16,17].Translating this to our setting, we need this property to be fulfilled for (R m (X K )R m (X K )) ′ .Therefore, R m (X K ) should contain constants and linear functions.However, this assumption is primarily made for technical reasons and can be relaxed.Indeed, even when R m (X K ) only contained constants, we were still able to construct positive and (R m (X K )R m (X K )) ′ -exact least squares quadrature formulas in all our examples.Future work will provide a theoretical justification for this.Due to the least-square ansatz, we may always assume that we have a positive and (R m (X K )R m (X K )) ′ -exact quadrature formula.With that ensured, we summarize the algorithm to construct a diagonal norm RBFSBP operators in the following steps: 1. Build P by setting the quadrature weights on the diagonal.2. Split Q into its known symmetric 1 2 B and unknown anti-symmetric part Q A . 3. Calculate Q A by using In the RBF context, one can always use cardinal functions as the basis.However, for simplicity reason is can be wise to use another basis representation, derived from the cardinal functions.
5. RBFSBP Operators.Next, we construct RBFSBP operators for a few frequently used kernels 1 .We consider a set of K points, X K = {x 1 , . . ., x K } ⊂ [x L , x R ], and assume that these include the boundaries x L and x R .Henceforth, we will consider the kernels listed in Table 1 and augment them with constants.The set of all RBF interpolants including constants (2.9) forms a K-dimensional approximation space, which we denote by R 1 (X K ).This space is spanned by the cardinal functions c k ∈ R 1 (X K ) which are uniquely determined by (2.15).The matching constraint is then simply K k=1 α k = 0.That is, Note that the right-hand sides of (5.2) and ( 5.3) both use K 2 elements to span the product space R 1 (X K )R 1 (X K ) and its derivative space (R 1 (X K )R 1 (X K )) ′ .However, these elements are not linearly independent and the dimensions of R 1 (X K )R 1 (X K ) and (R 1 (X K )R 1 (X K )) ′ are smaller than K 2 .Indeed, we can observe that c k c l = c l c k and the dimension of (5.2) is therefore bounded from above by (5.4) dim Finally, we point out that in the calculation of the operators P, Q and D below, we will round the numbers to the second decimal place.
2 This basis can be constructed using a simple Gauss elimination method.
We make the transformation to the basis representation span{b 1 , b 2 , b 3 } to simplify the determination of (R 1 (X 3 )R 1 (X 3 )) ′ .In this alternative basis representation, the product space R 1 (X 3 )R 1 (X 3 ) and its derivative space (R 1 (X 3 )R 1 (X 3 )) ′ are respectively given by Next, we have to find an (R 1 (X 3 )R 1 (X 3 )) ′ -exact quadrature formula with positive weights.For the chosen N = 4 equidistant grid points, the least-squares quadrature formula has positive weights and is (R This example was presented with less details in [20].

RBFSBP Operators using Gaussian Kernels. Next, we consider the Gaussian kernel
for the centers X 3 = {0, 1/2, 1}.The three-dimensional Gaussian RBF approximation space (5.1) is given by R 1 (X 3 ) = span{ c 1 , c 2 , c 3 } with cardinal functions (5.To include an example with non-equidistant points for the centers, we also build matrices and FSBP operators with Halton points X 3 for this case.A bit surprising, we need twice as many points than on an equidistant grid to get a positive exact quadrature formula.We obtain an exact quadrature using the nodes and weights x = i/7, T , with i = 0, • • • , 7, and P = diag (0.04, 0.12, 0.19, 0.13, 0.04, 0.10, 0.30, 0.08).The corresponding matrices Q and D are R 8×8 and are given by 5.3.RBFSBP Operators using Multiquadric Kernels.As the last example, we consider the SBPRBF operators using multiquadric kernels ϕ(r) = √ 1 + r 2 on [x L , x R ] = [0, 0.5] and centers X 3 = {0, 1/4, 1/2}.The (R 1 (X 3 )R 1 (X 3 )) ′ -exact least square ansatz yields the points and norm matrix P = diag (0.07, 0.18, 0.18, 0.07) .With this norm matrix, we obtain finally 6. Numerical Results.For all numerical tests presented in this work, we used an explicit SSP-RK methods.The step size ∆t was chosen to be sufficiently small.To guarantee stability, we applied weakly enforced boundary conditions using Simultanuous Approximation Terms (SATs), as is usually done in the SBP community [1,2], and for RBFs in [19].To avoid matrices with high condition number, we sometimes use a multi-block structure in our tests.In each block, a global RBF method is used and the blocks are coupled using SAT terms as in [3,21].We mainly use polyharmonic splines in the upcoming tests.
6.1.Advection with Periodic Boundary Conditions.In the first test, we consider the linear advection (6.1)  with a = 1 and periodic BCs.The initial condition is u(x, 0) = e −20x 2 from the introducing example (1.1) and the domain is [−1, 1].We are in the same setting as shown in Figure 1.We compare a classical collocation RBF method with our new RBFSBP methods, focus on cubic splines and consider the final time to be T = 2.In Figure 2a and Figure 2c, the solutions are plotted using collocation RBF method and the RBFSBP approach.In Figure 2a, we select K = 15 for both approximations.The collocation RBF method damp the Gaussian bump significantly while the RBFSBP method do better.The decrease can also be seen in the energy profile 2b where the collocation approach lose more.To obtain a comparable result between the collocation and RBFSBP methods, we double the number of interpolation points K in our second simulation for the collocation RBF method, cf. Figure 2c and Figure 2d.The RBFSBP method still performs better and demonstrates the advantage of the RBFSBP approach.Next, we focus only on RBFSBP methods and demonstrate the high accuracy of the approach by increasing the degrees of freedom.In Figure 3, we plot the result and the energy using Gaussian (ǫ = 1) and cubic kernels.We use K = 5 and I = 20 blocks.We obtain an highly accurate solution and the energy remains constant.
We have a smooth IC and an inflow BC at the left boundary x = 0. We apply cubic splines with constants as basis functions and the discretization (6.3) u t + aDu = P −1 S.
with the simultaneous approximate terms (SAT) S := [S 0 , 0, . . ., 0] T , S 0 := −(u 0 − g).In Figure 4a -Figure 4b, we show the solutions at time t = 0.5 with K = 5 and I = 15, 20 elements using equidistant point and randomly disturbed equidistant points.The numerical solutions using disturbed points in Figure 4a has wiggles but these are reduced by increasing the number of blocks, see Figure 4b.Note that that the wiggles are more pronounced if the point selection is not distributed symmetrically around the midpoints, e.g. for the Halton points in Figure 4c -Figure 4d.Next, we focus on the error behavior.As mentioned before, the RBF methods can reach spectral accuracy for smooth solutions.In Figure 5, the error behaviour for K = 3 − 7 basis functions using 20 blocks is plotted in a logarithmic scale.Spectral accuracy is indicated by the (almost) constant slope.
6.3.Advection-Diffusion. Next, the boundary layer problem from [44] is considered   with shape parameter are used together with constants.We expect to obtain better results using Gaussian kernels to structure of the steady state solution.In Figure 6, we show the for different times using K = 5 elements on equidistant grid points parameters κ = 0.2 and κ = overshoots can be seen in the more steep case for = 0.1.can be circumvented using more degrees of freedom and  which are avoided in this case.
6.4.2D Linear Advection.We conclude our examples with a 2D case and consider the linear advection equation: (6.4) ∂ t u(x, y, t) + a∂ x u(x, y, t) + b∂ y u(x, y, t) = 0 with constants a, b ∈ R.
6.4.1.Periodic Boundary Conditions.our first test, = b = 1 are used in (6.4).The initial condition is u(x, 0) = e −20((x−0.5)+(y−0.5) 2 ) for (x, y) ∈ [0, 1] and periodic boundary conditions, i. u(0, y, t) = u(1, y, t) and u(x, 0, t) = u(x, 1, t), are The coupling at the boundary was again done via SAT terms.We use cubic kernels (K = equipped with constants.Figure 7b illustrates the numerical solution at time T = 1.The bump has the domain at the right and entered again the left lower corner.It its initial at T = No visible between the numerical solution at T = and the initial condition can be seen.In Figure 7c the is reported time.We notice slide decrease energy when bump is the domain (at t = 0.5) due to weakly slightly dissipative SBP-SAT coupling.

Inflow Conditions.
In the last simulation, we consider (6.4) with a = 0.5, b = 1, initial condition u(x, y, 0) = e −20((x−0.25) 2 +(y−0.25) 2 ) for (x, y) ∈ [0, 1] 2 and zero inflow u(0, y, t) = 0, and u(x, 0, t) = 0. We again use cubic kernels (K = 13) equipped with constants.The boundary conditions are enforced weakly via SAT terms.The initial condition lies in the left corner, cf. Figure 7a.In Figure 8b, the numerical solution is shown.The bump moves in y direction with speed one and in x-direction with speed 0.5.Figure 8c shows a slight decrease of the energy over time due the bump leaving the domain.
7. Concluding Thoughts.RBF methods are a popular tool for numerical PDEs.However, despite their success for problems with sufficient inherent dissipation, stability issues are often observed for advection-dominated problems.In this work, we used the FSBP theory combined with a weak enforcement of BCs to develop provable energy-stable RBF methods.We found ).It should be stressed that b(P ) = 0 is necessary for the given D to satisfy the SBP property, but not sufficient.This follows directly from [20,Lemma 4.3] containing the fact that the derivatives of the basis functions are integrated exactly.In our implementation we solved (B.6) using Matlab's CVX [22].The results for different numbers and types of grid points x as well as kernels ϕ and polynomial degrees m − 1 can be found in Table 3.Our numerical findings indicate that in all cases classic global RBF methods do not satisfy the RBFSBP property.This can be noted from the residual b(P ) = P D + (DP ) T − B 2 corresponding to the minimizer P of (B.6) to be distinctly different from zero (machine precision in our implementation is around 10 −16 ).This result is not suprising and in accordance with the observations made in the literature [18,41].

REFERENCES
Energy profile developing in time

Definition 4 . 1 (
RBF Summation-by-Parts Operators).An operator D and only if a positive and (FF) ′ -exact quadrature formula exists on the same grid (the same requirement as for classical SBP operators).The differentiation matrix D ∈ R N ×N of a collocation RBF method can thus only satisfy the FSBP property if there exists a positive and (R m (Y N )R m (Y N )) ′ -exact quadrature formula on the grid Y N .The weights w ∈ R N of such a quadrature formula would have to satisfy (4.1) Gw = m, w > 0,

Figure 3 :
Figure 3: Gaussian and Cubic kernels with approximation space K = 5 and I = 20 blocks on equidistant points after 10 periods

Figure 4 :
Figure 4: Cubic kernel with approximation space K = 5 on equidistant, and Halton points

Figure 5 :
Figure 5: plots using cubic with approximation space K = 4 − 7 on equidistant points with I = blocks.For K = 5, the errors correspond to the printed in the red dotted line the right side of Figure 4.

Figure 6 :
Figure 6: Gaussian and Cubic kernels with approximation space K = 5 and I = 1 block on equidistant points at T = 2.

Table 1 :
Some frequently used RBFs