Cavoretto, R.

De Rossi, A.

English

Institutional Research Information System University of Turin

Rendiconti Sem. Mat. Univ. Pol. TorinoVol. 76, 2 (2018), 77 – 86R. Cavoretto* - A. De Rossi*AN ADAPTIVE ALGORITHM BASED ON RBF-PUCOLLOCATION FOR SOLVING 2D POISSON PROBLEMSAbstract.In this paper we present a new adaptive algorithm to solve 2D Poisson problems via aradial basis function partition of unity collocation method. The refinement process dependson an error indicator, which consists in comparing two approximate solutions computed on acoarser set and a finer one of collocation points. This technique is able to detect the domainareas that need to be refined, so that the accuracy of the method can be improved. Numericalresults illustrate the performance of the adaptive algorithm.1. IntroductionIn this work we investigate the problem of constructing adaptive methods via radialbasis functions (RBFs) for solving 2D Poisson problems. In particular, the use ofadaptive meshless algorithms has recently been considered by applying different RBF-based approaches, which include finite difference (RBF-FD) [4, 10], partition of unity(RBF-PU) [2] or, more in general, RBF collocation [5, 9] methods. More precisely,in [2] we proposed within a RBF-PU framework a new error estimator based on thecomparison of two collocation solutions evaluated on a coarser set and a finer one.Since this indicator was not extensively analyzed in our previous paper, here we presentin detail the new adaptive algorithm along with its error indicator. This study is furthersupported by numerical experiments carried out on some test problems.The paper is organized as follows. In Section 2 we review some basic informa-tion of the RBF-PU collocation method for solving elliptic PDE problems. In Section 3we describe the adaptive algorithm along with its refinement startegy. In Section 4 weshow some numerical results carried out to illustrate the performance of our adaptivescheme. Section 5 contains conclusions and future work.2. RBF-PU collocation methodLet Ω be an open and bounded domain on Rs and u : Ω R. Given a set XNx1, . . . ,xN Ω of collocation points, we partition the domain Ω into a number ofd subdomains or patches Ω j such that dj 1Ω j Ω with some mild overlaps amongthem [8]. As subdomains we take hyperspherical patches on Rs of fixed radius suchthat Ω j dj 1 is a covering of Ω. Along with the subdomains we define a partitionof unity by considering a family of compactly supported, nonnegative and continuous*Member of the INdAM Research group GNCS.7778 R. Cavoretto - A. De RossiRBF φε rInverse MultiQuadric C (IMQ) 1 ε2r2 1 2MultiQuadricC (MQ) 1 ε2r2 1 2Mat e´ rn C6 (M6) e εr ε3r3 6ε2r2 15εr 15Mat e´ rn C4 (M4) e εr ε2r2 3εr 3Table 1: Some examples of popular RBFs table 1, 2 and 3 (also in the text).functions wj, with supp wj Ω j, and such thatdj 1wj x 1, x Ω.A possible choice is given by Shepard’s weightwj xw˜ j xdk 1w˜k x, j 1, . . . ,d,where w˜ j are compactly supported functions on Ω j (see [3, 11]).After choosing the weights wj dj 1, the global RBF-PU approximant is formedby a weighted sum of local RBF approximants u˜ ju˜ xdj 1wj x u˜ j x , x Ω.(2.1)withu˜ j xNji 1c ji φε x xji 2 ,(2.2)where Nj identifies the number of nodes in Ω j, i.e. the points x ji XNj XN Ω j, cjiis an unknown real coefficient, 2 denotes the Euclidean norm, and φ : R 0 R isa RBF depending on a positive shape parameter ε such thatφε x z 2 φ ε x z 2 , x,z Ω.In Table 1 we list some examples of RBFs with their smoothness degrees (see e.g.[1, 12] for details). These radial functions are commonly used for solving PDEs.Now, given a linear elliptic differential operatorL , our aim is to find the approxi-mate solution of the elliptic PDE problem equipped with Dirichlet boundary conditionsLu x f x , x Ω,u x g x , x Ω.(2.3)An adaptive algorithm for RBF-PU collocation 79The problem (2.3) is then discretized on the global set XN of collocation points,which is split into a set XNI of interior points and a set XNB of boundary points suchthat XN XNI XNB , where NI and NB denote the number of interior and boundarydiscretization points, respectively.Assuming that the problem (2.3) has an approximate solution of the form (2.1)(see [7]), we getL u˜ xidj 1L wj xi u˜ j xi f xi , xi Ωu˜ xidj 1wj xi u˜ j xi g xi , xi Ω.(2.4)Focusing on a Laplace problem, i.e. taking L ∆, the elliptic operator L canexpanded as∆ wj xi u˜ j xi ∆wj xi u˜ j xi 2∇wj xi ∇u˜ j xi w j xi ∆u˜ j xi , xi Ω.(2.5)Defined then the vector u˜ j u˜ j xj1 , . . . , u˜ j xjNjT of local nodal values, weknow that the local coefficient vector c j cj1, . . . ,cjNjT is such that c j A 1j u˜ j. Sowe obtain∆u˜ j A∆j A 1j u˜ j, ∇u˜ j A∇j A1j u˜ j,(2.6)where A∆j and A∇j , j 1, . . . ,Nj, are the matrices with entriesA∆j ki ∆φ xjk xji 2 ,andA∇j ki ∇φ xjk xji 2 .Associated with each subdomain Ω j, we consider the following diagonal matricesW∆j diag ∆wj xj1 , . . . ,∆wj xjNj ,W∇j diag ∇wj xj1 , . . . ,∇wj xjNj ,Wj diag wj xj1 , . . . ,wj xjNj .In order to get the discrete operator Pj, we differentiate (2.4) by using a product deriva-tive rule and then apply the relations given in (2.6). Making use of (2.5) and includingthe boundary conditions, the discrete local Laplacian can thus be expressed as followsPj kiP¯j ki, xji Ω,δki, x ji Ω,80 R. Cavoretto - A. De Rossiwhere δki is the Kronecker delta andP¯j W∆j A j 2W∇j A∇j WjA∆j A1j .By assembling the local matrices Pj into the global matrix P, i.e.Pj kidj 1Pj ηk j ,ηi j , k, i 1, . . . ,N,we can derive the global discrete operator, and finally solve the (global) sparse linearsystem of the formPy u,(2.7)where u u1, . . . ,uN T is defined byuif xi , xi Ω,g xi , xi Ω.and the numerical solution y u˜ x1 , . . . , u˜ xN T is obtained by inverting the collo-cation matrix P in (2.7).3. Adaptive algorithmIn this section we present the iterative algorithm proposed to adaptively solve the PDEproblem via the RBF-PU collocation method. It can be summarized as follows.Step 1. We define two sets XN of grid collocation points of size N01 and N02 , suchthat N 01 N02 (the symbol0 identifies the initial phase, or more in general theiteration). To make clearer the description, we denote these sets as XN 01and XN 02,respectively.Step 2. For k 0,1, . . ., we iteratively determine the two collocation solutions ofthe form (2.1), i.e. u˜N k1and u˜N k2, computed on N k1 and Nk2 collocation points,respectively.Step 3. We compare the two approximate solutions by evaluating the error on the(coarser) set containing N k1 points, i.e.u˜N k2xi u˜N k1xi , xi XN k1.Note that here we suppose that the solution given with N k2 collocation points is moreaccurate than that containing N k1 points only.An adaptive algorithm for RBF-PU collocation 810 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91x1x 20 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91x1x 20 0.2 0.4 0.6 0.8 1x100.10.20.30.40.50.60.70.80.91x 20 0.2 0.4 0.6 0.8 1x100.10.20.30.40.50.60.70.80.91x 2Figure 1: Examples of refinement strategy regarding the sets XN k1(left) and XN k2(right) in the adaptive algorithm.Step 4. Fixed a tolerance tol, we find all points xi XkN1 such thatu˜N2 k xi u˜N1 k xi tol.(3.8)Step 5. To refine the data point distribution, we compute the separation distanceqX kN112mini jxi x j 2, xi XN k1.(3.9)Step 6. For k 0,1, . . . we update the two sets XN k 11and XN k 12of discretizationpoints as follows. For each point xi XN k1, such that the condition (3.8) is satisfied,we add to xi at most:• four points (the blue circles depicted in the top-left frame of Figure 1), thusgenerating the set XN k 11;• eight points (the red squares shown in the top-right frame of Figure 1), thuscreating the set XN k 12.82 R. Cavoretto - A. De RossiHowever, when the point xi is on or close to the boundary, the number of involvedpoints is obviously reduced, as shown in Figure 1 (bottom). In both cases the newpoints are given by properly either adding or subtracting the value of (3.9) to the com-ponents of xi. Moreover, we remark that in the illustrative example of Figure 1 thepoint xi is marked by a black cross, while the new sets are such that XN k1XN k2, fork 1,2, . . ..Step 7. The iterative procedure stops when there are no points anymore that satisfythe condition (3.8), returning the set XN k2(where k denotes the last iteration).4. Numerical resultsIn this section we summarize the results derived from application of our adaptive re-finement algorithm, which is implemented in MATLAB. All the results are carried outon a laptop with an Intel(R) Core(TM) i7-6500U CPU 2.50 GHz processor and 8GBRAM.In the following we focus on two test problems of the form (2.3) defined on thedomain Ω 0,1 2. The exact solutions of such Poisson problems areP1 : u1 x1,x2120exp 4x1 cos 2x1 x2 ,P2 : u2 x1,x2 exp 8 x1 0.5 2 x2 0.05 2 .A graphical representation of these analytic solutions is shown in Figure 2.Figure 2: Graphs of exact solutions u1 (left) and u2 (right) of Poisson problems.In the tests we illustrate the performance of the new adaptive scheme applied toRBF-PU collocation method by using M6 as RBF (cf. Table 1) with ε 3. The twostarting sets defined in Section 3 consist of N1 0 289 and N2 0 1089 grid or quasi-random Halton [6] collocation points, while the tolerances used in these experimentsAn adaptive algorithm for RBF-PU collocation 83are given by tol 10 5 and tol 10 4. Note that the method is basically not mesh-based, so we have no problem about possible “distortion” of the original structure ofnodes. In fact, although as expected the level of achievable accuracy with non griddedpoints is slightly lower, the method effectively works also with scattered/random nodes.To measure the quality of our results, we give the Root Mean Square Error (RMSE),i.e.,RMSE1NevalNevali 1u zi u˜ zi 2,which is evaluated on a grid of Neval 40 40 evaluation points. Then, in order toanalyze the stability of the method, we evaluate the Condition Number (CN) of thesparse collocation matrix P in (2.7) by using the MATLAB command condest. As toefficiency we report the CPU times computed in seconds.In Tables 2–3 we present the results obtained, also indicating the final numberNfin of discretization points required to achieve the tolerances fixed. Additionally, wereport the “refined grids” after applying iteratively our adaptive algorithm. Precisely,in Figures 3–4 we plot the final collocation points obtained after applying the iterativeprocess for test problems P1 and P2.0 0.2 0.4 0.6 0.8 1x100.10.20.30.40.50.60.70.80.91x 20 0.2 0.4 0.6 0.8 1x100.10.20.30.40.50.60.70.80.91x 2Figure 3: Final distribution of discretization points obtained by applying the adaptivealgorithm with gridded points via M6, ε 3 and tol 10 5, for problems P1 (left) andP2 (right).Test Problem Nfin RMSE CN timeP1 1541 5.15e 6 4.85e 07 8.4P2 1445 3.88e 6 1.03e 07 4.7Table 2: Results obtained by starting from (interior) gridded points for M6 with ε 3and tol 10 5.84 R. Cavoretto - A. De Rossi0 0.2 0.4 0.6 0.8 1x100.10.20.30.40.50.60.70.80.91x 20 0.2 0.4 0.6 0.8 1x100.10.20.30.40.50.60.70.80.91x 2Figure 4: Final distribution of discretization points obtained by applying the adaptivealgorithm with Halton points via M6, ε 3 and tol 10 4, for problems P1 (left) andP2 (right).Test Problem Nfin RMSE CN timeP1 1387 2.85e 5 3.53e 10 6.4P2 1369 2.44e 5 4.18e 09 2.9Table 3: Results obtained by starting from (interior) Halton points for M6 with ε 3and tol 10 4.Analyzing the numerical results, we note that the adaptive algorithm increasesthe number of points in the regions where the solution behavior varies. As regardsefficiency, then, the numerical scheme converges in few seconds in each of the testscarried out. Finally, the CN is quite similar in both cases and not so high as it oftenhappens for traditional collocation RBF methods (see [6]).Finally, in order to assess the advantage of our adaptive scheme, as a compari-son we report the results obtained by applying the RBF-PU method on uniform gridswithout using adaptive refinement but with a comparable number of nodes. In partic-ular, for problem P1 we take N 392 1521 obtaining a RMSE = 3.26e 4, whereasfor problem P2 we consider N 382 1444 collocation points reaching a RMSE =2.77e 5. As evident from these experiments, comparing the results with the onesshown in Table 2, the adaptive algorithm enables us to increase the precision of ournumerical method.5. Conclusions and future workIn this paper we presented an adaptive algorithm to solve time-independent ellipticPDEs such as Poisson problems. This refinement strategy is applied to the RBF-PUcollocation method. In particular, we proposed an adaptive approach based on a refine-An adaptive algorithm for RBF-PU collocation 85ment technique, which consisted in comparing two approximate solutions computedon a coarser set and a finer one of collocation points. This process allowed us to detectthe domain areas that needed to be refined so as to improve the accuracy of the method.Numerical results showed the performance of our algorithm on some test problems.As future work we propose to extend our adaptive algorithm for solving othertypes of differential problems, including also the case of possible discontinuities.AcknowledgmentsThe authors sincerely thank the anonymous referee for the insightful comments andsuggestions. Moreover, they acknowledge support from the Department of Mathemat-ics “G. Peano” of the University of Torino via 2018 project “Mathematics for applica-tions”. This work was partially supported by GNCS–INdAM 2018 project “Methods,algorithms and applications of multivariate approximation”. This paper is dedicated toProf. C. Dagnino on the occasion of her retirement.References[1] BUHMANN M.D., Radial basis functions: Theory and implementation, Cambridge University Press,Cambridge, 2003.[2] CAVORETTO R. AND DE ROSSI A., Adaptive Meshless Refinement Schemes for RBF-PUM Colloca-tion, Applied Mathematics Letters 90 (2019), 131–138.[3] CAVORETTO R., DE ROSSI A. AND PERRACCHIONE E., Optimal Selection of Local Approximants inRBF-PU Interpolation, Journal of Scientific Computing 74 (2018), 1–22.[4] DAVYDOV O. AND OANH D.T., Adaptive Meshless Centres and RBF Stencils for Poisson Equation,Journal of Computational Physics 230 (2011), 287–304.[5] DRISCOLL T.A. AND HERYUDONO A.R.H., Adaptive Residual Subsampling Methods for RadialBasis Function Interpolation and Collocation Problems, Computers & Mathematics with Applications53 (2007), 927–939.[6] FASSHAUER G.E.,Meshfree approximation methods withMATLAB, World Scientific, Singapore 2007.[7] HERYUDONO A., LARSSON E., RAMAGE A. AND SYDOW L.V., Preconditioning for Radial BasisFunction Partition of Unity Methods, Journal of Scientific Computing 67 (2016), 1089–1109.[8] LARSSON E., SHCHERBAKOV V. AND HERYUDONO A., A Least Squares Radial Basis Function Par-tition of Unity Method for Solving PDEs, SIAM Journal on Scientific Computing 39 (2017), A2538–A2563.[9] LIBRE N.A., EMDADI A., KANSA E.J., SHEKARCHI M. AND RAHIMIAN M., A Fast AdaptiveWavelet Scheme in RBF Collocation for Nearly Singular Potential PDEs, CMES. Computer Modelingin Engineering & Sciences 38 (2008), 263–284.[10] OANH D.T., DAVYDOV O. AND PHU H.X., Adaptive RBF-FD Method for Elliptic Problems withPoint Singularities in 2D, Applied Mathematics and Computation 313 (2017), 474–497.[11] SAFDARI-VAIGHANI A., HERYUDONO A. AND LARSSON E., A Radial Basis Function Partition ofUnity Collocation Method for Convection-Diffusion Equations, Journal of Scientific Computing 64(2015), 341–367.[12] WENDLAND H., Scattered data approximation, Cambridge University Press, Cambridge 2005.86 R. Cavoretto - A. De RossiAMS Subject Classification: 65D15, 65M70Roberto CAVORETTO, Alessandra DE ROSSIDepartment of Mathematics “G. Peano”, University of TorinoVia Carlo Alberto 10, 10123 Torino, ITALYe-mails: roberto.cavoretto@unito.it, alessandra.derossi@unito.itLavoro pervenuto in redazione il 16-5-19.

An adaptive algorithm based on RBF-PU collocation for solving 2D Poisson problems

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An adaptive algorithm based on RBF-PU collocation for solving 2D Poisson problems

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