TECHCAL TRASACTOS 7/207 CZASOPSMO TECHCZE 7/207 MECHACS DO: 0.4467/2353737XCT.7.5.6656 Artur Krowa (rowa@mech.p.edu.pl) nsttute of Computer Scence, Faculty of Mechancal Engneerng, Cracow Unversty of Technology Jordan Podgórs Student of Appled nformatcs, Faculty of Mechancal Engneerng, Cracow Unversty of Technology The adaptaton of the cross valdaton aproach for rbf-based collocaton methods Adaptacja podejsca rzyżowego sprawdzana do metod bazujących na funcjach rbf Abstract The paper shows the adaptaton of the cross valdaton approach, nown from nterpolaton problems, for estmatng the value of a shape parameter for radal bass functons. The latter are nvolved n two collocaton technques used on an unstructured grd to fnd approxmate soluton of dfferental equatons. To obtan accurate results, the shape parameter should be chosen as a result of a trade-off between accuracy and condtonng of the system. The cross valdaton approach called leave one out taes these aspects nto consderaton. The numercal examples that summarze the nvestgatons show the usefulness of the approach. Keywords: radal bass functons, shape parameter, cross-valdaton approach Streszczene W artyule poazano adaptację algorytmu rzyżowego sprawdzana, znanego z zagadneń statysty nterpolacj, do wyznaczena wartośc współczynna ształtu w radalnych funcjach bazowych. Funcje te użyto w dwóch typach techn oloacyjnych stosowanych na neregularnej satce do przyblżonego rozwązywana równań różnczowych. Aby otrzymać rezultaty o odpowednej doładnośc, współczynn ształtu pownen być dobrany na baze ompromsu pomędzy doładnoścą a uwarunowanem uładu równań. Przedstawony algorytm, zwany leave one out, berze te aspety pod uwagę. Podsumowanem artyułu są numeryczne testy, tóre poazują użyteczność tego podejśca. Słowa luczowe: radalne funcje bazowe, współczynn ształtu, algorytm cross-valdaton 47
. ntroducton n recent years a sgnfcant development of numercal methods for solvng dfferental equatons wth the use of an unstructured grd has been observed. All these technques are called meshless or meshfree methods as opposed to well-nown mesh based methods such as fnte element, fnte dfference, or fnte volume method. The meshless methods can overcome some drawbacs of mesh based technques assocated wth grd dstorton, remeshng n adaptaton approaches and handlng problems characterzed by complcated geometres. There are several formulatons of meshless methods [, 2]. Among them, one can dstngush methods that apply nterpolant composed of radal bass functons (RF) and use the collocaton technque n order to dscretze a dfferental problem the Kansa method and RF-based pseudospectral method (RF-PS). There are many papers devoted to these methods [3 7]. The problem of choosng a respectve value of the shape parameter for RF s the ssue whch appears n almost all of these papers. Ths parameter s responsble for the flatness of RF and nfluences the accuracy of the methods as well as condtonng of the system of algebrac equatons that follows from the dscretzaton procedure. To acheve hgh accuracy, the value of the parameter should be large but ths leads to an ll-condtoned system, whch cannot be easly solved [8]. To estmate a respectve value, a trade-off s needed. Ths value s estmated mostly on the base of researcher s experence, but there are also a few more sophstcated approaches [9, 0]. The present paper shows the adaptaton of a nd of cross valdaton algorthm to ths end. The latter s called leave one out and t has been used n statstcs and nterpolaton problems [5]. The paper presents an easy mplementaton of the approach n the Kansa method and demonstrates that the same value of the parameter s vald for the RF-PS method. The layout of the paper s as follows: n secton 2 two RF-based numercal technques are brefly descrbed, n secton 3 the use of the leave one out algorthm s demonstrated and fnally n secton 4 the numercal tests are shown. 2. RF-based collocaton methods for partal dfferental equatons There are several numercal methods for solvng boundary-value problem. The latter can be wrtten n a general form as: Lu= f n Ω, () u = g on Ω where L and denote lnear dfferental operators mposed on the sought functon u n the doman Ω and on the boundares Ω, respectvely, and f, g are nown functons. Among the methods that tae advantage of rregularly dstrbuted nodes for doman dscretzaton, one can dstngush collocaton technques that employs nterpolant consstng of RF. Such nterpolant has a general form, whch s as follows: 48
u( x) = αjϕ x x j j= where x, =,, represent a set of rregularly dstrbuted nodes n the doman as well as on the boundary. Among them one can dstngush nteror nodes x, =,..., and the nodes mposed on the boundary x, =,...,. n Eq. (2) ϕj( x) = ϕ( x x j ) represents RF and α j are the nterpolaton coeffcents. One of the approach whch falls nto the mentoned category called the Kansa method, ntroduces functon (2) to problem () and by the collocaton technque transforms dfferental problem nto a set of algebrac equatons. Another one uses the RF nterpolant to determne dscrete approxmaton of dervatves ncluded n, thereby obtanng algebrac approxmaton of the consdered problem the RF- PS method. The detals of these methods are gven below. (2) 2.. Kansa method y ntroducng nterpolaton functon descrbed by Eq. (2) nto Eq. () and by collocatng at each node n the doman ncludng boundares, one gets: j= j= α α Lϕ x x = f x, =,...,, j j x j ϕ ( x x j ) x = = g x,,..., Usng matrx notaton one can put Eq. (3) n the followng way: A α=b (4) A L f n Eq. A = b A, = g, where ( AL ) = Lϕ x x j ( j ), =,...,, j=,..., x ( A ) = ϕ x x j ( j ), =,...,, j=,..., x, f = f( x ), =,...,, g = g( x ), =,..., The nterpolaton coeffcents are obtaned from Eq. f only A s nvertble: (3) α =A b (5) The study on the nvertblty of the Kansa matrx (A ) can be found n several papers. One can conclude that although there are numercal examples showng that the matrx can be sngular for arbtrary center locatons [], these cases are rare and many other wors [2] ndcate a successful applcaton of the method. Snce the nterpolaton coeffcents are determned, the approxmate soluton s descrbed by nterpolant (2). 49
2.2. RF-PS method RF-PS s a combnaton of RF nterpolaton wth the pseudospectral technque. n ths approach, the nterpolant n the form of (2) s used to determne the dscrete approxmaton of dfferental operator from Eq. (). To ths end, the nterpolaton condtons are taen nto account: j= αϕ j ( x x j )= u, =,..., (6) t allows for presentng the coeffcents α n terms of the values of the sought functon, whch can be put n the matrx notaton n the followng way: α = A u (7) T where α = α α, u = u u and Aj = ϕ( x x j ),, j=,...,. Then, by mposng an approprate dfferental operator on the nterpolant and evaluatng t at each nteror as well as boundary node, one gets: u = A α (8) L T u L L = A (9) where u L, u are dscrete representaton of approprate dervatves and A L, A are the same matrces that appear n Eq. (4). Usng Eq. (7) one can express dervatves u L and u n Eqs. (8) and (9) n terms of the sought functon values from the whole doman as: α u = A A u (0) T where u =[ ul, u ] and A matrx s composed of A L, A n the the same way as n the Kansa method (Eq. (4)). Matrx A A s a dscrete form of dfferental operators L, and s called as dfferentaton matrx n the nomenclature of pseudospectral methods. Wth the use of ths matrx Eq. () can be easly dscretzed: A A u= b () and solved for unnown functon values, whch yelds: u= AA b (2) From the above t can be clearly seen that the solvablty of the problem usng the RF-PS s condtoned by the nevtablty of the same matrx as n the Kansa approach. As one can notce, the approach presented s smlar to the Kansa method. The man dfference between the RF-PS and Kansa method s that n the latter we ntroduce the 50
nterpolaton functon drectly nto dfferental equaton obtanng the nterpolaton coeffcents. Wth these coeffcents, the nterpolaton functon approxmates the soluton at any pont of the doman. n the RF-PS we use the same nterpolaton functon to derve a dscrete approxmaton of a dfferental operator at each node and then ths approxmaton n the form of a dfferentaton matrx s used to dscretze the equaton. Fnally, n the RF-PS, the functon values at the nodes are obtaned as the soluton. Snce the RF-PS operates drectly wth functon values (does not need to evaluate the nterpolant) t s more effcent n non-lnear problems and n tme dependent problems, where a nd of teratons are requred to obtan the approxmate functon values. 3. Adoptng leave-one-out algorthm for boundary-value-problem t was found [5, 6, 8] that the shape parameter has a sgnfcant nfluence on accuracy. A larger value of ths parameter theoretcally should mae the soluton more accurate but leads to an ll-condtoned system, whch may not be accurately solved. Therefore, an mportant ssue n usng RF based methods s the choce of the approprate value of c. One of the approaches that can be employed comes from the nterpolaton problem: A α =u (3) where A, α and u are the RF nterpolaton matrx, vector of nterpolaton coeffcents and vector of nodal functon values, respectvely. Eq. s a result of the applcaton of the nterpolaton condtons. n ths case values of u are nown. The optmal value of c depends on the number of nodes and on the pattern of ther dstrbuton, on the rght-hand-sde vector and precson of computaton. All these factors are taen nto consderaton n the approach as t s reported n [3]. The approach s based on cross-valdaton and s called leave-one-out. n ths algorthm, an optmal value of c s obtaned by mnmzng the error of an nterpolant based on the data from whch one of the nodes was left out. The error at the th node, whch was left out can be obtaned as: E = u u [ ] ( x ) (4) [ where u s the functon value at ths node and u ] [ ] ( x) = αj ϕ x x nterpolant to the data u = [u j=,,u -,u +,, u ]. j ( j ) s the RF T Removng n turn each of the nodes, the vector of errors E = E E can be composed. The norm of ths vector ndcates the qualty of the ft, whch depends on the shape parameter. y repeatng calculatons for dfferent values of c, one can choose the optmal one whch mnmzes the E norm. Snce the mplementaton of hs strategy s rather expensve, Rppa [3] showed that E can be computed n a smpler way as: 5
E = α A (5) where α s the th coeffcent n the nterpolant u based on a full set of nodes and A s the th element n the dagonal of the nverse of the nterpolaton matrx. n the present paper, the above method for estmatng the optmal c s adopted to methods of solvng dfferental equatons presented n sectons 2. and 2.2. Here, we follow drectly the dea presented by Rppa, understandng the problem descrbed by Eq. () as a nd of nterpolaton problem, but defned for dervatves. At frst, let us consder the method from secton 2., where the dscretzed boundaryvalue-problem s approxmated by Eq.(4). We can consder ths system of equatons as the nterpolaton condtons such as Eq.(3), but wrtten for dervatve of the sought functon. n ths case the nterpolant appled at the nteror nodes assumes the form of: u ( x) = Lu( x) = α Lϕ x x L = and nterpolant for approxmaton of boundary values s as follows: u ( x) = u( x) = α ϕ x x = (6) (7) One can mae use of the leave one out algorthm to obtan the soluton of ths nterpolaton problem n the case, where the th node s omtted: (8) α [ ] [ ] [ = A ] b Wth the obtaned coeffcents, the nterpolant for dervatve at the th node s evaluated [ ] [ ] [ ] ul ( x ) = α Lϕ x x or u ( x ) = α =, yeldng the error at the th node as: x =, [ ] ( ) ϕ x x x (9) E = b u [ ] ( x ) or E = b u [ ] ( x ) (20) L where b s the th element of the rght-hand-sde vector ntroduced by Eq.(4). Mang use of Rppa s acceleraton (Eq.(5)), the above error can be computed faster by: α E = = A b (2) A ( A ) where A s the nverse of dervatve nterpolaton matrx based on full set of nodes, ntroduced by Eq.(4). 52
For the method presented n secton 2.2, Eq. () can be consdered as the nterpolaton problem defned for the dervatves and therefore can be the startng pont for dervaton of the formula for error n the context of leave one out algorthm. Unfortunately, n the RF-PS [ approach we are not able to fnd drectly the value of u ] L ( x ). y solvng the system n the form of Eq. (), but defned by omttng the th node, we obtan u, =,, -, +,,, whch can be used by Eq. (0) to approxmate the dervatves at all the nodes wthout the th node. Therefore, formula (20) cannot be drectly obtaned t requres the value of the dervatve at the th node. [ ] [ ] [ ] [ ] To overcome ths nconvenence, wth the values of u = A ( A ) b at all nodes [ wthout the th node, one should mae a step bac and use Eq. (7) to calculate a ] yeldng: α [ ] [ ] [ ] [ ] [ ] [ ] [ ] = = = A u A A A b A b [ where A ] s a Kansa matrx derved wthout the th node. t s obvous that the same coeffcents as those used n Eq. (8) are obtaned, whch allows for evaluatng the nterpolant for dervatve at the th node and leads to the same formula for the error as n the Kansa approach (Eq.(2)). (22) 4. umercal tests To show the usefulness of the approach proposed n the last secton several equatons n 2D space have been solved wth the Kansa and RF-PS methods. Due to the lmted space, the results of two of them are presented. Example. Posson equaton wth Drchlet boundary condtons: 2 uxy (, ) = sn( π x)sn( π y), ( x, y) Ω = [ 0, ] [ 0, ] uxy (, ) = 0, ( x, y) Ω (23) for whch the analytcal soluton has the form of: u= sn( π x)sn( π y) (24) 2π 2 Example 2. 2D modfed Helmholtz equaton wth non-homogeneous boundary condtons: 2 uxy (, ) u( x, y) = ( π 2 + )( ysn( π x) + xcos( π y)), ( x, y) Ω = [ 0, ] [ 0, ] u( 0, y) = 0, u(, y) = cos( π y), u( x, 0) = x, ux (, ) = sn( π x) x (25) 53
whose exact soluton has the form: uxy (, ) = ysn( π x) + xcos( y) (26) As a measure of the qualty of results, an error norm n the followng form has been ntroduced ε= n e 2 e 2 ( u u ) ( u ) 00%, where u n s numercal soluton, u e the = = exact one. The obtaned results are shown n Tables 4. (regular grd) Table. Results for example solved by Kansa s method c opt ε mn leave one out algorthm c ε 8 2.2846 5.3802e-04.0670 5.6308e-04 2 0.988.5387e-04 0.958 8.7747e-04 69 0.967 6.0927e-05 0.927.279e-04 225 0.896 3.4936e-05 0.886 5.8035e-05 (rrreg. grd) 8.8035.835e-04.0370 4.6253e-04 2.3676 8.3723e-05 0.9367 2.7855e-04 69 0.9868 3.365e-05 0.9367 3.8024e-05 225 0.825.3580e-05 0.8265.6603e-05 Table 2. Results for example solved by the RF-PS method (regular grd) c opt ε mn leave one out algorthm c ε 8.0720 5.398e-04.0670 5.639e-04 2 0.988.5665e-04 0.958 8.866e-04 69 0.967.0287e-04 0.927.8572e-04 225 0.8866 3.9077e-05 0.886 5.0806e-05 (rreg.grd) 8.7935.9638e-04.0370 4.6253e-04 2.2775 8.3239e-05 0.9367 2.7856e-04 69 0.978 3.2367e-05 0.9367 3.6655e-05 225 0.876.4629e-05 0.8265 2.0200e-05 t s obvous that the presented algorthm for fndng a good value of the shape parameter gves the same results for Kansa s method as well as for the RF-PS when appled on the same grd, snce t maes use of the same matrx. Therefore, approprate columns presentng 54
a good value of c possess the same values, comparng between two dscretzaton methods, but for clear comparson of the results they are ncluded n both tables. For comparson, the optmal value of c obtaned on the base of the exact soluton s also ncluded n the tables. y analyzng the results one can conclude that the presented approach gves the values of c that lead to acceptable results. (regular grd) Table 3. Results for example 2 solved by Kansa s method c opt ε mn leave one out algorthm c ε 8 2.4349 4.9988e-04.3075 9.000e-03 2.886 2.606e-04.622 2.5000e-03 69.3626.430e-04.092 5.826e-04 225.973 9.0023e-05 0.9067 4.6734e-04 (rrreg. grd) 8.8587.2793e-04.593 6.5335e-04 2.5230.077e-04 0.967 3.800e-03 69.7 9.9409e-05.722.59e-04 225.009 7.0534e-05 0.9969 4.8329e-04 Table 4. Results for example 2 solved by the RF-PS method (regular grd) c opt ε mn leave one out algorthm c ε 8 2.3547 7.825e-04.3075 9.000e-03 2.6733 3.095e-04.622 2.474e-03 69.5680 2.0440e-04.092 3.7698e-03 225.0620.564e-04 0.9067 5.3632e-04 (rreg.grd) 8.8937.8087e-04.593 6.6200e-04 2.4878.8640e-04 0.967 3.7853e-03 69 0.9768.707e-04.722 4.732e-04 225 0.8766 7.902e-05 0.9969 3.334e-04 5. Concluson n the paper, the problem of accuracy of two meshless collocaton methods that employ RF nterpolaton s consdered. t s well-nown that the accuracy of such methods depends on the value of the shape parameter ncluded n RF. The paper apples a nd of cross valdaton approach, nown form nterpolaton problems, to fnd respectve value of ths parameter. 55
To ths end, the system of algebrac equatons followng from the applcaton of the Kansa or RF-PS method s treated as a nd of nterpolaton problem. The leave one out approach taes nto consderaton several dscretzaton and computatonal parameters to fnd the value of c, whch s a great value of ths approach. The use of ths algorthm requres to set a range, whch s searched for the optmal value of c and many evaluatons of system matrx. These can be consdered as some weanesses of the algorthm that should be mproved n future wor. References [] elytscho T., Krongauz Y., Organ D., Flrmng M., Krysl P., Meshless methods: an overvew and recent developments, Computer Methods n Appled Mechancs and Engneerng, vol. 39, 996, 3-47. [2] Lu G.R., Meshlees Methods Movng beyond the Fnte Element Method, CRC Press, oca Raton, Florda 2003. [3] Kansa E., Multquadrcs A scattered data approxmaton scheme wth applcatons to computatonal flud dynamcs : Surface approxmatons and partal dervatve estmates, Computers and Mathematcs wth Applcatons, vol. 9, 990, 27-45. [4] Kansa E., Multquadrcs A scattered data approxmaton scheme wth applcatons to computatonal flud dynamcs : Solutons to parabolc, hyperbolc, and ellptc partal dfferental equatons, Computers and Mathematcs wth Applcatons, vol. 9, 990, 47-6. [5] Fasshauer G.E., Meshfree Approxmaton Methods wth Matlab, World Scentfc Publshng, Sngapore, 2007. [6] Cheng A.H.D., Multquadrcs and ts shape parameter a numercal nvestgaton of error estmate, condton number and round-off error by arbtrary precson computaton, Engneerng analyss wth boundary elements, vol. 36, 202, 220-239. [7] Ferrera A.J.M, A formulaton of the multquadrc radal bass functon method for the analyss of lamnated composte plates, Compost Structures, vol. 59, 2003, 385-92. [8] Schabac R., Error estmates and condton numbers for radal bass functon nterpolaton, Advances n Computatonal Mathematcs, vol. 3, 995, 25-264. [9] Krowa A., Radal bass functon-based pseudospectral method for statc analyss of thn plates, Engneerng Analyss wth oundary Elements, vol. 7, 206, 50-58. [0] Krowa A., On choosng a value of shape parameter n Radal ass Functon collocaton methods, umercal Methods for Partal Dfferental Equatons, submtted for publcaton. [] Hon Y.C., Schabac R., On nonsymmetrc collocaton by radal bass functons, Appl. Math. Comput., vol. 9, 200, 77-86. [2] Chen W., Fu Z.J., Chen C.S., Recent Advances n Radal ass Functon Collocaton Methods, Sprnger, 204. [3] Rppa S., An algorthm for selectng a good value for the parameter c n radal bass functon nterpolaton, Adv. n Comput. Math., vol., 999, 93-20. 56