User Manual: TASMANIAN Sparse Grids
Contents
1. For a wavelet grid it returns rule_wavelet 3 20 function getOneDRuleDescription const char getOneDRuleDescription const Returns a short string description of the one dimensional rule used 13 3 21 function getNumPoints int getNumPoints const Return the total number of abscissas associated with the grid 3 22 function getPoints void getPoints double amp pnts const Return the abscissas associated with the grid pnts on input it is either NULL or a non NULL pointer that would be deleted on output returns an array of size getNumDimensions x getNumPoints of values that rep resent the abscissas The first abscissa is located in the first getNumDimensions number of entries the second abscissa is located in the second getNumDimensions number of entries and so on 3 23 function getWeights void getWeights double amp weights const Return the quadrature weights associated with the abscissas as in equation 1 1 weights on input it is either NULL or a non NULL pointer that would be deleted on output it is an array of size getNumPoints of the quadrature weights associated with the abscissas The first weight is associated with the first abscissa returned by getPoints the second weight is associated with the second abscissa and so on 14 3 24 function getInterpolantWeights void getInterpolantWeights const double x doublex amp weights const Returns the inte2. A 1 Global Grids AZ Local Polynomials sc oad ERA AREA A oa PES ARAS A 3 Wavelet Grid LIST OF FIGURES vi LIST OF TABLES A l Summary of the available quadrature rules For each rule we have the enumer ated type as rule_ followed by the string given to the tasgrid and MATLAB interfaces crepas rd br ra A vii ABSTRACT This documents serves to explain the functionality of the Sparse Grid module of the Toolkit for Adaptive Stochastic Modeling And Non Intrusive Approximation TASMANIAN The document covers the three main components the libtasmaniansparsegrids library the tasgrid wrapper and the MATLAB interface ACKNOWLEDGEMENTS The ORNL is operated by UT Battelle LLC for the United States Department of Energy under Contract DE AC05 000R22725 1 Introduction Sparse Grids is a family of algorithms for constructing multidimensional quadrature and interpo lation rules from tensor products of one dimensional such rules 8 10 18 19 22 Given a function f x R gt R a quadrature rule that integrates f a over the domain I C R with respect to the weight p x is defined as by the abscissas 2 CT and weights w _ CR so that Jona N gt wif ei 1 1 The integration weight p x is a tensor of one dimensional weight p x pi 11 p1 x12 pi ta and the domain T is the tensor of a one dimensional domain 1 I An interpolation rule is defined by abscissas 1 Y
3. grid that are not yet associated with values of the interpolated function The matrix file will have getNumPoints number of rows and getNumDimensions number of columns The first abscissa will be on the first row the second on the second row and so on print write out the same data as in the outputfile but to the cout stream 4 12 Command loadvalues tasgrid loadvalues lt optionl gt lt valuel gt lt option2 gt lt value2 gt Reads a grid from a file and the values of the interpolated function is read from a matrix file The function values are given to the grid via the loadValues function and the modified grid is written back to the same input file gridfile this is the file with an already created grid On exit it will contain the grid with loaded values inputfile is a matrix file with getNumNeededPoints number of rows and getNumOutputs number of columns The first row contains the values of the interpolated function associated with the first needed abscissa The second row corresponds to the second abscissa and so on 23 4 13 Command evaluate tasgrid evaluate lt optionl gt lt valuel gt lt option2 gt lt value2 gt Reads a grid from a file and a list of points of interest form a matrix file The interpolant is evaluated at all the points and the result is written to a matrix file gridfile this is the file with an already created grid and loaded values inputfile is a matrix file with points
4. scissa 1s stored on one row of the matrix 5 5 function tsgRecycleGrid newp tsgRecycleGrid 1Grid depth order type anisotropy Calls tasgrid with the recycle command INPUTS IGrid is an object created by tsgMakeGrid depth same as in tsgMakeGrid and depth order same as in tsgMakeGrid and order 29 type same as in tsgMakeGrid and type anisotropy same as in tsgMakeGrid and anisotropy transformAB same as in tsgMakeGrid and inputfile OUTPUTS newp is an optional output MATLAB matrix containing the set of abscissas that are not yet asso ciated with values from the interpolated function 5 6 function tsgGetQuadrature weights points tsgGetQuadrature 1Grid Calls tasgrid with the getquadrature command INPUTS IGrid is an object created by tsgMakeGrid OUTPUTS weights is a MATLAB matrix containing the quadrature weights associated with the points ponits is a MATLAB matrix that contains the abscissas associated with the sparse grid Each ab scissa 1s stored on one row of the matrix 5 7 function tseGetInterpolationWeights weights tsgGetInterpolationWeights 1Grid points Calls tasgrid with the getinterweights command INPUTS IGrid is an object created by tsgMakeGrid points isa MATLAB matrix with dim number of columns and arbitrary number of rows Each row represents one point of interest OUTPUTS weights isa MATLAB matrix of getNumPoitns number
5. 4 for more details Advanced Build Options Open the Makefile in an editor and adjust the options CC specifies the compiler command The code was written for the GNU C compiler GCC The default command is g however that can be changed to force a specific version of the compiler or even a different compiler OpenMP is used throughout the code for multicore parallelism It can be optionally enabled by spec ifying the COMPILE_OPTIONS fopenmp or alternatively disabled by removing the op tions OPTC specifies standard GCC compiler options refer to the GCC manual for details Known Problems Mac users have reported problems with OpenMP and some versions of GCC Mac users that want to use OpenMP should make sure to have the latest available version of GCC versions 4 7 and newer tend to resolve most issues 3 LIBTASMANIANSPARSEGRIDS libtsg All of the sparse grids functionality is included in the libtsg C library Code that interfaces with the library should include the TasmanianSparseGrid hpp which introduces the TasGrid namespace and the definition of the TasmanianSparseGrid class WARNING The code performs virtually no sanity check on the validity of input Wrong input would most likely result in a crash 3 1 Constructor TasmanianSparseGrid TasmanianSparseGrid This is the only class constructor called by default makes an empty grid Before any op erations can be performed a grid has to be made with one of
6. as opposed to the RAM and the Grid object is the pointer Also the grids are associated by name as opposed to a memory address e If multiple users are sharing the same temporary folder then it would be useful if they come up with a naming convention that prevents two users from using the same grid name For example instead of both users creating a grid named mygrid the users should name their grids johngridl and janegridl e All of the grid data for all of the grids is stored in the same folder Anyone with access to the temporary folder has full access to all of the sparse grid data which is a potential security issue e If two users have separate copied of tsgGetPaths then they can use separate storage folders without any of the multi user considerations This is true even if all other files are shared including the tasgrid executable and libtsg library 34 A Types of One Dimensional Rules A 1 Global Grids All global grids use Lagrange polynomial for interpolation Gauss rules are most suitable for optimal integration with respect to a given weight The non Gauss rules are more suitable for interpolation However depending on the structure of f x it is perfectly possible for a non Gauss rule to produce a more accurate integral or for a Gauss rule to produce a more accurate interpolant For more details on the various quadrature rules and their properties see 1 5 7 14 17 24 Clenshaw Curtis Also known as nested Cheb
7. matrix or a matrix of size 2 x 1 with the a and P parameters See alpha and beta option for tasgrid and const double alpha_beta parameter for libtsg is either an empty MATLAB matrix or a matrix of dim rows and 2 columns that contains the a and b parameters associated with the transformation of the domain This is the same as inputfile given to tasgrid with the makegrid command See Table A 1 for the various types of transformation 28 OUTPUTS IGrid is an object that saves the grid name and some additional parameters the object is used by other functions to access the files associated with the grid ponits is an optional output MATLAB matrix that contains the abscissas associated with the sparse grid Each abscissa is stored on one row of the matrix 5 4 function tsgMakeQuadrature weights points tsgMakeQuadrature dim oned depth order type anisotropy alpha_beta transformAB Calls tasgrid with the makequadrature command The grid is not written to a file and only the abscissas and weights are returned The inputs are the same as tsgMakeGrid however no sGridName and out are needed INPUTS Same as tsgMakeGrid except no sGridName is needed as the grid isn t stored permanently and out is assumed to be zero OUTPUTS weights is a MATLAB matrix containing the quadrature weights associated with the points ponits is a MATLAB matrix that contains the abscissas associated with the sparse grid Each ab
8. of interest The file can have arbitrary number of rows and getNumDimensions number of columns Each row corresponds to one point of interest outputfile is an optional matrix file that is written on exit The file contains the values of the inter polant at the points provided by the inputfile The file has the same number of rows and getNumOutputs number of columns Each row contains the values of the interpolant at the corresponding point of interest print write out the same data as in the outputfile but to the cout stream 4 14 Command integrate tasgrid integrate lt optionl gt lt valuel gt lt option2 gt lt value2 gt Reads a grid from a file and integrates the interpolant over the domain The result is written to a matrix file gridfile this is the file with an already created grid and loaded values outputfile is an optional matrix file that is written on exit The file contains the integrals of the inter polant over the domain The file has one row and getNumOutputs number of columns print write out the same data as in the outputfile but to the cout stream 24 4 15 Command refine tasgrid refine lt optionl gt lt valuel gt lt option2 gt lt value2 gt Reads a grid from a file and improves the interpolant by adding a new set of abscissas Refer to setRefinement for details gridfile this is the file with an already created grid and loaded values tolerance is a positive real numb
9. the makeGlobalGrid makeLo calPolynomialGrid or make WaveletGrid functions or alternatively the grid can be read from a stream file using the read functions in order to read a grid it must first be written to the file with the write function The user can also call getVersion and getLicense functions at any time Calling any other function will result in a Segfault 3 2 Destructor TasmanianSparseGrid TasmanianSparseGrid This is the destructor that releases any dynamical memory used by the class this instance of the class can no longer be used 3 3 function getVersion const char getVersion const Returns the version of the library which is a simple hard coded string 3 4 function getLicense const charx getlicense const Returns a short string indicating the license of the library This is a simple hard coded string 3 5 function makeGlobalGrid void makeGlobalGrid int dimensions int outputs int depth TypeDepth type TypeOneDRule oned const int anisotropic_weights 0 const double xalpha_beta 0 This function creates a sparse grid induced by one of the global quadrature and interpolation rules dimensions outputs depth type See Appendix A for a full list of the rules The parameters are described as follows is a positive integer specifying the dimension of the grid There is no hard restriction on how big the dimension can be however
10. 2 fejer 2 f fla de f f x dx 1 1 f 1 f 0 Yes Gauss Legendre rule_gausslegendre gauss legendre fo f a de J f w dx dd N A No Gauss Chebyshev type 1 rule_gausschebyshev1 gauss chebyshev 1 fy f x 12 8dx f f x b x 05 x a dx 1 1 N A No Gauss Chebyshev type 2 rule_gausschebyshev2 gauss chebyshev 2 fo F 2 Fda J f a b x a a Fdx 1 1 N A No Gauss Gegenbauer rule_gaussgegenbauer gauss gegenbauer ie f x 1 2 da f f x b x a a edx 1 1 Must specify a No Gauss Jacobi rule_gaussjacobi gauss jacobi Si FA 1 z dz f f b x x a dz 1 1 Must specify a 3 No Gauss Laguerre rule_gausslaguerre gauss laguerre jn ate de f f a a ajee 0 0 dar 0 oo Must specify a No Gauss Hermite rule_gausshermite gauss hermite JS fla a e dx JE fa a aJ2e 0 0 dy oo 00 Must specify a No Local Polynomials rule pwpolynomial local polynomial Si f x dx f f x dx 1 1 N A Yes Local Polynomials with Zero Boundary rule_pwpolynomial0 local polynomial zero 1 b J Fede J Hoja en f 1 f 0 Yes Local Wavelets rule_wavelet local wavelet J fa de J fade 1 1 N A Yes Table A 1 Summary of the available quadrature rules For each rule we have the enumerated type as rule_ followed by the string given to the tasgrid and MATLAB interfaces that at a given depth we can only use a polynomial of the same order There are two variations of th
11. C T and basis functions x as N f x X cgile 1 2 i 1 where the coefficients c are chosen so that N gt ola i l The coefficients c can be found by solving a system of linear equations i e c _ are the result of a linear transformation applied to f x _ In general the conditioning of this linear map is of consideration however a suitable choice of abscissas x and functions x results in a well conditioned problem An alternative definition of the interpolant is given by f a a gt hi x f z i 1 where the functions h x are constructed by applying the dual of the linear transformation to the vector of basis functions _ For a specific point 7 the approximation becomes F 5 gt hi f xi Ze gif 2 1 3 where g h Z are the interpolation weights One dimensional integration and interpolation rules can be transformed from a domain I to ro where re is the image of I under a linear map defined by a and b For example if T 1 1 then an arbitrary interval a b is the image of T under the linear transformation b a baa T q 3 Analogously the integration weight p 1 can be transformed into p x Extending the result in multiple dimensions it is trivial to define a map for every dimension and have T co Q T922 Q Q res and the integration weights p x pi xi pi 2 prlta For a full list of the transformations supported by this co
12. Nu merische Mathematik 40 1982 pp 407 422 35 15 A KLIMKE AND B WOHLMUTH Algorithm 847 Spinterp piecewise multilinear hier archical sparse grid interpolation in matlab ACM Transactions on Mathematical Software TOMS 31 2005 pp 561 579 3 38 16 X MA AND N ZABARAS An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations Journal of Computational Physics 228 2009 pp 3084 3113 3 38 17 R S MARTIN AND J WILKINSON The implicitql algorithm Numerische Mathematik 12 1968 pp 377 383 35 18 F NOBILE R TEMPONE AND C G WEBSTER An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data SLAM Journal on Numerical Analysis 46 2008 pp 2411 2442 2 3 7 8 19 F NOBILE R TEMPONE AND C G WEBSTER A sparse grid stochastic collocation method for partial differential equations with random input data SLAM Journal on Numeri cal Analysis 46 2008 pp 2309 2345 2 3 7 20 K PETRAS Smolyak cubature of given polynomial degree with few nodes for increasing dimension Numerische Mathematik 93 2003 pp 729 753 35 21 J SHEN AND L L WANG Sparse spectral approximations of high dimensional problems based on hyperbolic cross SIAM Journal on Numerical Analysis 48 2010 pp 1087 1109 7 22 S A SMOLYAK Quadrature and interpolation formulas for tensor products of certain
13. ORNL REPORT Unlimited Release Printed August 2013 User Manual TASMANIAN Sparse Grids M Stoyanov Prepared by Oak Ridge National Laboratory One Bethel Valley Road Oak Ridge Tennessee 37831 The Oak Ridge National Laboratory is operated by UT Battelle LLC for the United States Department of Energy under Contract DE AC05 000R22725 Approved for public release further dissemination unlimited oax RIDGE NATIONAL LABORATORY MANAGED BY UT BATTELLE FOR THE DEPARTMENT OF ENERGY DOCUMENT AVAILABILITY Reports produced after January 1 1996 are generally available free via the U S Department of Energy DOE Information Bridge Web site http www osti gov bridge Reports produced before January 1 1996 may be purchased by members of the public from the following source National Technical Information Service 5285 Port Royal Road Springfield VA 22161 Oak Ridge TN 37831 Telephone 703 605 6000 1 800 553 6847 TDD 703 487 4639 Fax 703 605 6900 E mail info ntis gov Web site http www ntis gov support ordernowabout htm Reports are available to DOE employees DOE contractors Energy Technology Data Exchange ETDE representatives and International Nuclear Information System INIS representatives from the following source Office of Scientific and Technical Information P O Box 62 Oak Ridge TN 37831 Telephone 865 576 8401 Fax 865 576 5728 E mail reports osti gov Web site http www osti gov contact
14. PADS Se 24 414 Command IDEAS castrado eas 24 LIS Command Tenne or OEE RRS EPR KS PR As WR EROS RNA 25 4 16 Command IA ic heh Oo ed ea eee nee eae EERU pee eee 25 4 17 Matrix File Forniat 3 scn0 3 6 00 a Bbw ns SH ws op ed BG oo ee 26 gt MATLAB Interia cionado A 27 314 Tunction USOC AINE seess on adas sa 27 5 2 functions tsgReadMatrix and tsgWriteMatrixO oo ooooooooocooroomooo 28 gt f nc on ts Mako s eie nasasama gise a AS weed ead oa ee gaa 28 54 function tseMakeQuadratite lt br dees bye eee ed aa 29 5 3 f nction tseRecycleGidO csc des reges ii desler cae adhe idea 29 30 Rincon e QUISE recensie oak EE a AAA AR 30 5 7 function tseGetinterpolationWeighis 2 0 6c ci ees abe dwesa ad ideews 30 5 8 function tseGetNeededP omits siii A A 31 5 9 Tonction teeLoad Vanies 6640s oo kde AAA SOE 31 5 10 function Se YUCA AA ade RA SOR de ee Hees 31 5 11 function teelntestalet copii tics iria ad BA hee ees 32 5 12 function tsRenneGid eseese hressa a ennai hooks aae A Ea iaa 32 5 13 function eer rie si conse ease eee ke esa ous RA RRA RAR 33 314 function tseDeleteGrid srs bos ee kok wen A id A EES KEE ean FES 33 3 15 fonction tee DeleteGrid By Namal lt 65549095 ada 33 5 16 function tse LisiGridshy Name socorro rogar ta ae ade ob 33 5 17 fanc on eer Kaine cias a A A AAA 34 ALE MAA 34 E A 34 Appendix A Types of One Dimensional Rules os oss isos ccccccccccccves secs cass sseeeesesssec 35 1v
15. S EYEE GES a HY ES 13 3 20 function cetOneDRuleDeschiption s vomita 13 3 21 function CoINUIMPOUNS puto EE KEE EER ESS ER 14 3 22 ss AS a aa r dhe eee T eee 14 J23 TIRSO CET A Ph a ee KS AR 14 3 24 function getInterpolantWeights circa rr dde 15 3 25 function getNumNeededPoints ccoo 15 3 20 function getNeededPoints RR ERA 15 3 27 function loadNeededPoints oooooooooooonrrrr morron ooo s 16 3 26 function evadir a a aaa 16 3 22 TICO IESO cra a AA ia 16 3 30 TURCO PONES IAS ADS AAA A AA AAA 16 3 31 function setRefinement e ES 17 iii E e ee baa a ee tk ha nE REEE RAS EE RER ACERS LARA D ARREARS RTE RE CES 17 4 TASGRID coi a A ARRE wads teen eed seer ieee exe es 18 Od Basie idee so ck oe oh ooh eee OE ea de ese ae cea eee ees 18 4 2 o SIND res 5 2 EG ch E ASA EGS 3 E EERE 18 43 Command VEO eiere oor ew tee erin detente beeen ew eta es eee 18 AA ACOMMANG SESE oo ss ey ba eae onset eae eee RE Eee Shee eee 18 4 5 Command MAKETAS AA SES Rea AA AAA 19 4 6 Command smakequadrature lt a 21 47 Command ICONO renies riter Ses eRe RA ERATOR E EERIE S TI EERE 21 4 8 Command getg adrat re s ee Ses bes ood e DE E A aa a A a a 22 49 Command DEDOS si n cao bh KS ee ken EENE PENE A RS ERNE GREER 22 4 10 Command getintefrweights 2 osc nel ee haga ve ow ddda AA onsen eos 22 411 Command cetmeedsdpoints y s s aasre a 23 4 12 Command Loa dvalues coso rrrrrrrra road Saad 23 413 Command AA DHA ADS CHRO A DS P
16. a will be on the first row the second on the second row and so on is an optional file The grid can be saved in this file for future use is an optional matrix file however unlike regular matrix files the entries must be integers otherwise the behavior of the code becomes unpredictable The matrix file must have dimensions number of rows and only one column If a global grid is being created then the file will specify the anisotropy_weights array given to makeGlobalGrid If a tensor grid is being created then this file will specify the order array given to makeFullTensorGrid write out the same data as in the outputfile but to the cout stream 20 4 6 Command makequadrature tasgrid makequadrature lt optionl gt lt valuel gt lt option2 gt lt value2 gt This command works the same as makegrid except for the outputfile command outputfile 4 7 is an optional matrix file At the end of the program tasgird will write in the file the quadra ture weights and abscissas associated with the grid The matrix file will have getNumPoints number of rows and dimensions plus one number of columns The first abscissa will be on the first row the second on the second row and so on On each row the first column is the weight and the rest of the columns are the associated abscissa Command recycle tasgrid recycle lt optionl gt lt valuel gt lt option2 gt lt value2 gt This command creates a new gr
17. classes of functions Dokl Akad Nauk SSSR 4 1963 pp 240 243 English translation 2 5 T 23 M STOYANOV Hierarchy direction selective approach for locally adaptive sparse grids Tech Rep ORNL TM 2013 384 Oak Ridge National Laboratory 2013 17 24 A H STROUD AND D SECREST Gaussian quadrature formulas vol 374 Prentice Hall Englewood Cliffs NJ 1966 35 25 W SWELDENS AND P SCHRODER Building your own wavelets at home in Wavelets in the Geosciences Springer 2000 pp 72 107 38 26 G ZHANG AND M GUNZBURGER Error analysis of a stochastic collocation method for parabolic partial differential equations with random input data SIAM Journal on Numerical Analysis 50 2012 pp 1922 1940 3 27 G ZHANG M GUNZBURGER AND W ZHAO A sparse grid method for multi dimensional backward stochastic differential equaitons Journal of Computational Mathematics 31 2013 pp 221 248 3 40 v1 0 oax RIDGE NATIONAL LABORATORY MANAGED BY UT BATTELLE FOR THE DEPARTMENT OF ENERGY
18. d loadNeededPoints const double vals Provides the values of the function to be interpolated evaluated at the corresponding abscissas vals is an array of size getNumOutputs x getNumNeededPoints The first getNumOutputs entries correspond to the outputs of the interpolated function at the first abscissa point The second set of getNumOutputs entries correspond to the second abscissa and so on 3 28 function evaluate void evaluate const double x double y const Finds the value of the interpolant at the provided point x as defined by equation 1 2 The result is written into y x an array of size getNumDimensions that indicate the point where the interpolant should be evaluated y an already allocated array of size getNumOutputs On exit the entries of y are overwritten with the values of the interpolant at the point x 3 29 function integrate void integrate double y const Integrates the interpolant over the domain and returns the result in y y an already allocated array of size getNumOutputs On exit the entries of y are overwritten with the values of the integral of the interpolant over the domain 3 30 function printStats void printStats Prints short description of the sparse grid The output is written to standard output i e cout 16 3 31 function setRefinement void setRefinement double tolerance TypeRefinement criteria Improves the accuracy of the sparse grid bas
19. de see Appendix A The goal of Sparse Grids is to select a subset from all possible tensor product abscissas x so that maximum accuracy is achieved for the smallest number of function evaluations f x This is usually achieved by balancing the error in different dimensions For more information on the properties of sparse grids see 2 4 6 8 10 12 13 15 16 18 19 22 26 27 The TASMANIAN Sparse Grid code implements a number of different quadrature rules and function basis The rules are grouped in three categories e Global Grids suitable for globally smooth functions Quadrature is based on a number of available rules see Appendix A and interpolation is based on global Lagrange polynomials e Local Polynomial Grids suitable for non smooth functions with locally sharp behavior Interpolation is based on hierarchical piece wise polynomials with local support and user specified order These grids are suitable for local refinement e Wavelet Grids are similar to the local polynomials however when coupled with local refinement often times wavelet grids provide the same accuracy with fewer abscissas For a given grid the code can perform three tasks e generate a set of abscissas x _ and weights w for a quadrature rule of type 1 1 e generate a set of abscissas x _ and for the user specified function values A 8 i 1 P i 1 the code can create an interpolant of type 1 2 The interpolant can be evaluated
20. dimensional quadrature and interpolation rules 4 3 Command version Prints the version of the library and executable 4 4 Command test tasgrid test Performs a series of basic functionality tests For different grids different parameters and all possible quadrature rules tasgrid will perform a test to make sure that it can integrate or interpolate appropriate functions to a high degree of precision The output of the command should be a list of 18 the tests and the Pass or Fail result A failure of a test in an indication that something went wrong in the build process or there is a bug in the code Note that the wavelet tests take a long time and hence they are performed last Unless one is interested in using wavelet grids the wavelet tests can be skipped 4 5 Command makegrid tasgrid makegrid lt optionl gt lt valuel gt lt option2 gt lt value2 gt This command creates a new sparse grid It uses the following options that can be given in any order onedim specifies whether to use global local polynomial or wavelet grids as well as the underlying one dimensional quadrature and interpolation rule The available values for the rules are summarized in Table A 1 clenshaw curtis creates a global grid with Clenshaw Curtis rule chebyshev creates a global grid with Chebyshev rule fejer 2 creates a global grid with Fejer type 2 rule gauss legendre creates a global grid with Gauss Legendre rule gauss chebyshe
21. e local polynomial rule that assume zero and non zero boundary Also note that quadrature rules based on local polynomials may have abscissas associated with zero weights For more details on the local polynomials see 10 15 16 A 3 Wavelet Grid Same as local polynomials however it assumes that order is either 1 or 3 The boundary is not assumed to be zero and only one type of local refinement is possible Functionality would be expanded in the future For more details on the wavelet grid see 10 13 25 38 REFERENCES 1 M ABRAMOWITZ AND I A STEGUN Handbook of Mathematical Functions With Formu lars Graphs and Mathematical Tables vol 55 Dover Publications 1964 35 2 S ACHARJEE AND N ZABARAS A non intrusive stochastic galerkin approach for model ing uncertainty propagation in deformation processes Computers amp structures 85 2007 pp 244 254 3 3 N AGARWAL AND N R ALURU A domain adaptive stochastic collocation approach for analysis of mems under uncertainties Journal of Computational Physics 228 2009 pp 7662 7688 3 4 V BARTHELMANN E NOVAK AND K RITTER High dimensional polynomial interpo lation on sparse grids Advances in Computational Mathematics 12 2000 pp 273 288 3 5 P J DAVIS AND P RABINOWITZ Methods of numerical integration Courier Dover Publi cations 2007 35 6 M ELDRED C WEBSTER AND P CONSTANTINE Evaluation of non intrusive approaches for wiene
22. ed on the loaded values of the interpolant After calling setRefinement the needed points are updated and evaluate and integrate will work with the old interpolant until the new needed values are loaded If setRefinement is called twice in a row without loadNeededValues then any data from the first call will be cleared and only the second refinement would persist tolerance is a positive number with the desired tolerance criteria is an enumerated type from type_classic type parent first type_direction_selective type_fds For the three types of local refinement see 23 Note that the inputs have different meaning depending on the grid e Global Grid will ignore both parameters and will only increase the accuracy isotropically in every direction by adding the next levels of the one dimensional rule e Local Polynomial Grid will compare the relative magnitude of the surpluses c divided by the largest provided value f x to the tolerance The algorithm will refine only in the neigh borhood of the abscissas where the ratio is large for at least one of the outputs The criteria defines whether or not all direction should be refined and whether or not the parents should be added before the children where the family is described by the hierarchy 23 e Wavelet Grid will compare the surpluses c to the tolerance and will refine only in the neigh borhood of the abscissas associated with large surpluses The criteria is ignored i e
23. er Science and Mathematics Division Oak Ridge National Laboratory One Bethel Valley Road P O Box 2008 MS 6367 Oak Ridge TN 37831 6164 stoyanovmk ornl gov CONTENTS LIS DOR FIGURES 03d rada vi LISTOF TABLES uccrerri AA enkdezes vii ABS TACT 30 ai AA AAA 1 ACKNOWLEDGEMENTS siii Ad 1 1 Introduction csommsras ea da ea 2 2 COMPUSO uscar a RRA e E acancnan bie 5 3 LIBTASMANIANSPARSEGRIDS libtsg cccccccccccccccccccccssssssscces 6 3 1 Constructor Tasmanians parse 0 AAA ORS RRS 6 3 2 Destructor TasmanianSparseGrid ci ad 6 3 3 function Get Version eseri ary A AR ad 6 3 4 Tonction cet Acense A A A Ad 6 3 5 function makeGlobalGridO oooooooooorooooorornooooo s 7 3 6 function makelLocalPolynomialGnd cocromicinar oran es 8 3 7 function makeWaveletGrid oooooooooooorrr nooo 9 3 8 function makeFullTensorGnd codo dt BS ae 10 3 9 functions recycle FOG sI ARO de Aaaa AA E Gaa EARE 10 3 10 TURCO WENE 2 ersi le GES ye AE ee a e A A ee 11 3 11 tunctlon TEO cir a eS bar da 11 3 12 TURCO WEE e A a aaa 11 3 13 TUCU read tc id ne RO eee ita 11 3 14 tinction setTranstormAB or cercei neraet id dees wh at ew eee 12 3 15 function clear TranstormABO ii Bb AG See ne ee ee due Pace a 12 316 function et TransformABO rd ee eGo a bis la 12 3 17 function getNumDimensions s osos 2 ucen ce estes apes ey whe ea RA 13 3 18 funcion COIN UIMOUIDUNS se isa aaae aaa RO 13 3 19 function eee RUSO A a ew S
24. er that is given to the setRefinement command refinement is a string specifying the refinement criteria classic corresponds to type_classic parents corresponds to type_parent_first direction corresponds to type_direction_selective fds corresponds to type_fds outputfile is an optional matrix file At the end of the program tasgird will write in the file the needed abscissas i e the ones that are not yet associated with values of the interpolated function The matrix file will have getNumPoints number of rows and getNumDimensions number of columns The first abscissa will be on the first row the second on the second row print write out the same data as in the outputfile but to the cout stream 4 16 Command summary tasgrid summary gridfile lt filename gt Reads the grid in the provided file and prints short summary about the grid gridfile this is the file with an already created grid 25 4 17 Matrix File Format A matrix file is a simple text file that describes a two dimensional array of real numbers The file contains two integers on the first line indicating the number of rows and columns Those are followed by the actual entries of the matrix one row at a time The file containing 3 4 1 0 2 0 3 0 4 0 50 OO O g 85 0 9 0 10 0 11 0 12 0 represents the matrix 1 2 3 4 5 6 7 8 9 10 11 12 A matrix file may contain only one row or column e g 1 2 13 0 14 0 All files used by ta
25. for any arbitrary x and it can also be integrated over the domain e generate a set of abscissas x and for any arbitrary x it can also generate the interpola tion weights h x _ for an approximation of type 1 3 In addition local grids support iterative refinement where additional abscissas are chosen based on the provided f x _ to improve the approximation of the provided interpolant The code consists of three main components e libtasmaniansparsegrids a for short libtsg which is a library written in C that imple ments the TasmanianSparseGrid class The class provides an interface for manipulation of the grid See Section 3 e tasgrid which is an executable that provides a command line interface to libtsg The exe cutable reads and writes data to text files and every command generally reads an instance of TasmanianSparseGrid class from a text file calls a function from the class and writes the modified class back to a text file See Section 4 e MATLAB Interface for short tsg m which is a series of MATLAB functions that call the executable tasgrid and read the result into MATLAB matrices See Section 5 2 Compilation Quick Build Inside the folder with the source files type make The code doesn t require any external libraries and uses the simple GNU Make engine Hence it will most likely compile just fine To verify the build you should run tasgrid test and make sure all the test pass See Section
26. for large dimensions the number of abscissas associated with a sparse grid grows fast i e the curse of dimensionality and hence the grid may require prohibitive amount of memory is a non negative integer specifying the number of outputs for the function that would be interpolated If outputs is zero then the grid can only generate quadrature and interpolation weights i e problems 1 1 and 1 3 There is no hard restriction on how many outputs can be handled however note that the code requires at least outputs x number of abscissas storage and hence for large number of outputs memory management may have adverse effect on performance is a non negative or strictly positive integer that controls the density of abscissa points For grids of type_level and type_hyperbolic depth is strictly positive and it corresponds to the notion of sparse grid level see 18 19 22 For grids of type_basis the depth specifies the largest total degree polynomial that can be integrated or interpolated exactly Gauss based rules imply integration while non Gauss rules imply interpolation see Appendix A There is no hard restriction on how big depth can be however it has direct effect on the number of abscissas and hence performance and memory requirements is an enumerated type from type_level type_hyperbolic type_basis which guides the tensor selection to balance the precision in different directions type_level classical Smolyak Sperse G
27. ghts lt optionl gt lt valuel gt lt option2 gt lt value2 gt Reads a grid from a file and a list of points of interest form a matrix file For each point in the matrix file tasgrid computes the corresponding interpolation weights as in equation 1 3 The result is written to an output matrix file 22 gridfile this is the file with an already created grid and loaded values inputfile is a matrix file with points of interest The file can have arbitrary number of rows and getNumDimensions number of columns Each row corresponds to one point of interest outputfile is an optional matrix file that is written on exit The file contains the interpolation weights as sociated with the points provided by the inputfile The file has the same number of rows and getNumPoints number of columns Each row contains the interpolation weights associated with the corresponding point of interest print write out the same data as in the outputfile but to the cout stream 4 11 Command getneededpoints tasgrid getneededpoints lt optionl gt lt valuel gt lt option2 gt lt value2 gt Reads a grid from a file extracts the abscissas associated with the grid that are not yet associ ated with values of the interpolated function Those abscissas are written to a matrix file gridfile this is the file with an already created grid outputfile is an optional matrix file The program will write in the file the abscissas associated with the
28. html NOTICE This report was prepared as an account of work sponsored by an agency of the United States Government Neither the United States Government nor any agency thereof nor any of their employees nor any of their con tractors subcontractors or their employees make any warranty express or implied or assume any legal liability or responsibility for the accuracy completeness or usefulness of any information apparatus product or process disclosed or represent that its use would not infringe privately owned rights Reference herein to any specific commercial product pro cess or service by trade name trademark manufacturer or otherwise does not necessarily constitute or imply its endorsement recommenda tion or favoring by the United States Government any agency thereof or any of their contractors or subcontractors The views and opinions expressed herein do not necessarily state or reflect those of the United States Government any agency thereof or any of their contractors Printed in the United States of America This report has been reproduced directly from the best available copy Computer Science and Mathematics Division USER MANUAL TASMANIAN SPARSE GRIDS M Stoyanov Date Published August 2013 Prepared by OAK RIDGE NATIONAL LABORATORY Oak Ridge Tennessee 37831 6283 managed by UT BATTELLE LLC for the U S DEPARTMENT OF ENERGY under contract DE ACOS 000R22725 Comput
29. id and it specifies the type_level enumerated type basis calls makeGlobalGrid and it specifies the type_basis enumerated type hyperbolic calls makeGlobalGrid and it specifies the type_hyperbolic enumerated type tensor calls makeFullTensorGrid specifies the order of the local polynomials or wavelets and is used only with local polyno mial and wavelet grids For polynomials the value is 1 for automatically using the max imum possible order or a non negative integer that restricts the maximum order Wavelet grids accept only orders 1 and 3 specifies the parameter of the one dimensional quadrature rule The value is a real number and it is used by gauss gegenbauer gauss jacobi gauss hermite and gauss laguerre rules specifies the 3 parameter of the one dimensional quadrature rule The value is a real number and it is used by the gauss jacobi rule is an optional matrix file that specifies the transformation from the canonical domain to a custom domain The matrix file should have dimensions number of rows and 2 columns The first column is the a parameter and the second column is the b parameter and each row corresponds to one dimension For detail on the matrix file format see subsection 4 17 is an optional matrix file At the end of the program tasgird will write in the file the abscissas associated with the grid The matrix file will have getNumPoints number of rows and dimensions number of columns The first absciss
30. id with most of the parameters taken from an existing grid specified by the gridfile option Furthermore the command will try to use existing values of the interpolated function if they have already been loaded This command calls one of the recy cle Grid functions The recycled grid created will use the same number of dimensions and outputs as well as the same base one dimensional quadrature or interpolation rule including the same values for the a and P parameters gridfile depth type order inputfile outputfile anisotropyfile print on input this is the file with an already created grid On exit the file will be overwritten with the new grid same as in makegrid if the loaded grid is a global grid then the value is one of the strings level basis or hyperbolic which has the same meaning as in makegrid If the grid file contains a tensor grid then type is ignored same as in makegrid same as in makegrid is an optional matrix file At the end of the program tasgird will write in the file the needed abscissas i e the ones that are not yet associated with values of the interpolated function The matrix file will have getNumPoints number of rows and getNumDimensions number of columns The first abscissa will be on the first row the second on the second row same as in makegrid write out the same data as in the outputfile but to the cout stream 21 4 8 Command getquadrature ta
31. les will be created and deleted in this folder 27 5 2 functions tsgRead Matrix and tsgWriteMatrix Those functions are used internally to read from or write to matrix files The user shouldn t call those functions directly 5 3 function tsgMakeGrid lGrid points tsgMakeGrid sGridName dim out oned depth order type anisotropy alpha_beta transformAB Calls tasgrid with the makegrid command This function creates an Grid object that can be used to refer to the grid by other functions sGridName dim out oned depth order type anisotropy alpha_beta transformAB INPUTS a user provided name that will be appended to all the file names associated with this grid If a grid with this name already exists it will be overwritten and the Grid object associated with the old grid will be invalid the dimension of the grid same as dimensions the number of outputs of the grid same as outputs the one dimensional integration and interpolation rule same as onedim same as depth is used by local polynomial and local wavelet grids only Specifies the order of the polyno mial same as order is used by global grids only Specifies the type option is either an empty MATLAB matrix or a matrix of integers with size dim x1 that describe the anisotropic weights see anisotropicfile option for tasgrid and const int anisotropic_weights parameter for libtsg is either an empty MATLAB
32. ll controls the number of abscis sas however it no longer has the same relationship to levels or total degree polynomial order is either NULL or an array of one or two doubles The first entry of the array is a and the second is and those values are referenced only if the quadrature rule requires them One dimensional rules of type rule_gaussgegenbauer rule_gaussjacobi rule_gausslaguerre and rule_gausshermite require an a and in addition rule_gaussjacobi requires p see Table A 1 function makeLocalPolynomialGrid void makeLocalPolynomialGrid int dimensions int outputs int depth int order TypeOneDRule boundary Creates a grid based on local hierarchical piece wise polynomial function basis The main focus of hierarchical grids is the ability to do local iterative refinement to daptively obtain an interpolant that clusters abscissa in regions of sharp behavior and puts fewer abscissa in regions of smooth behavior Local grids can be used for integration however in many cases this would result in abscissas associated with zero weights dimensions outputs same as makeGlobalGrid same as makeGlobalGrid however due to the non trivial form of the coefficients c large number of outputs comes with bigger computational cost in addition to the larger storage cost of 2 x outputs x number of abscissas depth is a positive integer that specifies the initial number of levels for the grid order is an i
33. nteger bigger than 2 which specifies the largest order of polynomial to be used For a one dimensional interpolation rule a polynomial of order cannot be used before level l 1 ie before depth l Thus if order is larger than depth only lower degree local polynomials would be used If order is set to 1 the largest possible order would be selected automatically on the fly boundary is an enumerated type with value either rule pwpolynomial or rule_pwpolynomial0 The difference is that the latter type assumes that the interpolated function is zero at the boundary 3 7 function makeWaveletGrid void makeWaveletGrid int dimensions int outputs int depth int order 1 Creates a grid based on local hierarchical wavelet basis It is very similar to the local poly nomial rule however local refinement using wavelet functions would sometimes require fewer function evaluations dimensions same as in makeGlobalGrid and makeLocalPolynomialGrid outputs same as in makeLocalPolynomialGrid depth same as in makeLocalPolynomialGrid order an integer equal to either 1 or 3 The wavelet grids use the corresponding order of wavelet even for grid with depth 1 However a wavelet grid of a given depth would have more abscissas than a corresponding local polynomial grid 3 8 function makeFullTensorGrid void makeFullTensorGrid int dimensions int outputs int order TypeOneDRule oned const double
34. of columns and the same number of rows as points The weights contain the interpolation weights associated with each point of interest in points 30 5 8 function tsgGetNeededPoints newp tsgGetNeededPoints 1Grid Calls tasgrid with the getneededpoints command INPUTS IGrid is an object created by tsgMakeGrid OUTPUTS newp is a MATLAB matrix of getNumNeededPoitns number of rows and out number of columns Each rows is an abscissa that is not yet associated with a value of the interpolated function 5 9 function tsgLoad Values tsgLoadValues 1Grid values Calls tasgrid with the loadvalues command INPUTS IGrid is an object created by tsgMakeGrid values isa MATLAB matrix with getNumNeededPoitns number of rows and out number of columns Each row represents the values of the function at one point OUTPUTS there are no output variables however the files associated with the grid are modified 5 10 function tsgEvaluate result tsgEvaluate 1Grid points Calls tasgrid with the evaluate option INPUTS IGrid is an object created by tsgMakeGrid 31 points isa MATLAB matrix with rows representing points of interest OUTPUTS result is a MATLAB matrix with rows representing the values of the interpolant at the point of interest 5 11 function tsgIntegrate result tsgIntegrate 1Grid Calls tasgrid with the integrate option INPUTS IGrid is an object created b
35. r askey generalized polynomial chaos in Proceedings of the 10th AIAA Non Deterministic Approaches Conference number AIAA 2008 1892 Schaumburg IL vol 117 2008 p 189 3 7 S ELHAY AND J KAUTSKY Algorithm 655 Igpack Fortran subroutines for the weights of interpolatory quadratures ACM Transactions on Mathematical Software TOMS 13 1987 pp 399 415 35 8 T GERSTNER AND M GRIEBEL Numerical integration using sparse grids Numerical al gorithms 18 1998 pp 209 232 2 3 7 9 Dimension adaptive tensor product quadrature Computing 71 2003 pp 65 87 3 10 M GRIEBEL Adaptive sparse grid multilevel methods for elliptic pdes based on finite differ ences Computing 61 1998 pp 151 179 2 3 38 11 M GRIEBEL AND J HAMAEKERS Sparse grids for the schr dinger equation M2AN Math Model Numer Anal 41 2007 pp 215 247 7 12 M GUNZBURGER C TRENCHEA AND C WEBSTER A generalized stochastic collocation approach to constrained optimization for random data identification problems Tech Rep ORNL TM 2012 185 Oak Ridge National Laboratory 2012 3 13 M GUNZBURGER C WEBSTER AND G ZHANG An adaptive wavelet stochastic colloca tion method for irregular solutions of partial differential equations with random input data Tech Rep ORNL TM 2012 186 Oak Ridge National Laboratory 2012 3 38 39 14 J KAUTSKY AND S ELHAY Calculation of the weights of interpolatory quadratures
36. rid 10 3 10 function write void write std ofstream ofs const Writes out the grid in text format to the ofstream 3 11 function read bool read std ifstream amp ifs const Reads a grid that has already been written to the stream The function returns True if the reading was successful or False if errors with the file format were encountered The function will write error information to the standard output stream 3 12 function write void write const charx filename const Opens a file with filename and calls void write std ofstream amp ofs const with the associated stream At the end the file is closed 3 13 function read bool read const charx filename Opens a file with filename and calls bool read std ifstream amp ifs const with the associated stream At the end the file is closed 11 3 14 function setTransformAB void setTransformAB const double xa const double xb By default integration and interpolation are performed on a canonical interval described in Table A 1 Optionally the library can transform the canonical interval into a custom one defined by the a and b parameters for every direction The transformation is applied as a post processing step to the abscissas and weights a is an array of real numbers of size getNumDimensions that defines the a parameter associ ated with every dimension b is an array of real numbers of size getNumDimensions tha
37. rid selection The sum of the level indexes over all of the directions has to be less then depth 18 19 22 type_hyberbolic hyperbolic cross section The product of the level indexes over all of the directions has to be less then depth 8 11 21 type_basis takes into consideration the accuracy of the one dimensional rule Any rule combined with type_basis is guaranteed to integrate or interpolate any polynomial of total degree no more than depth oned anisotropic alpha_beta 3 6 is an enumerated type from any of the global rules in Table A 1 Those are rule_clenshawcurtis rule_chebyshev rule_fejer2 rule_gausslegendre rule_gausschebyshevl rule_gausschebyshev2 rule_gaussgegenbauer rule_gaussjacobi rule_gausslaguerre rule_hermite anisotropic_weights is either NULL or an array of integers of size dimensions See 18 for the meaning of the weights Note that in the literature the weights are assumed to be real numbers The code assumes that the weights are rational numbers and that they add up to 1 thus the anisotropic_weights array contains only the numerators of the rational numbers This is done so that type_level and type_basis grids can be constructed using only integer based arithmetic type_hyperbolic grids still use double precision arithmetic that may be un stable 1 e the number of abscissa may be heavily influenced by rounding error In addition after introducing anisotropic weights the value of depth sti
38. rpolantion weight associated with the abscissa and the point defined by x as in equation 1 3 x is an array of dimension getNumDimensions representing the point of interest to evaluate the interpolant weights on input it is either NULL or a non NULL pointer that would be deleted on output returns an array of size getNumPoints of the interpolation weights associated with the abscissas The first weight is associated with the first abscissa returned by getPoints the second weight is associated with the second abscissa and so on 3 25 function getNumNeededPoints int getNumNeededPoints const Interpolation described in equation 1 2 requires the user to provide the values of the interpo lated function at the abscissa points This functions returns the number of abscissas that are still not associated with function values 3 26 function getNeededPoints void getNeededPoints doublex amp pnts const pnts on input it is either NULL or a non NULL pointer that would be deleted on output returns an array of size getNumDimensions x getNumNeededPoints of entries that represent the abscissas that still need to be associated with function values The first abscissa is located in the first getNumDimensions number of entries the second abscissa is located in the second getNumDimensions number of entries and so on If getNumNeeded Points returns 0 then pnts is returned NULL 15 3 27 function loadNeededPoints voi
39. s Legendre Optimal integration rule for functions on a bounded domain and constant weight 1 i f x dx A basis type grid is guaranteed to integrate exactly any polynomial of degree no more than the value of the depth property Gauss Chebyshev Type 1 Optimal integration rule for functions on a bounded domain and weight p x 1 x torai f Foa ar A basis type grid is guaranteed to integrate exactly any polynomial of degree no more than the value of the depth property Gauss Chebyshev Type 2 Optimal integration rule for functions on a bounded domain and weight p x 1 x 03 1 1 Fome f FOA an 1 A basis type grid is guaranteed to integrate exactly any polynomial of degree no more than the value of the depth property Gauss Gegenbauer Generalized Gauss Chebyshev rule Optimal integration rule for functions on a bounded domain and weight p x 1 x toned f HON Aar A basis type grid is guaranteed to integrate exactly any polynomial of degree no more than the value of the depth property 36 Gauss Jacobi Generalized Gauss Gegenbauer rule Optimal integration rule for functions on a bounded domain and weight p 1 x 1 2 1 1 Fomes sea 2 0 oar 1 1 A basis type grid is guaranteed to integrate exactly any polynomial of degree no more than the value of the depth property Gauss Laguerre This is the generalized Gauss Laguerre quadrature Op
40. sgrid getquadrature lt optionl gt lt valuel gt lt option2 gt lt value2 gt Reads a grid from a file computes the quadrature associated with the grid and writes it to a matrix file gridfile this is the file with an already created grid outputfile is an optional matrix file The program will write in the file the quadrature weights and abscissas associated with the grid The matrix file will have getNumPoints number of rows and getNumDimensions 1 number of columns The first abscissa will be on the first row the second on the second row and so on On each row the first column is the weight and the rest of the columns are the associated abscissa print write out the same data as in the outputfile but to the cout stream 4 9 Command getpoints tasgrid getpoints lt optionl gt lt valuel gt lt option2 gt lt value2 gt Reads a grid from a file extracts the abscissas associated with the grid and writes them out in a matrix file gridfile this is the file with an already created grid outputfile is an optional matrix file The program will write in the file the abscissas associated with the grid The matrix file will have getNumPoints number of rows and getNumDimensions number of columns The first abscissa will be on the first row the second on the second row and so on print write out the same data as in the outputfile but to the cout stream 4 10 Command getinterweights tasgrid getinterwei
41. sgrid have the above format with the exception of the gridfile that contains saved sparse grids and the anisotropyfile that is a matrix with one column and it should contain only integers 26 5 MATLAB Interface The MATLAB interface to tasgrid consists of several functions that call various tasgrid commands and read and write matrix files Unlike most MATLAB interfaces this is code does not use mex files but rather system commands and text files Before using the interface you must manually edit the tsgGetPath m file e The MATLAB interface requires that MATLAB is able to call external commands and the tasgrid executable in particular e The MATLAB interface also requires access to a folder where the files can be written e Each grid has a user specified name that is a string which gets appended at the end of the file name e The tsgDeleteGrid tsgDeleteGridByName and tsgListGridsByName allow for cleaning the files in the temporary folder e Every function comes with help comments that can be accessed by typing help tsgFunctionName e Note that it is recommended to add the folder with the MATLAB interface to your MATLAB path 5 1 function tsgGetPaths sFiles sTasGrid tsgGetPaths You must edit the two strings in this file sTasGrid is a string containing the path to the tasgrid executable including the name of the exe cutable sFiles is the path to a folder where MATLAB has read write permission Fi
42. t defines the b parameter associ ated with every dimension 3 15 function clearTransformA B const charx clearTransformAB const Scales back all abscissas and weights to the canonical interval Since the transformation is a post processing step the clearTransformAB function simply removes the values set by setTrans formAB 1 e the grid is not actually recomputed 3 16 function getTransformAB void getTransformAB double ta doublex amp b const Returns the transform parameters a on input it is either a NULL pointer or a non NULL pointer that will be deleted on output returns a pointer to an array of size getNumDimensions that contains the values of the a parameter for every direction b on input it is either a NULL pointer or a non NULL pointer that will be deleted on output returns a pointer to an array of size getNumDimensions that contains the values of the b parameter for every direction 12 3 17 function getNumDimensions int getNumDimensions const Returns the value of the dimension parameter used by the make Glid function call 3 18 function getNumOutputs int getNumOutputs const Returns the value of the outputs parameter used by the make Glid function call 3 19 function getOneDRule TypeOneDRule getOneDRule const For a global grid returns the value of the oned parameter For a local polynomial grid returns the value of the boundary parameter
43. timal integration rule for functions on a bounded domain and weight p 1 x e lA ES En izda A basis type grid is guaranteed to integrate exactly any polynomial of degree no more than the value of the depth property Note Interpolation with this rule is extremely unstable due to the infinite range of the domain Gauss Hermite This is the generalized Gauss Hermite quadrature Optimal integration rule for functions on a z bounded domain and weight p x z e f tonades f fle lelee ae A basis type grid is guaranteed to integrate exactly any polynomial of degree no more than the value of the depth property Note Interpolation with this rule is extremely unstable due to the infinite range of the domain A 2 Local Polynomials The local polynomial rules are best for interpolation of a function with locally sharp gradients The domain of interpolation is T4 1 1 Local polynomial grids allow for different types of local adaptivity The maximum degree of the polynomials is specified by the order property Note 37 Canonical Generalized Domain of Canonical Additional Nested Integral Integral Interpolation Assumptions Clenshaw Curtis rule_clenshawcurtis clenshaw curtis fo f de J f w dx i N A Yes Chebyshev rule_chebyshev chebyshev ee f x dx f f x dx 1 1 N A No Fejer type 2 rule_fejer
44. tion tsgListGridsByName tsgListGridsByName Reads the temp folder and finds all the grids regardless whether or not they are associated with grid objects 33 5 17 function tsgExample tsgExample This function contains sample code that replicated the C example This is a demonstration on the proper way to call the MATLAB functions 5 18 Saving a Grid You can save the Grid object just like any other MATLAB object However a saved grid has two components the Grid object and the files associated with the grid that are stored in the folder specified by tsgGetPath The files in the temporary folder will be persistent until either tsgDelete Grid is called or the files are manually deleted The only exception is that the tsgExample function will overwrite any grids with names tsgExample2 or tsgExample5 Note that modifying tsgGetPath may result in the code not being able to find the needed files and hence the grid object may be invalidated 5 19 Avoiding Some Problems e Make sure to call tsgDeleteGrid as soon as you are done with a grid this will avoid clutter in the temporary folder e If you clear an Grid object without calling tsgDeleteGrid i e you exit MATLAB without saving then make sure to use tsgListGridsByName and tsgDeleteGridByName to safely delete the lost grids e Working with the MATLAB interface is very similar to working with dynamical memory where the data is stored on the disk
45. v 1 creates a global grid with Gauss Chebyshev rule of type 1 gauss chebyshev 2 creates a global grid with Gauss Chebyshev rule of type 2 gauss gegenbauer creates a global grid with Gauss Gegenbauer rule gauss jacobi creates a global grid with Gauss Jacobi rule gauss laguerre creates a global grid with Gauss Laguerre rule gauss hermite creates a global grid with Gauss Hermite rule local polynomial creates a local polynomial grid local polynomial zero creates a local polynomial grid with zero boundary local wavelet creates a wavelet grid dimensions specifies the dimension of the problem The value should be a positive integer and it gets passed directly to make Grid see the previous section outputs specifies the number of outputs of the function that needs to be interpolated The values should be a positive integer that gets passed directly to make Grid see the previous section To set zero outputs use the makequadrature command depth specifies the depth parameter of the make Grid function see the previous section In case of a tensor grid and if no anisotropyfile is give makeFullTensor will be called with order array having all entries equal to depth 19 type order alpha beta inputfile outputfile gridfile anisotropyfile print specifies the selection criteria of the sparse grid and it is used only by global and tensor grids Available values are level calls makeGlobalGr
46. wavelet grids currently only implement type classic refinement 3 32 Examples The file example cpp in the Examples folder has sample code that demonstrates proper use of the TasmanianSparseGrid class In addition there is also a Makefile that compiles the example 17 4 TASGRID The tasgrid executable is a command line interface to libtsg It provides the ability to create and manipulate sparse grids save and load them into files and optionally interface with another program via text files For the most part tasgrid reads a grid from a file calls one or more of the functions described in the previous section and then saves the resulting grid In addition tasgrid provides a set of basic functionality tests 4 1 Basic Usage tasgrid lt command gt lt optionl gt lt valuel gt lt option2 gt lt value2 gt The first input to the executable is the command that specifies the action that needs to be taken The command is followed by options and values Every command is associated with a number of options If other options are provided then they are ignored Tasgrid has some basic error checking and if it encounters and error in the input tasgrid will print an error message as well as some help for the input 4 2 Command h help Prints information about the usage of tasgrid Note that many commands and options have a long and short name and the help command will list both In addition it will also list the available one
47. x alpha_beta 0 Full tensor grids are not sparse grids however full tensors share many of the properties of global grids and hence the capability is included into the code mostly for testing purposes Note that in most situations full tensor grids would require a much larger number of abscissas to achieve the same accuracy as than a sparse grid dimensions same as in makeGlobalGrid outputs same as in makeGlobalGrid order an array of non negative integers of size dimensions The array indicates the level of one dimensional rule to be used in every direction oned same as in makeGlobalGrid alpha_beta same as in makeGlobalGrid 3 9 functions recycle Grid void recycleGlobalGrid int depth TypeDepth type const int anisotropic _weights 0 void recycleLocalPolynomialGrid int depth int order 1 void recycleWaveletGrid int depth int order 1 void recycleFullTensorGrid int order The recycle functions recreate a new grid with similar properties but different number of ab scissas and hence different accuracy The recycle functions modify some of the parameters of the grids but keep all the ones that are not specified i e dimensions outputs oned boundary Recycle will also try to use any values of f x loaded in the old grid The code discards values of f a that are associated with abscissas in the old grid but are not part of the new g
48. y tsgMakeGrid OUTPUTS result is a MATLAB matrix with one row that represents the integral of the interpolant over the canonical domain associated with the quadrature rule i e oned 5 12 function tsgRefineGrid newp tsgRefineGrid lGrid tolerance criteria Calls tasgrid with the refine option INPUTS IGrid is an object created by tsgMakeGrid tolerance the tolerance for the refinement same as tolerance criteria the type of refinement same as refinement OUTPUTS newp is a MATLAB matrix of getNumNeededPoitns number of rows and out number of columns Each rows is an abscissa that is not yet associated with a value of the interpolated function 32 5 13 function tsgPrintStats tsgPrintStats 1Grid Calls tasgrid with the summary option 5 14 function tsgDeleteGrid tsgDeleteGrid 1Grid Delete all files associated with the grid The Grid object is no longer valid for further reference INPUTS IGrid is an object created by tsgMakeGrid OUTPUTS deletes all the files associated with the grid and the Grid object is no longer valid 5 15 function tsgDeleteGridByName tsgDeleteGrid sGridName Delete all files associated with the grid with the name sGridName This function should be called whenever the Grid object has been lost similar to a memory leak INPUTS sGridName the name of the grid to be deleted OUTPUTS deletes all the files associated with the grid 5 16 func
49. yshev It uses nested Chebyshev points and integration weights on T 1 1 The rule is known to produce heavy bias on the main axis When used with the basis type the axis growth is slowed which is equivalent to 20 A basis type grid is guaranteed to interpolate exactly any polynomial of degree no more than the depth property the rule will also interpolate some higher order polynomials depending on the depth and dimension Chebyshev The Chebyshev rule on T4 1 1 with non nested points that grow one point at a level This is the only rule where level and basis types grids are identical A basis type grid is guaranteed to interpolate exactly any polynomial of degree no more than the depth property Chebyshev two point growth The Clenshaw Curtis rule on y 1 1 is split into more levels so that every level adds ex actly two points to the previous one A basis type grid is guaranteed to interpolate exactly any polynomial of degree no more than the depth property Note this is not a stable rule and it should be used with great caution Fejer type 2 The Fejer type 2 rule on I 1 1 that is nested and uses Chebyshev points and weights Unlike the Clenshaw Curtis rule Fejer type 2 assumes that f a vanishes at the two end points A basis type grid is guaranteed to interpolate exactly any polynomial of degree no more than the depth property so long as the polynomial vanishes at the boundary of the domain 35 Gaus
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