Actual source code: dt.c
petsc-3.4.0 2013-05-13
1: /* Discretization tools */
3: #include <petscdt.h> /*I "petscdt.h" I*/
4: #include <petscblaslapack.h>
5: #include <petsc-private/petscimpl.h>
6: #include <petscviewer.h>
10: /*@
11: PetscDTLegendreEval - evaluate Legendre polynomial at points
13: Not Collective
15: Input Arguments:
16: + npoints - number of spatial points to evaluate at
17: . points - array of locations to evaluate at
18: . ndegree - number of basis degrees to evaluate
19: - degrees - sorted array of degrees to evaluate
21: Output Arguments:
22: + B - row-oriented basis evaluation matrix B[point*ndegree + degree] (dimension npoints*ndegrees, allocated by caller) (or NULL)
23: . D - row-oriented derivative evaluation matrix (or NULL)
24: - D2 - row-oriented second derivative evaluation matrix (or NULL)
26: Level: intermediate
28: .seealso: PetscDTGaussQuadrature()
29: @*/
30: PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
31: {
32: PetscInt i,maxdegree;
35: if (!npoints || !ndegree) return(0);
36: maxdegree = degrees[ndegree-1];
37: for (i=0; i<npoints; i++) {
38: PetscReal pm1,pm2,pd1,pd2,pdd1,pdd2,x;
39: PetscInt j,k;
40: x = points[i];
41: pm2 = 0;
42: pm1 = 1;
43: pd2 = 0;
44: pd1 = 0;
45: pdd2 = 0;
46: pdd1 = 0;
47: k = 0;
48: if (degrees[k] == 0) {
49: if (B) B[i*ndegree+k] = pm1;
50: if (D) D[i*ndegree+k] = pd1;
51: if (D2) D2[i*ndegree+k] = pdd1;
52: k++;
53: }
54: for (j=1; j<=maxdegree; j++,k++) {
55: PetscReal p,d,dd;
56: p = ((2*j-1)*x*pm1 - (j-1)*pm2)/j;
57: d = pd2 + (2*j-1)*pm1;
58: dd = pdd2 + (2*j-1)*pd1;
59: pm2 = pm1;
60: pm1 = p;
61: pd2 = pd1;
62: pd1 = d;
63: pdd2 = pdd1;
64: pdd1 = dd;
65: if (degrees[k] == j) {
66: if (B) B[i*ndegree+k] = p;
67: if (D) D[i*ndegree+k] = d;
68: if (D2) D2[i*ndegree+k] = dd;
69: }
70: }
71: }
72: return(0);
73: }
77: /*@
78: PetscDTGaussQuadrature - create Gauss quadrature
80: Not Collective
82: Input Arguments:
83: + npoints - number of points
84: . a - left end of interval (often-1)
85: - b - right end of interval (often +1)
87: Output Arguments:
88: + x - quadrature points
89: - w - quadrature weights
91: Level: intermediate
93: References:
94: Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 221--230, 1969.
96: .seealso: PetscDTLegendreEval()
97: @*/
98: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
99: {
101: PetscInt i;
102: PetscReal *work;
103: PetscScalar *Z;
104: PetscBLASInt N,LDZ,info;
107: /* Set up the Golub-Welsch system */
108: for (i=0; i<npoints; i++) {
109: x[i] = 0; /* diagonal is 0 */
110: if (i) w[i-1] = 0.5 / PetscSqrtReal(1 - 1./PetscSqr(2*i));
111: }
112: PetscRealView(npoints-1,w,PETSC_VIEWER_STDOUT_SELF);
113: PetscMalloc2(npoints*npoints,PetscScalar,&Z,PetscMax(1,2*npoints-2),PetscReal,&work);
114: PetscBLASIntCast(npoints,&N);
115: LDZ = N;
116: PetscFPTrapPush(PETSC_FP_TRAP_OFF);
117: PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("I",&N,x,w,Z,&LDZ,work,&info));
118: PetscFPTrapPop();
119: if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEQR error");
121: for (i=0; i<(npoints+1)/2; i++) {
122: PetscReal y = 0.5 * (-x[i] + x[npoints-i-1]); /* enforces symmetry */
123: x[i] = (a+b)/2 - y*(b-a)/2;
124: x[npoints-i-1] = (a+b)/2 + y*(b-a)/2;
126: w[i] = w[npoints-1-i] = (b-a)*PetscSqr(0.5*PetscAbsScalar(Z[i*npoints] + Z[(npoints-i-1)*npoints]));
127: }
128: PetscFree2(Z,work);
129: return(0);
130: }
134: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
135: Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
136: PETSC_STATIC_INLINE PetscErrorCode PetscDTFactorial_Internal(PetscInt n, PetscReal *factorial)
137: {
138: PetscReal f = 1.0;
139: PetscInt i;
142: for (i = 1; i < n+1; ++i) f *= i;
143: *factorial = f;
144: return(0);
145: }
149: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
150: Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
151: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
152: {
153: PetscReal apb, pn1, pn2;
154: PetscInt k;
157: if (!n) {*P = 1.0; return(0);}
158: if (n == 1) {*P = 0.5 * (a - b + (a + b + 2.0) * x); return(0);}
159: apb = a + b;
160: pn2 = 1.0;
161: pn1 = 0.5 * (a - b + (apb + 2.0) * x);
162: *P = 0.0;
163: for (k = 2; k < n+1; ++k) {
164: PetscReal a1 = 2.0 * k * (k + apb) * (2.0*k + apb - 2.0);
165: PetscReal a2 = (2.0 * k + apb - 1.0) * (a*a - b*b);
166: PetscReal a3 = (2.0 * k + apb - 2.0) * (2.0 * k + apb - 1.0) * (2.0 * k + apb);
167: PetscReal a4 = 2.0 * (k + a - 1.0) * (k + b - 1.0) * (2.0 * k + apb);
169: a2 = a2 / a1;
170: a3 = a3 / a1;
171: a4 = a4 / a1;
172: *P = (a2 + a3 * x) * pn1 - a4 * pn2;
173: pn2 = pn1;
174: pn1 = *P;
175: }
176: return(0);
177: }
181: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
182: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
183: {
184: PetscReal nP;
188: if (!n) {*P = 0.0; return(0);}
189: PetscDTComputeJacobi(a+1, b+1, n-1, x, &nP);
190: *P = 0.5 * (a + b + n + 1) * nP;
191: return(0);
192: }
196: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
197: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapSquareToTriangle_Internal(PetscReal x, PetscReal y, PetscReal *xi, PetscReal *eta)
198: {
200: *xi = 0.5 * (1.0 + x) * (1.0 - y) - 1.0;
201: *eta = y;
202: return(0);
203: }
207: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
208: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapCubeToTetrahedron_Internal(PetscReal x, PetscReal y, PetscReal z, PetscReal *xi, PetscReal *eta, PetscReal *zeta)
209: {
211: *xi = 0.25 * (1.0 + x) * (1.0 - y) * (1.0 - z) - 1.0;
212: *eta = 0.5 * (1.0 + y) * (1.0 - z) - 1.0;
213: *zeta = z;
214: return(0);
215: }
219: static PetscErrorCode PetscDTGaussJacobiQuadrature1D_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
220: {
221: PetscInt maxIter = 100;
222: PetscReal eps = 1.0e-8;
223: PetscReal a1, a2, a3, a4, a5, a6;
224: PetscInt k;
229: a1 = pow(2, a+b+1);
230: #if defined(PETSC_HAVE_TGAMMA)
231: a2 = tgamma(a + npoints + 1);
232: a3 = tgamma(b + npoints + 1);
233: a4 = tgamma(a + b + npoints + 1);
234: #else
235: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"tgamma() - math routine is unavailable.");
236: #endif
238: PetscDTFactorial_Internal(npoints, &a5);
239: a6 = a1 * a2 * a3 / a4 / a5;
240: /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
241: Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
242: for (k = 0; k < npoints; ++k) {
243: PetscReal r = -cos((2.0*k + 1.0) * PETSC_PI / (2.0 * npoints)), dP;
244: PetscInt j;
246: if (k > 0) r = 0.5 * (r + x[k-1]);
247: for (j = 0; j < maxIter; ++j) {
248: PetscReal s = 0.0, delta, f, fp;
249: PetscInt i;
251: for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
252: PetscDTComputeJacobi(a, b, npoints, r, &f);
253: PetscDTComputeJacobiDerivative(a, b, npoints, r, &fp);
254: delta = f / (fp - f * s);
255: r = r - delta;
256: if (fabs(delta) < eps) break;
257: }
258: x[k] = r;
259: PetscDTComputeJacobiDerivative(a, b, npoints, x[k], &dP);
260: w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
261: }
262: return(0);
263: }
267: /*@C
268: PetscDTGaussJacobiQuadrature - create Gauss-Jacobi quadrature for a simplex
270: Not Collective
272: Input Arguments:
273: + dim - The simplex dimension
274: . npoints - number of points
275: . a - left end of interval (often-1)
276: - b - right end of interval (often +1)
278: Output Arguments:
279: + points - quadrature points
280: - weights - quadrature weights
282: Level: intermediate
284: References:
285: Karniadakis and Sherwin.
286: FIAT
288: .seealso: PetscDTGaussQuadrature()
289: @*/
290: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt dim, PetscInt npoints, PetscReal a, PetscReal b, PetscReal *points[], PetscReal *weights[])
291: {
292: PetscReal *px, *wx, *py, *wy, *pz, *wz, *x, *w;
293: PetscInt i, j, k;
297: if ((a != -1.0) || (b != 1.0)) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
298: switch (dim) {
299: case 1:
300: PetscMalloc(npoints * sizeof(PetscReal), &x);
301: PetscMalloc(npoints * sizeof(PetscReal), &w);
302: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, x, w);
303: break;
304: case 2:
305: PetscMalloc(npoints*npoints*2 * sizeof(PetscReal), &x);
306: PetscMalloc(npoints*npoints * sizeof(PetscReal), &w);
307: PetscMalloc4(npoints,PetscReal,&px,npoints,PetscReal,&wx,npoints,PetscReal,&py,npoints,PetscReal,&wy);
308: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, px, wx);
309: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 1.0, 0.0, py, wy);
310: for (i = 0; i < npoints; ++i) {
311: for (j = 0; j < npoints; ++j) {
312: PetscDTMapSquareToTriangle_Internal(px[i], py[j], &x[(i*npoints+j)*2+0], &x[(i*npoints+j)*2+1]);
313: w[i*npoints+j] = 0.5 * wx[i] * wy[j];
314: }
315: }
316: PetscFree4(px,wx,py,wy);
317: break;
318: case 3:
319: PetscMalloc(npoints*npoints*3 * sizeof(PetscReal), &x);
320: PetscMalloc(npoints*npoints * sizeof(PetscReal), &w);
321: PetscMalloc6(npoints,PetscReal,&px,npoints,PetscReal,&wx,npoints,PetscReal,&py,npoints,PetscReal,&wy,npoints,PetscReal,&pz,npoints,PetscReal,&wz);
322: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, px, wx);
323: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 1.0, 0.0, py, wy);
324: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 2.0, 0.0, pz, wz);
325: for (i = 0; i < npoints; ++i) {
326: for (j = 0; j < npoints; ++j) {
327: for (k = 0; k < npoints; ++k) {
328: PetscDTMapCubeToTetrahedron_Internal(px[i], py[j], pz[k], &x[((i*npoints+j)*npoints+k)*3+0], &x[((i*npoints+j)*npoints+k)*3+1], &x[((i*npoints+j)*npoints+k)*3+2]);
329: w[(i*npoints+j)*npoints+k] = 0.125 * wx[i] * wy[j] * wz[k];
330: }
331: }
332: }
333: PetscFree6(px,wx,py,wy,pz,wz);
334: break;
335: default:
336: SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
337: }
338: if (points) *points = x;
339: if (weights) *weights = w;
340: return(0);
341: }
345: /* Overwrites A. Can only handle full-rank problems with m>=n
346: * A in column-major format
347: * Ainv in row-major format
348: * tau has length m
349: * worksize must be >= max(1,n)
350: */
351: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
352: {
354: PetscBLASInt M,N,K,lda,ldb,ldwork,info;
355: PetscScalar *A,*Ainv,*R,*Q,Alpha;
358: #if defined(PETSC_USE_COMPLEX)
359: {
360: PetscInt i,j;
361: PetscMalloc2(m*n,PetscScalar,&A,m*n,PetscScalar,&Ainv);
362: for (j=0; j<n; j++) {
363: for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
364: }
365: mstride = m;
366: }
367: #else
368: A = A_in;
369: Ainv = Ainv_out;
370: #endif
372: PetscBLASIntCast(m,&M);
373: PetscBLASIntCast(n,&N);
374: PetscBLASIntCast(mstride,&lda);
375: PetscBLASIntCast(worksize,&ldwork);
376: PetscFPTrapPush(PETSC_FP_TRAP_OFF);
377: LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info);
378: PetscFPTrapPop();
379: if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
380: R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */
382: /* Extract an explicit representation of Q */
383: Q = Ainv;
384: PetscMemcpy(Q,A,mstride*n*sizeof(PetscScalar));
385: K = N; /* full rank */
386: LAPACKungqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info);
387: if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");
389: /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
390: Alpha = 1.0;
391: ldb = lda;
392: BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb);
393: /* Ainv is Q, overwritten with inverse */
395: #if defined(PETSC_USE_COMPLEX)
396: {
397: PetscInt i;
398: for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
399: PetscFree2(A,Ainv);
400: }
401: #endif
402: return(0);
403: }
407: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
408: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
409: {
411: PetscReal *Bv;
412: PetscInt i,j;
415: PetscMalloc((ninterval+1)*ndegree*sizeof(PetscReal),&Bv);
416: /* Point evaluation of L_p on all the source vertices */
417: PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);
418: /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
419: for (i=0; i<ninterval; i++) {
420: for (j=0; j<ndegree; j++) {
421: if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
422: else B[i*ndegree+j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
423: }
424: }
425: PetscFree(Bv);
426: return(0);
427: }
431: /*@
432: PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals
434: Not Collective
436: Input Arguments:
437: + degree - degree of reconstruction polynomial
438: . nsource - number of source intervals
439: . sourcex - sorted coordinates of source cell boundaries (length nsource+1)
440: . ntarget - number of target intervals
441: - targetx - sorted coordinates of target cell boundaries (length ntarget+1)
443: Output Arguments:
444: . R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]
446: Level: advanced
448: .seealso: PetscDTLegendreEval()
449: @*/
450: PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
451: {
453: PetscInt i,j,k,*bdegrees,worksize;
454: PetscReal xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
455: PetscScalar *tau,*work;
461: if (degree >= nsource) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Reconstruction degree %D must be less than number of source intervals %D",degree,nsource);
462: #if defined(PETSC_USE_DEBUG)
463: for (i=0; i<nsource; i++) {
464: if (sourcex[i] >= sourcex[i+1]) SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_ARG_CORRUPT,"Source interval %D has negative orientation (%G,%G)",i,sourcex[i],sourcex[i+1]);
465: }
466: for (i=0; i<ntarget; i++) {
467: if (targetx[i] >= targetx[i+1]) SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_ARG_CORRUPT,"Target interval %D has negative orientation (%G,%G)",i,targetx[i],targetx[i+1]);
468: }
469: #endif
470: xmin = PetscMin(sourcex[0],targetx[0]);
471: xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
472: center = (xmin + xmax)/2;
473: hscale = (xmax - xmin)/2;
474: worksize = nsource;
475: PetscMalloc4(degree+1,PetscInt,&bdegrees,nsource+1,PetscReal,&sourcey,nsource*(degree+1),PetscReal,&Bsource,worksize,PetscScalar,&work);
476: PetscMalloc4(nsource,PetscScalar,&tau,nsource*(degree+1),PetscReal,&Bsinv,ntarget+1,PetscReal,&targety,ntarget*(degree+1),PetscReal,&Btarget);
477: for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
478: for (i=0; i<=degree; i++) bdegrees[i] = i+1;
479: PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);
480: PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);
481: for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
482: PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);
483: for (i=0; i<ntarget; i++) {
484: PetscReal rowsum = 0;
485: for (j=0; j<nsource; j++) {
486: PetscReal sum = 0;
487: for (k=0; k<degree+1; k++) {
488: sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
489: }
490: R[i*nsource+j] = sum;
491: rowsum += sum;
492: }
493: for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
494: }
495: PetscFree4(bdegrees,sourcey,Bsource,work);
496: PetscFree4(tau,Bsinv,targety,Btarget);
497: return(0);
498: }