Actual source code: dt.c
petsc-3.4.2 2013-07-02
1: /* Discretization tools */
3: #include <petscconf.h>
4: #if defined(PETSC_HAVE_MATHIMF_H)
5: #include <mathimf.h> /* this needs to be included before math.h */
6: #endif
8: #include <petscdt.h> /*I "petscdt.h" I*/
9: #include <petscblaslapack.h>
10: #include <petsc-private/petscimpl.h>
11: #include <petscviewer.h>
15: /*@
16: PetscDTLegendreEval - evaluate Legendre polynomial at points
18: Not Collective
20: Input Arguments:
21: + npoints - number of spatial points to evaluate at
22: . points - array of locations to evaluate at
23: . ndegree - number of basis degrees to evaluate
24: - degrees - sorted array of degrees to evaluate
26: Output Arguments:
27: + B - row-oriented basis evaluation matrix B[point*ndegree + degree] (dimension npoints*ndegrees, allocated by caller) (or NULL)
28: . D - row-oriented derivative evaluation matrix (or NULL)
29: - D2 - row-oriented second derivative evaluation matrix (or NULL)
31: Level: intermediate
33: .seealso: PetscDTGaussQuadrature()
34: @*/
35: PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
36: {
37: PetscInt i,maxdegree;
40: if (!npoints || !ndegree) return(0);
41: maxdegree = degrees[ndegree-1];
42: for (i=0; i<npoints; i++) {
43: PetscReal pm1,pm2,pd1,pd2,pdd1,pdd2,x;
44: PetscInt j,k;
45: x = points[i];
46: pm2 = 0;
47: pm1 = 1;
48: pd2 = 0;
49: pd1 = 0;
50: pdd2 = 0;
51: pdd1 = 0;
52: k = 0;
53: if (degrees[k] == 0) {
54: if (B) B[i*ndegree+k] = pm1;
55: if (D) D[i*ndegree+k] = pd1;
56: if (D2) D2[i*ndegree+k] = pdd1;
57: k++;
58: }
59: for (j=1; j<=maxdegree; j++,k++) {
60: PetscReal p,d,dd;
61: p = ((2*j-1)*x*pm1 - (j-1)*pm2)/j;
62: d = pd2 + (2*j-1)*pm1;
63: dd = pdd2 + (2*j-1)*pd1;
64: pm2 = pm1;
65: pm1 = p;
66: pd2 = pd1;
67: pd1 = d;
68: pdd2 = pdd1;
69: pdd1 = dd;
70: if (degrees[k] == j) {
71: if (B) B[i*ndegree+k] = p;
72: if (D) D[i*ndegree+k] = d;
73: if (D2) D2[i*ndegree+k] = dd;
74: }
75: }
76: }
77: return(0);
78: }
82: /*@
83: PetscDTGaussQuadrature - create Gauss quadrature
85: Not Collective
87: Input Arguments:
88: + npoints - number of points
89: . a - left end of interval (often-1)
90: - b - right end of interval (often +1)
92: Output Arguments:
93: + x - quadrature points
94: - w - quadrature weights
96: Level: intermediate
98: References:
99: Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 221--230, 1969.
101: .seealso: PetscDTLegendreEval()
102: @*/
103: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
104: {
106: PetscInt i;
107: PetscReal *work;
108: PetscScalar *Z;
109: PetscBLASInt N,LDZ,info;
112: /* Set up the Golub-Welsch system */
113: for (i=0; i<npoints; i++) {
114: x[i] = 0; /* diagonal is 0 */
115: if (i) w[i-1] = 0.5 / PetscSqrtReal(1 - 1./PetscSqr(2*i));
116: }
117: PetscRealView(npoints-1,w,PETSC_VIEWER_STDOUT_SELF);
118: PetscMalloc2(npoints*npoints,PetscScalar,&Z,PetscMax(1,2*npoints-2),PetscReal,&work);
119: PetscBLASIntCast(npoints,&N);
120: LDZ = N;
121: PetscFPTrapPush(PETSC_FP_TRAP_OFF);
122: PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("I",&N,x,w,Z,&LDZ,work,&info));
123: PetscFPTrapPop();
124: if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEQR error");
126: for (i=0; i<(npoints+1)/2; i++) {
127: PetscReal y = 0.5 * (-x[i] + x[npoints-i-1]); /* enforces symmetry */
128: x[i] = (a+b)/2 - y*(b-a)/2;
129: x[npoints-i-1] = (a+b)/2 + y*(b-a)/2;
131: w[i] = w[npoints-1-i] = (b-a)*PetscSqr(0.5*PetscAbsScalar(Z[i*npoints] + Z[(npoints-i-1)*npoints]));
132: }
133: PetscFree2(Z,work);
134: return(0);
135: }
139: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
140: Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
141: PETSC_STATIC_INLINE PetscErrorCode PetscDTFactorial_Internal(PetscInt n, PetscReal *factorial)
142: {
143: PetscReal f = 1.0;
144: PetscInt i;
147: for (i = 1; i < n+1; ++i) f *= i;
148: *factorial = f;
149: return(0);
150: }
154: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
155: Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
156: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
157: {
158: PetscReal apb, pn1, pn2;
159: PetscInt k;
162: if (!n) {*P = 1.0; return(0);}
163: if (n == 1) {*P = 0.5 * (a - b + (a + b + 2.0) * x); return(0);}
164: apb = a + b;
165: pn2 = 1.0;
166: pn1 = 0.5 * (a - b + (apb + 2.0) * x);
167: *P = 0.0;
168: for (k = 2; k < n+1; ++k) {
169: PetscReal a1 = 2.0 * k * (k + apb) * (2.0*k + apb - 2.0);
170: PetscReal a2 = (2.0 * k + apb - 1.0) * (a*a - b*b);
171: PetscReal a3 = (2.0 * k + apb - 2.0) * (2.0 * k + apb - 1.0) * (2.0 * k + apb);
172: PetscReal a4 = 2.0 * (k + a - 1.0) * (k + b - 1.0) * (2.0 * k + apb);
174: a2 = a2 / a1;
175: a3 = a3 / a1;
176: a4 = a4 / a1;
177: *P = (a2 + a3 * x) * pn1 - a4 * pn2;
178: pn2 = pn1;
179: pn1 = *P;
180: }
181: return(0);
182: }
186: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
187: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
188: {
189: PetscReal nP;
193: if (!n) {*P = 0.0; return(0);}
194: PetscDTComputeJacobi(a+1, b+1, n-1, x, &nP);
195: *P = 0.5 * (a + b + n + 1) * nP;
196: return(0);
197: }
201: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
202: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapSquareToTriangle_Internal(PetscReal x, PetscReal y, PetscReal *xi, PetscReal *eta)
203: {
205: *xi = 0.5 * (1.0 + x) * (1.0 - y) - 1.0;
206: *eta = y;
207: return(0);
208: }
212: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
213: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapCubeToTetrahedron_Internal(PetscReal x, PetscReal y, PetscReal z, PetscReal *xi, PetscReal *eta, PetscReal *zeta)
214: {
216: *xi = 0.25 * (1.0 + x) * (1.0 - y) * (1.0 - z) - 1.0;
217: *eta = 0.5 * (1.0 + y) * (1.0 - z) - 1.0;
218: *zeta = z;
219: return(0);
220: }
224: static PetscErrorCode PetscDTGaussJacobiQuadrature1D_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
225: {
226: PetscInt maxIter = 100;
227: PetscReal eps = 1.0e-8;
228: PetscReal a1, a2, a3, a4, a5, a6;
229: PetscInt k;
234: a1 = pow(2, a+b+1);
235: #if defined(PETSC_HAVE_TGAMMA)
236: a2 = tgamma(a + npoints + 1);
237: a3 = tgamma(b + npoints + 1);
238: a4 = tgamma(a + b + npoints + 1);
239: #else
240: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"tgamma() - math routine is unavailable.");
241: #endif
243: PetscDTFactorial_Internal(npoints, &a5);
244: a6 = a1 * a2 * a3 / a4 / a5;
245: /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
246: Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
247: for (k = 0; k < npoints; ++k) {
248: PetscReal r = -cos((2.0*k + 1.0) * PETSC_PI / (2.0 * npoints)), dP;
249: PetscInt j;
251: if (k > 0) r = 0.5 * (r + x[k-1]);
252: for (j = 0; j < maxIter; ++j) {
253: PetscReal s = 0.0, delta, f, fp;
254: PetscInt i;
256: for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
257: PetscDTComputeJacobi(a, b, npoints, r, &f);
258: PetscDTComputeJacobiDerivative(a, b, npoints, r, &fp);
259: delta = f / (fp - f * s);
260: r = r - delta;
261: if (fabs(delta) < eps) break;
262: }
263: x[k] = r;
264: PetscDTComputeJacobiDerivative(a, b, npoints, x[k], &dP);
265: w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
266: }
267: return(0);
268: }
272: /*@C
273: PetscDTGaussJacobiQuadrature - create Gauss-Jacobi quadrature for a simplex
275: Not Collective
277: Input Arguments:
278: + dim - The simplex dimension
279: . npoints - number of points
280: . a - left end of interval (often-1)
281: - b - right end of interval (often +1)
283: Output Arguments:
284: + points - quadrature points
285: - weights - quadrature weights
287: Level: intermediate
289: References:
290: Karniadakis and Sherwin.
291: FIAT
293: .seealso: PetscDTGaussQuadrature()
294: @*/
295: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt dim, PetscInt npoints, PetscReal a, PetscReal b, PetscReal *points[], PetscReal *weights[])
296: {
297: PetscReal *px, *wx, *py, *wy, *pz, *wz, *x, *w;
298: PetscInt i, j, k;
302: if ((a != -1.0) || (b != 1.0)) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
303: switch (dim) {
304: case 1:
305: PetscMalloc(npoints * sizeof(PetscReal), &x);
306: PetscMalloc(npoints * sizeof(PetscReal), &w);
307: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, x, w);
308: break;
309: case 2:
310: PetscMalloc(npoints*npoints*2 * sizeof(PetscReal), &x);
311: PetscMalloc(npoints*npoints * sizeof(PetscReal), &w);
312: PetscMalloc4(npoints,PetscReal,&px,npoints,PetscReal,&wx,npoints,PetscReal,&py,npoints,PetscReal,&wy);
313: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, px, wx);
314: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 1.0, 0.0, py, wy);
315: for (i = 0; i < npoints; ++i) {
316: for (j = 0; j < npoints; ++j) {
317: PetscDTMapSquareToTriangle_Internal(px[i], py[j], &x[(i*npoints+j)*2+0], &x[(i*npoints+j)*2+1]);
318: w[i*npoints+j] = 0.5 * wx[i] * wy[j];
319: }
320: }
321: PetscFree4(px,wx,py,wy);
322: break;
323: case 3:
324: PetscMalloc(npoints*npoints*3 * sizeof(PetscReal), &x);
325: PetscMalloc(npoints*npoints * sizeof(PetscReal), &w);
326: PetscMalloc6(npoints,PetscReal,&px,npoints,PetscReal,&wx,npoints,PetscReal,&py,npoints,PetscReal,&wy,npoints,PetscReal,&pz,npoints,PetscReal,&wz);
327: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, px, wx);
328: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 1.0, 0.0, py, wy);
329: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 2.0, 0.0, pz, wz);
330: for (i = 0; i < npoints; ++i) {
331: for (j = 0; j < npoints; ++j) {
332: for (k = 0; k < npoints; ++k) {
333: 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]);
334: w[(i*npoints+j)*npoints+k] = 0.125 * wx[i] * wy[j] * wz[k];
335: }
336: }
337: }
338: PetscFree6(px,wx,py,wy,pz,wz);
339: break;
340: default:
341: SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
342: }
343: if (points) *points = x;
344: if (weights) *weights = w;
345: return(0);
346: }
350: /* Overwrites A. Can only handle full-rank problems with m>=n
351: * A in column-major format
352: * Ainv in row-major format
353: * tau has length m
354: * worksize must be >= max(1,n)
355: */
356: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
357: {
359: PetscBLASInt M,N,K,lda,ldb,ldwork,info;
360: PetscScalar *A,*Ainv,*R,*Q,Alpha;
363: #if defined(PETSC_USE_COMPLEX)
364: {
365: PetscInt i,j;
366: PetscMalloc2(m*n,PetscScalar,&A,m*n,PetscScalar,&Ainv);
367: for (j=0; j<n; j++) {
368: for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
369: }
370: mstride = m;
371: }
372: #else
373: A = A_in;
374: Ainv = Ainv_out;
375: #endif
377: PetscBLASIntCast(m,&M);
378: PetscBLASIntCast(n,&N);
379: PetscBLASIntCast(mstride,&lda);
380: PetscBLASIntCast(worksize,&ldwork);
381: PetscFPTrapPush(PETSC_FP_TRAP_OFF);
382: LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info);
383: PetscFPTrapPop();
384: if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
385: R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */
387: /* Extract an explicit representation of Q */
388: Q = Ainv;
389: PetscMemcpy(Q,A,mstride*n*sizeof(PetscScalar));
390: K = N; /* full rank */
391: LAPACKungqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info);
392: if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");
394: /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
395: Alpha = 1.0;
396: ldb = lda;
397: BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb);
398: /* Ainv is Q, overwritten with inverse */
400: #if defined(PETSC_USE_COMPLEX)
401: {
402: PetscInt i;
403: for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
404: PetscFree2(A,Ainv);
405: }
406: #endif
407: return(0);
408: }
412: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
413: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
414: {
416: PetscReal *Bv;
417: PetscInt i,j;
420: PetscMalloc((ninterval+1)*ndegree*sizeof(PetscReal),&Bv);
421: /* Point evaluation of L_p on all the source vertices */
422: PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);
423: /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
424: for (i=0; i<ninterval; i++) {
425: for (j=0; j<ndegree; j++) {
426: if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
427: else B[i*ndegree+j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
428: }
429: }
430: PetscFree(Bv);
431: return(0);
432: }
436: /*@
437: PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals
439: Not Collective
441: Input Arguments:
442: + degree - degree of reconstruction polynomial
443: . nsource - number of source intervals
444: . sourcex - sorted coordinates of source cell boundaries (length nsource+1)
445: . ntarget - number of target intervals
446: - targetx - sorted coordinates of target cell boundaries (length ntarget+1)
448: Output Arguments:
449: . R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]
451: Level: advanced
453: .seealso: PetscDTLegendreEval()
454: @*/
455: PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
456: {
458: PetscInt i,j,k,*bdegrees,worksize;
459: PetscReal xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
460: PetscScalar *tau,*work;
466: 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);
467: #if defined(PETSC_USE_DEBUG)
468: for (i=0; i<nsource; i++) {
469: 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]);
470: }
471: for (i=0; i<ntarget; i++) {
472: 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]);
473: }
474: #endif
475: xmin = PetscMin(sourcex[0],targetx[0]);
476: xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
477: center = (xmin + xmax)/2;
478: hscale = (xmax - xmin)/2;
479: worksize = nsource;
480: PetscMalloc4(degree+1,PetscInt,&bdegrees,nsource+1,PetscReal,&sourcey,nsource*(degree+1),PetscReal,&Bsource,worksize,PetscScalar,&work);
481: PetscMalloc4(nsource,PetscScalar,&tau,nsource*(degree+1),PetscReal,&Bsinv,ntarget+1,PetscReal,&targety,ntarget*(degree+1),PetscReal,&Btarget);
482: for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
483: for (i=0; i<=degree; i++) bdegrees[i] = i+1;
484: PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);
485: PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);
486: for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
487: PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);
488: for (i=0; i<ntarget; i++) {
489: PetscReal rowsum = 0;
490: for (j=0; j<nsource; j++) {
491: PetscReal sum = 0;
492: for (k=0; k<degree+1; k++) {
493: sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
494: }
495: R[i*nsource+j] = sum;
496: rowsum += sum;
497: }
498: for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
499: }
500: PetscFree4(bdegrees,sourcey,Bsource,work);
501: PetscFree4(tau,Bsinv,targety,Btarget);
502: return(0);
503: }