Some thoughts on linearity, nonlinearity, and partial separability

Transcription

1 Some thoughts on linearity, nonlinearity, and partial separability Paul Hovland Argonne Na0onal Laboratory Joint work with Boyana Norris, Sri Hari Krishna Narayanan, Jean Utke, Drew Wicke Argonne Na0onal Laboratory Assefaw H. Gebremedhin Purdue University 1

2 Outline q Some context: MMICC and M2ACS q Automa0c detec0on and use of par0al separability q Structure in nonlinear computa0ons 2

3 Context: Multifaceted, Integrative Mathematics Computa0onal Mathema0cs Cross- Cu'ng Projects Mathema0cal Mul0faceted Integrated Capability Centers Integrated Projects DOE Mission Efforts Op0miza0on PDEs Stochas0cs and Uncertainty Analysis Big Data Analy0cs Linear Algebra & Scalable Solvers Mul0faceted Mathema0cs for Complex Energy Systems Collaboratory of Mathema0cs for Mesoscopic Modeling of Materials DiaMonD: An Integrated Mul0faceted Approach to Mathema0cs at the Interfaces of Data, Models, and Decisions Future Collaboratory Math Projects SciDAC Partnerships DOE Applica0ons Exascale Co- Design Centers Mul0scale Math Exploratory Applied Mathema0cs Research Argonne Na0onal Laboratory Mathema0cs and Computer Science Division 6/10/13 3

4 Context: Multifaceted, Integrative Mathematics Mul0faceted Mathema0cs for Complex Energy Systems q Focuses on the grand challenges of analysis, design, planning, maintenance, and opera0on of electrical energy systems and related infrastructure in the presence of rapidly increasingly complexity of the systems. q Four mathema0cal areas iden0fied: q q Predic0ve modeling that accounts for uncertainty and errors Mathema0cs of decisions that allow hierarchical, data- driven and real- 0me decision making Scalable solu0on algorithms for op0miza0on and dynamic simula0on Integra0ve frameworks leveraging model reduc0on and mul0scale analysis Mathema0cal aspects include: discrete and con0nuous op0miza0on, dynamical systems, mul0- level techniques, data- driven methods, graph- theore0cal methods, and stochas0c and probabilis0c approaches for uncertainty and error. Mathema0cs addresses a broad class of applica0ons for complex energy systems including planning for power grid and related infrastructure; analysis and design for renewable energy integra0on; real- 0me broad- scale system monitoring and predic0on; and predic0ve control of cascading blackouts. Argonne Na0onal Laboratory Mathema0cs and Computer Science Division 6/10/13 4

5 Partial separability q Large op0miza0on problems oaen exhibit structure known as par4al separability (e.g., least squares problems) q Given where depends on variables. 5

6 Automation in partial separability identification and code transformation q Source transforma0ons prior to AD transforma0ons. Three AST traversals: 1. Iden0fica0on of par0al separability 2. Promo0on of elemental func0ons within loops and crea0on of a summa0on func0on 3. Genera0on of elemental ini0aliza0on loop q Iden0fy sparsity structure of extended Jacobian q Color q Compute compressed extended Jacobian q Assemble extended Jacobian into gradient 6

7 Automated partial separability detection! Original void deptfg(int nx, int ny,double *x, double *f, double c) { //Initialization of elementals fquad = 0.0; flin = 0.0; for(j = -1; j < ny; j++) { for(i = -1; i < nx; i++) { //Updates to fquad and flin //Assignment to dependent variable *f = area*(p5*fquad+flin); Post transformation With user annotations void deptfg(int nx, int ny,double *x, double *f, double c) { #pragma $adic_partialelemental fquad = 0.0; #pragma $adic_partialelemental flin = 0.0; for(j = -1; j < ny; j++) { for(i = -1; i < nx; i++) { //Updates to fquad and flin #pragma $adic_partiallyseparable, fquad, flin *f = area*(p5*fquad+flin); void deptfg(int nx, int ny,double *x, double *f, double c) { int ad_var_max = (ny - 1) * (nx -1); #pragma $adic_partialelemental for (i = 0; i < ad_var_max; i = i + 1) { flin[i] = 0.0; fquad[i] = 0.0; for(j = -1; j < ny; j++) { for(i = -1; i < nx; i++) { //Updates to fquad and flin #pragma $adic_partiallyseparable, fquad, flin ADIC_SPARSE_Create1DimArray(& adic_temp_f, ad_var_max); for (i = 0; i < ad_var_max; i = i + 1) { adic_temp_f[i] = (area * ((p5 * fquad[i]) + flin[i])); ADIC_SPARSE_Summation(f, adic_temp_f,ad_var_max); ADIC 2 + ColPack 7

8 Exploting Jacobian Sparsity! q Step 1: Determine the sparsity structure of A (ADIC2/SparsLinC) q Step 2: Obtain seed matrix by coloring A s graph (ColPack) q Step 3: Compute a compressed matrix B = AS (ADIC2) q Step 4: Recover entries of A from B (ColPack) A" S" B" m n" n p" m p" 8

9 Example: 2-D elastic-plastic torsion model optimization (MINPACK-2) q Finite- element discre0za0on to compute the stress field on an infinitely long cylindrical bar to which a fixed angle of twist per unit length has been applied q Unconstrained minimiza0on problem 9

10 Performance q Result: up to 1,000x 0me reduc0on over dense for MINPACK2 s elas0c- plas0c torsion problem. q If the Jacobian is par0ally separable User iden0fies the sparse- elemental func0ons Compute compressed- elemental Jacobians and combine SIAM Annual Mee0ng, July 13,

11 Ongoing work: Automate detection of partial separability q Linearity analysis to iden0fy intermediate variables. q SparsLinC to check dynamically whether the intermediate gradients are truly sparse. Yes: par0al separability is exploited. No: use the reverse mode. 11

12 Linearity Analysis 12

13 Linearity + Partial Separability 13

14 Conclusions, Part 1 q We can automa0cally use user- iden0fied par0al separability q We have made progress toward automa0cally detec0ng par0al sepability q Next steps involve run0me detec0on of sparsity or lack thereof SIAM Annual Mee0ng, July 13,

15 Structure in nonlinear computations q Observa0on: in nonlinear power grid analysis, all coupling due to network structure is expressed via linear constraints; iden0cal nonlinear constraint for each edge q Ques0on: does similar decoupling of local nonlineari0es and linear coupling occur in PDEs? q Answer: maybe 15

16 Structure in nonlinear PDEs: solid fuel ignition q Simple nonlinear PDE: Δu + λe u = 0 q Can be discre0zed using finite differences: for (j=ys; j<ys+ym; j++) {! for (i=xs; i<xs+xm; i++) {! if (i == 0 j == 0 i == mx-1 j == my-1) {! f[j][i] = 2.0*(hydhx+hxdhy)*x[j][i];! else {! u = x[j][i];! uxx = (2.0*u - x[j][i-1] - x[j][i+1])*hydhx;! uyy = (2.0*u - x[j-1][i] - x[j+1][i])*hxdhy;! f[j][i] = uxx + uyy - sc*petscexpscalar(u);!!!! 16

17 Structure in nonlinear PDEs Δu + λe u = 0 q Coupling occurs via (linear) differen0al operator q Can separate linear and nonlinear terms for (j=ys; j<ys+ym; j++) {! for (i=xs; i<xs+xm; i++) {! if (i == 0 j == 0 i == mx-1 j == my-1) {! f[j][i] = 2.0*(hydhx+hxdhy)*x[j][i];! else {! f[j][i] = lin(...) + nonlin(...);!!! double lin(double C,double S,double N,double W,double E,...) {! uxx = (2.0*C - W - E)*hydhx;! uyy = (2.0*C - S - N)*hxdhy;! return (uxx + uyy);!! double nonlin(double C,double S,double N,double W,double E,...) {! return (sc*petscexpscalar(c));!! 17

18 Structure in nonlinear PDEs q How ubiquitous is this structure (linear coupling of nonlinear terms)? q Can we exploit structure in solves as well as J computa0on? q Can we automate the detec0on of the linear and nonlinear terms and the genera0on of deriva0ve code for these terms? SIAM Annual Mee0ng, July 13,

19 Conclusions q We s0ll have work to do. SIAM Annual Mee0ng, July 13,