Update README.md

yuemingl · web-flow · commit cdaef1cae20e · 2018-10-19T23:12:21.000-07:00
diff --git a/README.md b/README.md
@@ -336,3 +336,57 @@ y_0=0.01624 y_1=0.08735 y_2=0.15765 y_3=0.19518 y_4=0.25469 y_5=0.29667 y_6=0.31
 y_0=0.02256 y_1=0.09240 y_2=0.15593 y_3=0.19116 y_4=0.25076 y_5=0.29644 y_6=0.31550 \lambda_0=0.05487 \lambda_1=0.06919 \lambda_2=-0.12387 \lambda_3=0.04207 \lambda_4=0.04428 \lambda_5=-0.05989 \lambda_6=0.03240 a=0.36223 b=0.55462 
 y_0=0.02314 y_1=0.09356 y_2=0.15671 y_3=0.19159 y_4=0.25059 y_5=0.29598 y_6=0.31499 \lambda_0=0.05373 \lambda_1=0.06689 \lambda_2=-0.12542 \lambda_3=0.04123 \lambda_4=0.04463 \lambda_5=-0.05896 \lambda_6=0.03342 a=0.36185 b=0.55631 
 ```
+
+```Java	
+package symjava.examples;	
+ import static symjava.symbolic.Symbol.*;	
+import symjava.matrix.*;	
+import symjava.symbolic.*;	
+ /**	
+ * Example for PDE Constrained Parameters Optimization	
+ *	
+ */	
+public class Example4 {	
+	public static void main(String[] args) {	
+		Func u =  new Func("u",  x,y,z);	
+		Func u0 = new Func("u0", x,y,z);	
+		Func q =  new Func("q",  x,y,z);	
+		Func q0 = new Func("q0", x,y,z);	
+		Func f =  new Func("f",  x,y,z);	
+		Func lamd = new Func("\\lambda ", x,y,z);	
+			
+		Expr reg_term = (q-q0)*(q-q0)*0.5*0.1;	
+ 		Func L = new Func("L",(u-u0)*(u-u0)/2 + reg_term + q*Dot.apply(Grad.apply(u), Grad.apply(lamd)) - f*lamd);	
+		System.out.println("Lagrange L(u, \\lambda, q) = \n"+L);	
+			
+		Func phi = new Func("\\phi ", x,y,z);	
+		Func psi = new Func("\\psi ", x,y,z);	
+		Func chi = new Func("\\chi ", x,y,z);	
+		Expr[] xs =  new Expr[]{u,   lamd, q   };	
+		Expr[] dxs = new Expr[]{phi, psi,  chi };	
+		SymVector Lx = Grad.apply(L, xs, dxs);	
+		System.out.println("\nGradient Lx = (Lu, Llamd, Lq) =");	
+		System.out.println(Lx);	
+			
+		Func du = new Func("\\delta{u}", x,y,z);	
+		Func dl = new Func("\\delta{\\lambda}", x,y,z);	
+		Func dq = new Func("\\delta{q}", x,y,z);	
+		Expr[] dxs2 = new Expr[] { du, dl, dq };	
+		SymMatrix Lxx = new SymMatrix();	
+		for(Expr Lxi : Lx) {	
+			Lxx.add(Grad.apply(Lxi, xs, dxs2));	
+		}	
+		System.out.println("\nHessian Matrix =");	
+		System.out.println(Lxx);	
+	}	
+}	
+```	
+Output in Latex:	
+ Lagrange=	
+![](https://github.com/yuemingl/SymJava/blob/master/images/ex4_L.png)	
+ Hessian=	
+![](https://github.com/yuemingl/SymJava/blob/master/images/ex4_hessian.png)	
+ Grad(L)=	
+![](https://github.com/yuemingl/SymJava/blob/master/images/ex4_grad.png)	
+ Example6: Finite Element Solver for Laplace Equation	
+ ![](https://raw.githubusercontent.com/yuemingl/SymJava/master/images/ex6.png)
