add NLsolve trust region updating scheme and change GN step to -J\fu …

…to avoid growing ill-conditioning

src/trustRegion.jl

@@ -25,6 +25,13 @@ states as `RadiusUpdateSchemes.T`. Simply put the desired scheme as follows:
     """
     Simple
 
+    """
+    `RadiusUpdateSchemes.NLsolve`
+
+    The same updating rule as in NLsolve's trust region implementation
+    """
+    NLsolve
+
     """
     `RadiusUpdateSchemes.Hei`
 
@@ -244,7 +251,9 @@ function SciMLBase.__init(prob::NonlinearProblem{uType, iip}, alg::TrustRegion,
     p3 = convert(eltype(u), 0.0)
     p4 = convert(eltype(u), 0.0)
     ϵ = convert(eltype(u), 1.0e-8)
-    if radius_update_scheme === RadiusUpdateSchemes.Hei
+    if radius_update_scheme === RadiusUpdateSchemes.NLsolve
+        p1 = convert(eltype(u), 0.5)
+    elseif radius_update_scheme === RadiusUpdateSchemes.Hei
         step_threshold = convert(eltype(u), 0.0)
         shrink_threshold = convert(eltype(u), 0.25)
         expand_threshold = convert(eltype(u), 0.25)
@@ -310,8 +319,9 @@ function perform_step!(cache::TrustRegionCache{true})
         cache.stats.njacs += 1
     end
 
-    linres = dolinsolve(alg.precs, linsolve; A = cache.H, b = _vec(cache.g),
-        linu = _vec(u_tmp), p, reltol = cache.abstol)
+    linres = dolinsolve(alg.precs, linsolve, A = J, b = _vec(fu), # cache.H, b = _vec(cache.g),
+        linu = _vec(u_tmp),
+        p = p, reltol = cache.abstol)
     cache.linsolve = linres.cache
     cache.u_tmp .= -1 .* u_tmp
     dogleg!(cache)
@@ -374,7 +384,7 @@ function trust_region_step!(cache::TrustRegionCache)
 
     # Compute the ratio of the actual reduction to the predicted reduction.
     cache.r = -(loss - cache.loss_new) / (dot(step_size, g) + dot(step_size, H, step_size) / 2)
-    @unpack r = cache
+    @unpack r = cache 
 
     if radius_update_scheme === RadiusUpdateSchemes.Simple
         # Update the trust region radius.
@@ -403,6 +413,30 @@ function trust_region_step!(cache::TrustRegionCache)
             cache.force_stop = true
         end
 
+    elseif radius_update_scheme === RadiusUpdateSchemes.NLsolve
+        # accept/reject decision
+        if r > cache.step_threshold # accept
+            take_step!(cache)
+            cache.loss = cache.loss_new
+            cache.make_new_J = true
+        else # reject
+            cache.make_new_J = false
+        end
+
+        # trust region update 
+        if r < cache.shrink_threshold # default 1 // 10
+            cache.trust_r *= cache.shrink_factor # default 1 // 2
+        elseif r >= cache.expand_threshold # default 9 // 10
+            cache.trust_r = cache.expand_factor * norm(cache.step_size) # default 2
+        elseif r >= cache.p1 # default 1 // 2
+            cache.trust_r = max(cache.trust_r, cache.expand_factor * norm(cache.step_size))
+        end
+
+        # convergence test
+        if iszero(cache.fu) || cache.internalnorm(cache.fu) < cache.abstol
+            cache.force_stop = true
+        end
+
     elseif radius_update_scheme === RadiusUpdateSchemes.Hei
         if r > cache.step_threshold
             take_step!(cache)