diff --git a/src/acc/libsmm_acc/libsmm_acc.cpp b/src/acc/libsmm_acc/libsmm_acc.cpp
index b3981144586..8ad7ad05601 100644
--- a/src/acc/libsmm_acc/libsmm_acc.cpp
+++ b/src/acc/libsmm_acc/libsmm_acc.cpp
@@ -199,14 +199,14 @@ kernel_map_iterator add_kernel_handle_to_jitted_kernels(
   ACC_DRV(function) kern_func, ACC_DRV(stream) stream, Triplet h_mnk, int& threads, int& grouping, bool& generated_acc_untuned) {
   kernel_map_iterator kernel_it = kernel_handles.end();
 
-  // Check if the kernel was already generated and failed
+  // Check if the kernel was already generated and failed or if it is too big
   if (failed_acc_kernels.find(h_mnk) != failed_acc_kernels.end()) return kernel_it;
 
   libsmm_acc_algo algo;
   int tile_m, tile_n, w, v, minblocks;
 
   // Check whether autotuned parameters are given for this kernel, and if so, retrieve them
-  if (ht.find(h_mnk) != ht.end()) {
+  if (ht.find(h_mnk) != ht.end() && false) {
     // Retrieve launching parameters
     const KernelParameters params = ht.at(h_mnk);
     algo = libsmm_acc_algo(params[0]); // enum {largeDB1, largeDB2, medium, small, tiny}
@@ -219,7 +219,7 @@ kernel_map_iterator add_kernel_handle_to_jitted_kernels(
     minblocks = params[7];
     generated_acc_untuned = false;
   }
-  else { // Use a default untuned kernel
+  else if (h_mnk[0] < 50 && h_mnk[1] < 50 && h_mnk[2] < 50) { // Use a default untuned kernel
     algo = medium;
     tile_m = 2;
     tile_n = 2;
@@ -227,9 +227,13 @@ kernel_map_iterator add_kernel_handle_to_jitted_kernels(
     v = 0;
     threads = 256;
     grouping = 30;
-    minblocks = 2;
+    minblocks = 1;
     generated_acc_untuned = true;
   }
+  else {
+    failed_acc_kernels.insert(h_mnk);
+    return kernel_it;
+  }
 
   // JIT and validate the kernel
   jit_kernel(kern_func, algo, tile_m, tile_n, w, v, threads, grouping, minblocks, h_mnk[0], h_mnk[1], h_mnk[2]);
diff --git a/src/core/dbcsr_config.F b/src/core/dbcsr_config.F
index 9c57d6a1cfd..0655a1cc81a 100644
--- a/src/core/dbcsr_config.F
+++ b/src/core/dbcsr_config.F
@@ -701,7 +701,7 @@ SUBROUTINE reset_accdrv_active_device_id()
       accdrv_active_device_id = default_accdrv_active_device_id
    END SUBROUTINE reset_accdrv_active_device_id
 
-   FUNCTION use_acc()
+   PURE FUNCTION use_acc()
       LOGICAL :: use_acc
 
       IF (has_acc .AND. dbcsr_cfg%run_on_gpu%val) THEN
diff --git a/src/mm/dbcsr_mm_sched.F b/src/mm/dbcsr_mm_sched.F
index abe3dd86024..db68526e398 100644
--- a/src/mm/dbcsr_mm_sched.F
+++ b/src/mm/dbcsr_mm_sched.F
@@ -567,7 +567,8 @@ SUBROUTINE stats_print_report(report, output_unit)
       INTEGER(KIND=int_8), DIMENSION(3)                  :: flops_homo
       INTEGER, ALLOCATABLE, DIMENSION(:)                 :: sort_idx
       CHARACTER(LEN=4)                                   :: generated_acc_untuned_label
-      LOGICAL                                            :: has_acc_untuned_kernel
+      LOGICAL                                            :: has_acc_untuned_kernel, &
+                                                            use_cpu_kernels
 
       IF (output_unit <= 0) RETURN
 
@@ -582,6 +583,8 @@ SUBROUTINE stats_print_report(report, output_unit)
       total_flops_homo = 0
       flops_homo(:) = 0
       has_acc_untuned_kernel = .FALSE.
+      use_cpu_kernels = .FALSE.
+
       DO i = 1, SIZE(sort_idx)
          j = sort_idx(i) + 1
          total = SUM(report%num_mnk_stacks(j, 4:6))
@@ -595,6 +598,10 @@ SUBROUTINE stats_print_report(report, output_unit)
             has_acc_untuned_kernel = .TRUE.
          END IF
 
+         IF (SUM(report%num_mnk_stacks(j, 4:5)) .GT. 0) THEN
+            use_cpu_kernels = .TRUE.
+         END IF
+
          WRITE (output_unit, "(A,I5,' x ',I5,' x ',I5,T30,I20,5X,F5.1,'%',4X,F5.1,'%',4X,F5.1,'% ',A)") &
             " flops ", report%num_mnk_stacks(j, 1:3), &
             flops, &
@@ -603,7 +610,13 @@ SUBROUTINE stats_print_report(report, output_unit)
       END DO
 
       IF (has_acc_untuned_kernel) THEN
-         DBCSR_WARN(" (*) ACC Untuned kernels, consider to run the tuning procedure")
+         CALL dbcsr_warn(__LOCATION__, &
+                         " (*) ACC Untuned kernels, consider to run the ACC tuning procedure for them")
+      END IF
+
+      IF (use_cpu_kernels .AND. use_acc()) THEN
+         CALL dbcsr_warn(__LOCATION__, &
+                         " Some kernels are running on the CPU, consider to run the ACC tuning procedure for them")
       END IF
 
       total = report%cpu_flop + report%smm_flop + report%acc_flop