diff --git a/Project.toml b/Project.toml
index b69ab6d..85ec644 100644
--- a/Project.toml
+++ b/Project.toml
@@ -20,6 +20,7 @@ SciMLOperators = "c0aeaf25-5076-4817-a8d5-81caf7dfa961"
 CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
 FiniteDiff = "6a86dc24-6348-571c-b903-95158fe2bd41"
 ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
+LinearSolve = "7ed4a6bd-45f5-4d41-b270-4a48e9bafcae"
 ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267"
 Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
 
@@ -27,6 +28,7 @@ Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
 BatchedRoutinesCUDAExt = ["CUDA"]
 BatchedRoutinesFiniteDiffExt = ["FiniteDiff"]
 BatchedRoutinesForwardDiffExt = ["ForwardDiff"]
+BatchedRoutinesLinearSolveExt = ["LinearSolve"]
 BatchedRoutinesReverseDiffExt = ["ReverseDiff"]
 BatchedRoutinesZygoteExt = ["Zygote"]
 
diff --git a/ext/BatchedRoutinesLinearSolveExt.jl b/ext/BatchedRoutinesLinearSolveExt.jl
new file mode 100644
index 0000000..a808279
--- /dev/null
+++ b/ext/BatchedRoutinesLinearSolveExt.jl
@@ -0,0 +1,128 @@
+module BatchedRoutinesLinearSolveExt
+
+using ArrayInterface: ArrayInterface
+using BatchedRoutines: BatchedRoutines, UniformBlockDiagonalOperator, getdata
+using LinearAlgebra: LinearAlgebra
+using LinearSolve: LinearSolve
+
+# Overload LinearProblem, else causing problems in the adjoint code
+function LinearSolve.LinearProblem(op::UniformBlockDiagonalOperator, b, args...; kwargs...)
+    return LinearSolve.LinearProblem{true}(op, b, args...; kwargs...)
+end
+
+# Default Algorithm
+function LinearSolve.defaultalg(
+        op::UniformBlockDiagonalOperator, b, assump::LinearSolve.OperatorAssumptions{Bool})
+    alg = if assump.issq
+        LinearSolve.DefaultAlgorithmChoice.LUFactorization
+    elseif assump.condition === LinearSolve.OperatorCondition.WellConditioned
+        LinearSolve.DefaultAlgorithmChoice.NormalCholeskyFactorization
+    elseif assump.condition === LinearSolve.OperatorCondition.IllConditioned
+        if LinearSolve.is_underdetermined(op)
+            LinearSolve.DefaultAlgorithmChoice.QRFactorizationPivoted
+        else
+            LinearSolve.DefaultAlgorithmChoice.QRFactorization
+        end
+    elseif assump.condition === LinearSolve.OperatorCondition.VeryIllConditioned
+        if LinearSolve.is_underdetermined(op)
+            LinearSolve.DefaultAlgorithmChoice.QRFactorizationPivoted
+        else
+            LinearSolve.DefaultAlgorithmChoice.QRFactorization
+        end
+    elseif assump.condition === LinearSolve.OperatorCondition.SuperIllConditioned
+        LinearSolve.DefaultAlgorithmChoice.SVDFactorization
+    else
+        error("Special factorization not handled in current default algorithm.")
+    end
+    return LinearSolve.DefaultLinearSolver(alg)
+end
+
+# GenericLUFactorization
+function LinearSolve.init_cacheval(alg::LinearSolve.GenericLUFactorization,
+        A::UniformBlockDiagonalOperator, b, u, Pl, Pr, maxiters::Int,
+        abstol, reltol, verbose::Bool, ::LinearSolve.OperatorAssumptions)
+    A_ = UniformBlockDiagonalOperator(similar(getdata(A), 0, 0, 1))
+    return LinearAlgebra.generic_lufact!(A_, alg.pivot; check=false)
+end
+
+function LinearSolve.do_factorization(
+        alg::LinearSolve.GenericLUFactorization, A::UniformBlockDiagonalOperator, b, u)
+    return LinearAlgebra.generic_lufact!(A, alg.pivot; check=false)
+end
+
+# LUFactorization
+function LinearSolve.init_cacheval(
+        alg::LinearSolve.LUFactorization, A::UniformBlockDiagonalOperator, b, u, Pl, Pr,
+        maxiters::Int, abstol, reltol, verbose::Bool, ::LinearSolve.OperatorAssumptions)
+    A_ = UniformBlockDiagonalOperator(similar(getdata(A), 0, 0, 1))
+    return LinearAlgebra.lu!(A_, alg.pivot; check=false)
+end
+
+function LinearSolve.do_factorization(
+        alg::LinearSolve.LUFactorization, A::UniformBlockDiagonalOperator, b, u)
+    return LinearAlgebra.lu!(A, alg.pivot; check=false)
+end
+
+# QRFactorization
+function LinearSolve.init_cacheval(
+        alg::LinearSolve.QRFactorization, A::UniformBlockDiagonalOperator, b, u, Pl, Pr,
+        maxiters::Int, abstol, reltol, verbose::Bool, ::LinearSolve.OperatorAssumptions)
+    A_ = UniformBlockDiagonalOperator(similar(getdata(A), 0, 0, 1))
+    return LinearAlgebra.qr!(A_, alg.pivot)
+end
+
+function LinearSolve.do_factorization(
+        alg::LinearSolve.QRFactorization, A::UniformBlockDiagonalOperator, b, u)
+    alg.inplace && return LinearAlgebra.qr!(A, alg.pivot)
+    return LinearAlgebra.qr(A, alg.pivot)
+end
+
+# CholeskyFactorization
+function LinearSolve.init_cacheval(alg::LinearSolve.CholeskyFactorization,
+        A::UniformBlockDiagonalOperator, b, u, Pl, Pr, maxiters::Int,
+        abstol, reltol, verbose::Bool, ::LinearSolve.OperatorAssumptions)
+    A_ = UniformBlockDiagonalOperator(similar(getdata(A), 0, 0, 1))
+    return ArrayInterface.cholesky_instance(A_, alg.pivot)
+end
+
+function LinearSolve.do_factorization(
+        alg::LinearSolve.CholeskyFactorization, A::UniformBlockDiagonalOperator, b, u)
+    return LinearAlgebra.cholesky!(A, alg.pivot; check=false)
+end
+
+# NormalCholeskyFactorization
+function LinearSolve.init_cacheval(alg::LinearSolve.NormalCholeskyFactorization,
+        A::UniformBlockDiagonalOperator, b, u, Pl, Pr, maxiters::Int,
+        abstol, reltol, verbose::Bool, ::LinearSolve.OperatorAssumptions)
+    A_ = UniformBlockDiagonalOperator(similar(getdata(A), 0, 0, 1))
+    return ArrayInterface.cholesky_instance(A_, alg.pivot)
+end
+
+function LinearSolve.solve!(cache::LinearSolve.LinearCache{<:UniformBlockDiagonalOperator},
+        alg::LinearSolve.NormalCholeskyFactorization; kwargs...)
+    A = cache.A
+    if cache.isfresh
+        fact = LinearAlgebra.cholesky!(A' * A, alg.pivot; check=false)
+        cache.cacheval = fact
+        cache.isfresh = false
+    end
+    y = LinearAlgebra.ldiv!(
+        cache.u, LinearSolve.@get_cacheval(cache, :NormalCholeskyFactorization),
+        A' * cache.b)
+    return LinearSolve.SciMLBase.build_linear_solution(alg, y, nothing, cache)
+end
+
+# SVDFactorization
+function LinearSolve.init_cacheval(
+        alg::LinearSolve.SVDFactorization, A::UniformBlockDiagonalOperator, b, u, Pl, Pr,
+        maxiters::Int, abstol, reltol, verbose::Bool, ::LinearSolve.OperatorAssumptions)
+    A_ = UniformBlockDiagonalOperator(similar(getdata(A), 0, 0, 1))
+    return ArrayInterface.svd_instance(A_)
+end
+
+function LinearSolve.do_factorization(
+        alg::LinearSolve.SVDFactorization, A::UniformBlockDiagonalOperator, b, u)
+    return LinearAlgebra.svd!(A; alg.full, alg.alg)
+end
+
+end
diff --git a/src/chainrules.jl b/src/chainrules.jl
index 15ac2c6..44f5e56 100644
--- a/src/chainrules.jl
+++ b/src/chainrules.jl
@@ -86,16 +86,12 @@ end
 # batched_mul rrule
 function CRC.rrule(::typeof(_batched_mul), A::AbstractArray{T1, 3},
         B::AbstractArray{T2, 3}) where {T1, T2}
-    function ∇batched_mul(_Δ)
+    ∇batched_mul = @closure _Δ -> begin
         Δ = CRC.unthunk(_Δ)
-        ∂A = CRC.@thunk begin
-            tmp = batched_mul(Δ, batched_adjoint(B))
-            size(A, 3) == 1 ? sum(tmp; dims=3) : tmp
-        end
-        ∂B = CRC.@thunk begin
-            tmp = batched_mul(batched_adjoint(A), Δ)
-            size(B, 3) == 1 ? sum(tmp; dims=3) : tmp
-        end
+        tmpA = batched_mul(Δ, batched_adjoint(B))
+        ∂A = size(A, 3) == 1 ? sum(tmpA; dims=3) : tmpA
+        tmpB = batched_mul(batched_adjoint(A), Δ)
+        ∂B = size(B, 3) == 1 ? sum(tmpB; dims=3) : tmpB
         return (NoTangent(), ∂A, ∂B)
     end
     return batched_mul(A, B), ∇batched_mul
@@ -103,7 +99,7 @@ end
 
 # constructor
 function CRC.rrule(::Type{<:UniformBlockDiagonalOperator}, data)
-    function ∇UniformBlockDiagonalOperator(Δ)
+    ∇UniformBlockDiagonalOperator = @closure Δ -> begin
         ∂data = Δ isa UniformBlockDiagonalOperator ? getdata(Δ) :
                 (Δ isa NoTangent ? NoTangent() : Δ)
         return (NoTangent(), ∂data)
@@ -113,7 +109,7 @@ end
 
 function CRC.rrule(::typeof(getproperty), op::UniformBlockDiagonalOperator, x::Symbol)
     @assert x === :data
-    ∇getproperty(Δ) = (NoTangent(), UniformBlockDiagonalOperator(Δ))
+    ∇getproperty = @closure Δ -> (NoTangent(), UniformBlockDiagonalOperator(Δ))
     return op.data, ∇getproperty
 end
 
diff --git a/src/factorization.jl b/src/factorization.jl
index 8727a2a..bc0561c 100644
--- a/src/factorization.jl
+++ b/src/factorization.jl
@@ -82,18 +82,19 @@ function Base.show(io::IO, mime::MIME"text/plain", F::GenericBatchedFactorizatio
     show(io, mime, first(F.fact))
 end
 
-for fact in (:qr, :lu, :cholesky)
+for fact in (:qr, :lu, :cholesky, :generic_lufact, :svd)
     fact! = Symbol(fact, :!)
-    @eval begin
-        function LinearAlgebra.$(fact)(op::UniformBlockDiagonalOperator, args...; kwargs...)
+    if isdefined(LinearAlgebra, fact)
+        @eval function LinearAlgebra.$(fact)(
+                op::UniformBlockDiagonalOperator, args...; kwargs...)
             return LinearAlgebra.$(fact!)(copy(op), args...; kwargs...)
         end
+    end
 
-        function LinearAlgebra.$(fact!)(
-                op::UniformBlockDiagonalOperator, args...; kwargs...)
-            fact = map(Aᵢ -> LinearAlgebra.$(fact!)(Aᵢ, args...; kwargs...), batchview(op))
-            return GenericBatchedFactorization(LinearAlgebra.$(fact!), fact)
-        end
+    @eval function LinearAlgebra.$(fact!)(
+            op::UniformBlockDiagonalOperator, args...; kwargs...)
+        fact = map(Aᵢ -> LinearAlgebra.$(fact!)(Aᵢ, args...; kwargs...), batchview(op))
+        return GenericBatchedFactorization(LinearAlgebra.$(fact!), fact)
     end
 end
 
diff --git a/test/integration_tests.jl b/test/integration_tests.jl
index c9a0ec0..b539ae0 100644
--- a/test/integration_tests.jl
+++ b/test/integration_tests.jl
@@ -1,5 +1,5 @@
 @testitem "LinearSolve" setup=[SharedTestSetup] begin
-    using FiniteDiff, LinearSolve, Zygote
+    using FiniteDiff, LinearAlgebra, LinearSolve, Zygote
 
     rng = get_stable_rng(1001)
 
@@ -13,12 +13,26 @@
             prob2 = LinearProblem(A2, b)
 
             if dims[1] == dims[2]
-                solvers = [LUFactorization(), QRFactorization(), KrylovJL_GMRES()]
+                solvers = [LUFactorization(), QRFactorization(),
+                    KrylovJL_GMRES(), svd_factorization(mode), nothing]
             else
-                solvers = [QRFactorization(), KrylovJL_LSMR()]
+                solvers = [
+                    QRFactorization(), KrylovJL_LSMR(), NormalCholeskyFactorization(),
+                    QRFactorization(LinearAlgebra.ColumnNorm()),
+                    svd_factorization(mode), nothing]
             end
 
-            @testset "solver: $(solver)" for solver in solvers
+            @testset "solver: $(nameof(typeof(solver)))" for solver in solvers
+                # FIXME: SVD doesn't define ldiv on CUDA side
+                if mode == "CUDA"
+                    @show solver, solver isa SVDFactorization
+                    if solver isa SVDFactorization || (solver isa QRFactorization &&
+                        solver.pivot isa LinearAlgebra.ColumnNorm)
+                        # ColumnNorm is not implemented on CUDA
+                        continue
+                    end
+                end
+
                 x1 = solve(prob1, solver)
                 x2 = solve(prob2, solver)
                 @test x1.u ≈ x2.u
diff --git a/test/shared_testsetup.jl b/test/shared_testsetup.jl
index c3417fc..d69bfc7 100644
--- a/test/shared_testsetup.jl
+++ b/test/shared_testsetup.jl
@@ -32,7 +32,18 @@ end
 
 get_stable_rng(seed=12345) = StableRNG(seed)
 
+# SVD Helper till https://github.com/SciML/LinearSolve.jl/issues/488 is resolved
+using LinearSolve: LinearSolve
+
+function svd_factorization(mode)
+    mode == "CPU" && return LinearSolve.SVDFactorization()
+    mode == "CUDA" &&
+        return LinearSolve.SVDFactorization(true, CUDA.CUSOLVER.JacobiAlgorithm())
+    error("Unsupported mode: $mode")
+end
+
 export @jet, @test_gradients, check_approx
 export GROUP, MODES, cpu_testing, cuda_testing, get_default_rng, get_stable_rng
+export svd_factorization
 
 end