The above error analysis is given for the case when the matrix A(t) that needs to be approximated is given implicitly by the vector fieldF (t,·) and in case when the subproblems are solved exactly. In this section, we will handle three further situations, where we lean on [KLW16]. First, we will show in Section 2.4.1 that the error bound simplifies if A(t) is given explicitly. Second, we will handle the case when the substeps within the integration procedure are solved inexactly, e.g. when a numerical integrator such as a Runge–Kutta method is applied for solving the subproblems, see Section 2.4.2. Third, we discuss in Section 2.4.3, how the error bound improves in case when a one-sided Lipschitz condition is imposed, in order to alleviate the Lipschitz condition onF .
2.4.1 The explicit case
We consider the case whereA(t) is a given time-dependent matrix to which we search an approximation matrixY(t) of rank r. In this case the matrix projector-splitting integrator solves
.
Y(t) = P(Y(t))A(t),. Y(t0) = Y0.
We have already seen in (2.13) in Section 2.1.2 that we can determine the solutions to the corresponding subproblems in closed form, where due to the independence of the solution in each subproblem, the integrator just uses the increments A(t1)− A(t0). If we can split
up the possibly full rank matrixA(t) into a low-rank matrix X(t)∈ M and a perturbation term R(t) of possibly full rank for all t, i.e.,
A(t) = X(t) + R(t),
then the derivative ofA(t), which is projected ontoTYM, satisfies
.
A(t) = M(t) +R(t),.
where, similarly to (2.20), X(t) solvesX(t) = M(t) with X(t. 0) = X0 ∈ M. In order to be
able to apply the error analysis from Section 2.3 to the present situation, we impose kR(t0)k ≤ δ and k
.
R(t)k ≤ ε, (2.38)
such that conditions (3) and (4) in Assumption 2.3 are satisfied. Note thatM(t)∈ TYM,
since X(t) is of low rank r. Further, since F (t, Y) = A(t) is independent of Y, the. Lipschitz constant isL = 0. Thus, we are in the situation of Theorem 2.4, where the error bound can be adapted appropriately for the present explicit case. We will not provide an error analysis here, just briefly comment on the steps within the proof of Lemma 2.5 and Theorem 2.4, where equations or estimates change due to the explicit case.
In the first step of Lemma 2.5, the intension is to rewrite the differential equation for Y(t) in terms of X(t). Now, since F (t, Y(t)) is solution independent, we have.
.
Y(t) = P(Y(t))A(t) = P(Y(t))M(t) + P(Y(t)). R(t) = M(t) + P(Y(t)). R(t).
2. Error analysis of the matrix projector-splitting integrator
Hence, at first glance, it seems that there is no need to insert X(t) = M(t) as is done in. (2.21), since we already have the desired form by means ofX(t). But here, we have to be. careful concerning the projected remainder term: we cannot bound it by simply taking the norm, since this would involve a dependence on singular values of the projection, which we want to avoid. Therefore, we do insert a zero by addingX(t) and subtracting M(t) as. is done in (2.21), since this enables us to bound the perturbation term∆(t, Y) within the modified right-hand side of (2.39):
k∆(t, Y(t))k = k I −P(Y(t))R(t, Y(t)) +. R(t, X(t)). k ≤ 2ε.
Steps2–4 hold for this explicit case without any changes. Instead, step 5 contains bounds for the perturbation termsEi±, where∆(t, Y(t)) comes into play. There, for our situation we have the estimate
kEi±k ≤
Z t1
t0
k∆(s, Y±i (s))k ds ≤ 2(t1− t0)ε
for each perturbation term. Therefore, the error stated in Lemma 2.5 for the explicit case changes to
kY1− X(t1)k ≤ 4(t1− t0)ε.
Following the lines of the proof of Theorem 2.4 with this result, the local error of the Lie–Trotter projector-splitting integrator in the explicit case is then given as
kΦA.(t1, t0, Y0)− Y1k ≤ 7(t1− t0)ε.
Then, the accumulation of the propagated local errors until final timetnand the assump-
tion (2.38) about the initial distance results in the global error of the projector-splitting integrator for the explicit case, which we find as
kA(tn)− Ynk ≤ δ + 7(tn− t0)ε, t0≤ tn= T.
Note that the error bound depends on the time interval [t0, tn], though it is independent
of the time step size h.
2.4.2 Inexact solution within the integration steps
Let us consider the case when the projector-splitting integrator does not compute the ex- act values Y±i (t1), but solves the three substeps inexactly. This occurs for example when
applying a numerical method, such as a Runge–Kutta method, for determining solutions within the projector-splitting integrator. Another situation, where we solve substeps ap- proximately will be discussed in Chapter 4, where we present the integrator for Tucker tensors. Independently of the reason, let us suppose that we solve the substeps inexactly and obtain b Y±i (t1) = ΦF± i (t1, t0, bY ± i (t0)) + bE ± i .
In the first step, in fact, instead of K(t1) a perturbed value K(t1) + EK(t1) is computed and so we find b Y+1(t1) = K(t1) + EK(t1) V0,> = K(t1) V0,>+ EK(t1) V0,> = Y+1(t1) + bE + 1,
and the error satisfies
b
E+1 = EK(t1) V0,>V0V0,>= P+1(Y+1(t0))bE + 1.
The latter equation is a natural condition from the way the differential equations for the factors U, S, V of Y = U S V> are actually solved in the algorithm. For the second substep we compute S(t1) + ES(t1) and obtain the approximate solution
b Y−1(t1) = U1 S(t1) + ES(t1) V0,>= U1S(t1) V0,>+ U1ES(t1) V0,> = Y−1(t1) + bE − 1, where b E−1 = U1U1,>U1ES(t1) V0,>V0V0,>= P−1(Y−1(t0))bE − 1.
Finally, in the last step instead ofL(t1)>, we computeL(t1)>+ EL, such that the solution
results in b Y+2(t1) = U1 L(t1)>+ EL(t1) = U1L(t1)>+ U1EL(t1) = Y+2(t1) + bE + 2, where b E+2 = U1U1,>U1EL(t1) = P+2(Y+2(t0))bE + 2.
So in one full step of the method, instead ofY1 we actually compute
b Y1 = ΦF+ 2 (t1, t0, ΦF − 1 (t1, t0, ΦF + 1 (t1, t0, Y 0) + bE+ 1) + bE − 1) + bE + 2 = ΦG+ 2(t1, t0, ΦG − 1(t1, t0, ΦG + 1(t1, t0, Y 0) + E+ 1 +bE + 1) + E − 1 +bE − 1) + E+2 +bE + 2.
Suppose now that the errors are bounded by
kbE±i k ≤ η, (2.40)
then the bounds for the total error in each step are given by
kE+1 +bE + 1k ≤ h(4BLh + 2ε + η), kE−1 +bE − 1k ≤ h(4BLh + 2ε + η), kE+ 2 +bE + 2k ≤ h(5BLh + 2ε + η).
2. Error analysis of the matrix projector-splitting integrator
Therefore, we need to adapt the assumption about the initial values in Lemma 2.5 appro- priately, such that the error estimate becomes
k bY1− X(t1)k ≤ h(9BLh + 4ε + 2η).
In this situation the error bound of Theorem 2.4 changes, with the same proof, to
kA(tn)− bY n
k ≤ c0δ + c1ε + c2h + c3η,
wherec0, c1 and c2 are as before, and
c3 = (2 + eLh0)(eL(T −t0)− 1)/L. (2.41)
In case when the reason for the inexact solution of the substeps comes from the application of a Runge–Kutta method of order p in order to solve the differential equations for K, S and L, the bound (2.40) of the additional errors is of size
η =O(hp).
2.4.3 A one-sided Lipschitz condition
We have seen in Theorem 2.4 that the constants depend on the Lipschitz constant L of F (t, Y). Now, in case of a stiff differential equation, L becomes very large, which is a drawback, since then the constants and therefore the error bounds increase. However, we can overcome this difficulty mildly by the one-sided Lipschitz condition [HNW93, Section IV.12], which stays moderate in the stiff case. Suppose that with respect to the Frobenius inner product h·, ·i we have the one-sided Lipschitz bound
hF (t, Y) − F (t, eY), Y− eYi ≤ `kY − eYk2 for all Y, eY ∈ Rn1×n2,
with`≤ L and possibly ` L. In this case, Lemma 2.5 is left unchangend: the bound of the perturbation term∆(t, Y) in (2.23) still depends on the Lipschitz constant L. However, the error propagation in the proof of Theorem 2.4 improves to
kΦF(t, s, A)− ΦF(t, s, eA)k ≤ e`(t−s)kA − eAk for all A, eA∈ Rn1×n2, t > s,
where the factor eL(t−s) from (2.35) is replaced by the smaller factor e`(t−s). Following
the proof with respect to this adaption, we obtain an error bound of the same form as in Theorem 2.4, but compared to (2.37) and (2.36) with improved constants
c0 = e`(T −t0), c1= (4 + 3e`h0)(e`(T −t0)− 1)/`, c2= (9 + 4e`h0)BL(e`(T −t0)− 1)/`.
Note that the constant c2 still depends on the Lipschitz constant L, which stems from
(2.23). In case of inexact solution, the additional constant c3 in the error bound (2.41) of
the previous section is reduced to