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Operator splitting methods for non-autonomous evolution equations comprising three parts, with application to the Bloch-Torrey equation of MRI

Barbara Kaltenbacher, Pablo Muñoz, Mechthild Thalhammer

arXiv:2610.04355v1math.NAcs.NA

TL;DR

Efficient solution of the non-autonomous Bloch-Torrey equation matters for MRI coefficient reconstruction and time-dependent control. The paper analyzes two three-part Lie–Trotter approaches and establishes first-order convergence, with numerical tests illustrating the theoretical findings.

  • Problem

    The Bloch-Torrey equation is a non-autonomous PDE whose efficient numerical solution is essential for MRI coefficient reconstruction.

    Source: §1 · p. 1

  • Method

    The paper analyzes three-part Lie–Trotter splitting approaches for non-autonomous evolution equations, including a setting with multiplication and advection-diffusion operators.

    Source: §2 · p. 11, §3 · p. 12

  • Results

    Both approaches have first-order global error estimates, and numerical tests show O(Δt) convergence for frozen and non-frozen treatments of the non-autonomous substeps.

    Source: §2 · p. 11, §5.2 · p. 18

  • Takeaways & Limitations

    The framework applies to evolution problems with time-dependent bilinear control alongside an advection-diffusion component.

    Source: §1 · p. 1

    For the advection-diffusion operator, estimates cover C(t)−C(s) and C′(t), but do not assert boundedness of C(t) itself.

    Source: Proof of Proposition 5 · p. 25

Abstract

from arXiv · show

In this paper we devise and analyze Lie Trotter splitting schemes for non-autonomous evolution equations, following two approaches: 1) A fully non-autonomous scheme, motivated by the potential use of explicit solutions of some subproblems. 2) A splitting scheme relying on the concept of Magnus integrators, where we freeze the operators at the beginning of each time interval as this suffices for first order convergence of the Lie-Trotter integrator. We derive first order convergence in a general setting and apply the analysis to the Bloch-Torrey equation of magnetic resonance imaging, where a decomposition into three subproblems is well-motivated by the physical role of the underlying operators. Numerical tests, complementing the time stepping scheme by a finite element space discretization, illustrate the theoretical findings.

1. Introduction

The paper develops a framework for analyzing Lie–Trotter splitting with three non-autonomous components and applies it to the Bloch-Torrey equation. The MRI motivation is efficient numerical solution for coefficient reconstruction and time-dependent control, with the equation’s physical effects supporting a three-part decomposition.

1. Introduction

  • Efficient Bloch-Torrey solution is essential for MRI coefficient reconstruction, while time-dependent controls are used to enhance image quality.The equation’s structure supports decomposition into three parts corresponding to distinct physical effects.Source: §1 · p. 1
  • The paper develops a framework for analyzing operator splitting with more than two parts and devises three-part schemes for the Bloch-Torrey equation.The introduction identifies time-dependent controls alongside advection-diffusion as a broader application setting.Source: §1 · pp. 1, 4
  • The Bloch-Torrey terms represent rotation, relaxation, diffusion, and motion, with diffusion and motion forming an advection-diffusion subproblem.The introduction describes rotation and relaxation as distinct physical effects and notes that standard numerical schemes exist for the advection-diffusion component.Source: §1 · pp. 2–3
  • The analysis is motivated by time-dependent bilinear control problems that also include an advection-diffusion component.The paper also outlines a general framework, operator estimates, Bloch-Torrey analysis, space discretization, and numerical experiments.Source: §1 · p. 4

2. General framework

The section formulates linear non-autonomous evolution equations with three time-dependent operators and presents two Lie–Trotter constructions: a fully non-autonomous reformulation and a left-endpoint-frozen Magnus-style scheme. It establishes first-order global convergence for both approaches under stated stability and boundedness conditions.

Approach 1 (Reformulation, Lie–Trotter splitting)

  • The first approach reformulates time dependence with an auxiliary scalar and successively composes non-autonomous subproblem solutions, using known exact subproblem solutions for Bloch–Torrey.The subproblems advance through scaled portions of the time interval in operator order.Source: §2 · pp. 4–6

Summary (Approach 2)

  • For the Magnus-style approach, freezing the defining operator at each interval’s initial time gives a first-order approximation, and stability estimates yield the desired global convergence rate.The proof separates the error from freezing time from the error of autonomous Lie–Trotter splitting.Source: §2 · pp. 5, 10, 11

Local error analysis

  • The error analysis uses commutators of the defining operators and variation-of-constants representations to control local defects and establish their dependence on the time increment.Temporal smoothness and suitable stability and regularity conditions are needed for the resulting estimates.Source: §2 · pp. 7, 10

Summary (Approach 1)

  • The fully non-autonomous Lie–Trotter approach has a second-order local error bound and first-order convergence under the required stability and regularity assumptions.The summary of this approach states that the global first-order rate follows from its local error bound; the paper later specifies the assumptions in Theorem 1.Source: §2 · p. 10

2. General framework

  • Both the reformulated non-autonomous Lie–Trotter method and the piecewise-constant Magnus integrator composed with autonomous Lie–Trotter splitting have first-order global error, O(τ_n).The global error constant is independent of the time increment and number of steps.Source: §2 · p. 11
  • The convergence result assumes a shared operator-domain intersection, stable evolution operators, bounded exact-solution values, and bounded pairwise commutators and their compositions with evolution operators.The commutator bounds are required on a suitably restricted subspace.Source: §2 · p. 11
  • For the first approach, sufficient differentiability and boundedness hypotheses apply; in the Bloch–Torrey equation, B′ is identically zero because B is time-independent.For the second approach, bounded F′ follows from bounded A′, B′, and C′; the Bloch–Torrey bound is made explicit in Proposition 5.Source: §2 · p. 11

3. A class of problems containing multiplication and diffusion-advection operators

The section places time-dependent multiplication operators alongside component-wise diffusion-advection and develops evolution and commutator estimates for this setting. Those estimates support a global Lie–Trotter error bound under stated coefficient and temporal-regularity conditions.

3. A class of problems containing multiplication and diffusion-advection operators

  • The model class combines time-dependent multiplication operators for physical effects with component-wise diffusion-advection, under uniformly positive diffusion and possibly time-dependent velocity.For the Bloch–Torrey equation, the multiplication terms model controlled rotation and relaxation.Source: §3 · p. 12

3.1.1. Estimate of 𝐸𝐴and 𝐸𝐵

  • The evolution operator for the multiplication subproblem is well defined under spatial regularity assumptions on A.The estimate applies for any m in the stated class, with its constant depending on the domain, time horizon, dimension, and norm of A.Source: §3.1.1 · p. 12

3.1.2. Estimate of 𝐸𝐶and of 𝐸𝐹

  • Energy estimates establish evolution-operator bounds for the diffusion-advection problem, including the combined problem with multiplication terms.The stated setting assumes bounded coefficients and velocity; the stronger spatial estimate also requires elliptic regularity.Source: §3.1.2 · p. 13

3.2.1. Estimate of the commutator [𝐶(𝑡), 𝐴(𝑡)]

  • The commutator analysis separates diffusion and advection contributions to bound the interaction between C(t) and A(t).The resulting commutator bound depends on the domain, time horizon, dimension, and the specified norms of A, D, and v.Source: §3.2.1 · p. 13
  • Under the coefficient and temporal-regularity conditions, the commutator estimates yield a global error estimate with a constant independent of the time-step index and increments.The corollary applies the general theorem to the specified multiplication and diffusion-advection operators.Source: §3.2.1 · p. 14

4. Application to the Bloch-Torrey equation

The Bloch–Torrey application specializes the operator decomposition and derives commutator and temporal-regularity conditions for its global error estimate. In particular, one magnetic-field component cancels from a commutator, while the commutators have different spatial-regularity costs across components.

4. Application to the Bloch-Torrey equation

  • The application uses Decomposition I, with B independent of time, and accounts for the skew-symmetry of A(t) and the vanishing last component of C(t).These features make the Bloch–Torrey commutators differ from those in the preceding general setting.Source: §4 · p. 14

4.0.1. Commutators for Bloch-Torrey

  • The b_z(t) term cancels in [B,A(t)], while [C(t),A(t)] loses one derivative and [C(t),B] loses two in v_e and one in v_perp and v_z.The [C(t),A(t)] commutator leaves v_e untouched.Source: §4.0.1 · p. 14

4.0.2. Temporal regularity for Bloch-Torrey

  • Under assumptions (27), (28), and temporal regularity (29), Corollary 2 gives a global error estimate with a constant independent of N and the time increments.The stationary diffusion part cancels from C(t)-C(s) and C′(t), and B is autonomous because R1 and R2 depend only on space.Source: §4.0.2 · pp. 15–16

5. Numerical implementation and tests

The numerical tests report first-order temporal convergence across three cases for both frozen and non-frozen splitting. The implementation uses conforming P1 finite elements for spatial discretization and manufactured solutions to assess the discretization.

5.1. Spatial discretization

  • The spatial discretization uses conforming first-order Lagrange finite elements, with error estimates stated under spatial regularity assumptions on the exact solution.The numerical tests use manufactured fields and a residual forcing term to study convergence as mesh size or time step is refined.Source: §5.1 · p. 16, §5.2 · p. 17

5.2. Numerical experiments

  • The temporal study fixes a 33,449-vertex mesh and refines the time step from 0.25 s to approximately 5×10^-4 s.The tested transport configuration includes heterogeneous diffusion and steady Poiseuille flow; the simpler uniform-diffusion or zero-flow configurations were also reported to have the same convergence order.Source: §5.2 · p. 18
  • All three cases exhibit O(Δt) convergence for both frozen and non-frozen splitting, consistent with first-order Lie splitting.The studies report endpoint L2(Ω) error for Cases B, C, and D on a fine mesh.Source: §5.2 · p. 18, Table 2 · p. 19, Table 1 · p. 19

Proof of Proposition 1

The proof regularizes the multiplication-driven problem with vanishing viscosity, obtains bounds uniform in the viscosity parameter, and passes to a limit to recover the desired solution regularity.

Proof of Proposition 1

  • A vanishing-viscosity approximation is used to obtain spatial regularity for u_A(t).For each positive viscosity parameter, existence follows under the stated coefficient regularity and homogeneous boundary conditions.Source: Proof of Proposition 1 · p. 21
  • Testing the regularized equation with a Laplacian power gives an energy bound independent of the viscosity parameter.The estimate uses a sufficiently smooth domain and a Leibniz-rule estimate.Source: Proof of Proposition 1 · p. 21
  • A sequence of vanishing viscosity parameters yields a limit in L∞(0,T;H^m), which is then shown to satisfy the ODE in the stated sense.The proof uses the uniform bound to extract the sequence and checks the limiting equation against test functions.Source: Proof of Proposition 1 · pp. 21–22

Proof of Proposition 2

The proof obtains the stated H2 estimate by testing (19) with the divergence of D∇∂_t u_F, then using elliptic regularity and an earlier L2 estimate.

Proof of Proposition 2

  • The estimates use Cauchy–Schwarz in L2(0,T) and ℝ^d, Young’s inequality, and the spectral matrix norm.Source: Proof of Proposition 2 · p. 23
  • Testing (19) with ∇⋅(D∇∂_t u_F) yields the H2(Ω) estimate.The resulting expression is estimated analogously to the preceding argument; elliptic regularity (20) and the earlier L2(Ω) estimate complete the assertion.Source: Proof of Proposition 2 · p. 23
  • The proof relies on elliptic regularity (20) together with an earlier L2(Ω) estimate to conclude the assertion.Source: Proof of Proposition 2 · p. 23

Proof of Proposition 3

Under assumption (21), the proof estimates the displayed expression using Sobolev embedding, with constants depending only on the dimension and domain.

Proof of Proposition 3

Proof of Proposition 5

The proof establishes time-regularity estimates for the operators in Proposition 5, distinguishing time-dependent terms from autonomous diffusion and reaction components.

Proof of Proposition 5

  • The time-dependent operator L(t) consists of diffusion C_diff and advection C_adv(t), while D is time-independent.C_diff φ=∇⋅(D∇φ) and C_adv(t)φ=−v(t)⋅∇φ.Source: Proof of Proposition 5 · p. 24
  • A′(t) exists almost everywhere in the L2(Ω;ℝ4) operator norm and A(t) is Lipschitz in time, with a bound proportional to ‖∂_t b‖L∞(0,T;L∞(Ω;ℝ3)).The Lipschitz bound carries the factor C_Aγ.Source: Proof of Proposition 5 · p. 24
  • B is autonomous because R1 and R2 are time-independent, so B(t)−B(s)=0 and B′(t)=0 without additional temporal regularity on them.Source: Proof of Proposition 5 · p. 24
  • C′(t) exists almost everywhere and C(t) varies at most linearly in time, bounded by ‖∂_t v‖L∞(0,T;L∞(Ω;ℝ3)).The stationary diffusion part cancels in these estimates; boundedness of C(t) itself from H1(Ω;ℝ3)×L2(Ω) to L2(Ω;ℝ4) is not asserted.Source: Proof of Proposition 5 · p. 25
  • For Approach 2, F′(t) is bounded by the time derivatives of b and v in L∞(Ω;ℝ3), with the b term scaled by C_Aγ.Here F(t)=A(t)+B+C(t) and B′=0.Source: Proof of Proposition 5 · p. 25
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