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Adjoint-based gradient methods for inverse design in a multiple fragmentation model

Arijit Das

Abstract

We study an inverse design problem for the linear multiple fragmentation equation arising in particle dynamics. Our objective is to reconstruct an unknown initial size distribution that evolves, under a prescribed fragmentation law, into a desired size distribution at a specified final time. We first establish the existence of global mass-conserving solutions for a broad class of fragmentation kernels with unbounded rates, and subsequently prove the continuous dependence and uniqueness of these solutions under additional assumptions on the fragmentation kernels. We then formulate the inverse design problem as an optimal control problem constrained by the fragmentation dynamics and prove the existence of the optimal control problem. Also derive the corresponding continuous adjoint equation and propose a gradient-type iterative reconstruction method. For the numerical implementation, we develop finite volume schemes for both the forward and adjoint equations, including a weighted finite volume scheme designed to enhance mass conservation and accuracy. Two benchmark problems, involving linear and nonlinear fragmentation rates with known analytical solutions, are used to assess the accuracy and efficiency of the proposed approach and to compare the performance of the two discretizations in both forward simulations and inverse reconstructions.

Adjoint-based gradient methods for inverse design in a multiple fragmentation model

Abstract

We study an inverse design problem for the linear multiple fragmentation equation arising in particle dynamics. Our objective is to reconstruct an unknown initial size distribution that evolves, under a prescribed fragmentation law, into a desired size distribution at a specified final time. We first establish the existence of global mass-conserving solutions for a broad class of fragmentation kernels with unbounded rates, and subsequently prove the continuous dependence and uniqueness of these solutions under additional assumptions on the fragmentation kernels. We then formulate the inverse design problem as an optimal control problem constrained by the fragmentation dynamics and prove the existence of the optimal control problem. Also derive the corresponding continuous adjoint equation and propose a gradient-type iterative reconstruction method. For the numerical implementation, we develop finite volume schemes for both the forward and adjoint equations, including a weighted finite volume scheme designed to enhance mass conservation and accuracy. Two benchmark problems, involving linear and nonlinear fragmentation rates with known analytical solutions, are used to assess the accuracy and efficiency of the proposed approach and to compare the performance of the two discretizations in both forward simulations and inverse reconstructions.
Paper Structure (20 sections, 9 theorems, 110 equations, 9 figures, 3 tables)

This paper contains 20 sections, 9 theorems, 110 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions and organization of the work
  3. Existence of the forward problem
  4. Prior estimation of the moments
  5. Uniform boundedness:
  6. Time equicontinuity
  7. Space equicontinuity
  8. Existence of global solutions:
  9. Continuous Dependence on Initial Data and Uniqueness
  10. Derivation of adjoint equation and gradient descent method
  11. Derivation of the adjoint equation
  12. Gradient descent method
  13. Numerical approach to the inverse design problem
  14. Finite volume scheme for forward fragmentation problem (FVS)
  15. Weighted finite volume scheme for forward equation (WFVS)
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $b$ and $S$ be nonnegative functions, with $b$ continuous on $(0,\infty)^2$ and $S$ continuous on $(0,\infty)$. Assume further that $S$ has compact support and that $b$ satisfies condition 0_4. If the initial data $f_{0} \in X_{0,r}^{+}$ for some $r\ge 1$, then the initial value problem 0_1--0_2

Figures (9)

  • Figure 1: Flowchart of adjoint based method
  • Figure 2: Comparison between exact and approximated target function by FVS & WFVS in linear scale.
  • Figure 3: Comparison between exact and approximated initial datum by FVS & WFVS in logarithmic scale.
  • Figure 4: Change of target error and initial datum error by FVS.
  • Figure 5: Change of target error and initial datum error by WFVS.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Definition 2.1
  • Lemma 2.1: Existence theorem for compactly supported kernel
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1: Existence theorem
  • proof
  • ...and 8 more