#  SoftMAC: Differentiable Soft Body Simulation with Forecast-based Contact
Model and Two-way Coupling with Articulated Rigid Bodies and Clothes

Min Liu1, Gang Yang2, Siyuan Luo3, Chen Yu2, and Lin Shao2 1Min Liu is with
the Machine Learning Department, Carnegie Mellon University, USA.
<EMAIL_ADDRESS>Yang, Chen Yu and Lin Shao are with the Department of
Computer Science, National University of Singapore, Singapore.
<EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>Luo is
with miHoYo, China<EMAIL_ADDRESS>
###### Abstract

Differentiable physics simulation provides an avenue for tackling previously
intractable challenges through gradient-based optimization, thereby greatly
improving the efficiency of solving robotics-related problems. To apply
differentiable simulation in diverse robotic manipulation scenarios, a key
challenge is to integrate various materials in a unified framework. We present
SoftMAC, a differentiable simulation framework coupling soft bodies with
articulated rigid bodies and clothes. SoftMAC simulates soft bodies with the
continuum-mechanics-based Material Point Method (MPM). We provide a forecast-
based contact model for MPM, which greatly reduces artifacts like penetration
and unnatural rebound. To couple MPM particles with deformable and non-
volumetric clothes meshes, we also propose a penetration tracing algorithm
that reconstructs the signed distance field in local area. Based on simulators
for each modality and the contact model, we develop a differentiable coupling
mechanism to simulate the interactions between soft bodies and the other two
types of materials. Comprehensive experiments are conducted to validate the
effectiveness and accuracy of the proposed differentiable pipeline in
downstream robotic manipulation applications. Supplementary materials and
videos are available on our project website at
https://sites.google.com/view/softmac.

##  I INTRODUCTION

Interactions between diverse materials are prevalent in the realm of robotic
manipulation. For instance, the collision between glass and wine during a
pouring task, or the perpetual interplay between a tortilla and its fillings
when crafting a taco. As a number of differentiable physics engines emerge to
tackle learning and control problems [1, 2, 3, 4, 5, 6], it is tempting to
unify different materials in a single simulator, thus supporting a wide range
of manipulation tasks. Nevertheless, developing a general physics simulation
is non-trivial, as different types of materials often need different physical
models due to the dominant physical behaviors [7].

Figure 1:  To pour water into a glass in differentiable physics simulation, we
need realistic contact model for soft-rigid coupling and correct gradient
calculation. Then same applies to the interactions between soft bodies and
clothes.

Depending on their dynamical behaviors, most daily objects can be simulated
as: 1) soft bodies, like elastic, plastic, elasto-plastic objects and liquid;
2) rigid bodies, either articulated or not; 3) clothes, as well as other
objects with similar geometric and physical properties (e.g., tortilla). The
interactions between these three modalities encompasses a wide spectrum of
manipulation tasks. Rigid-clothes coupling has been extensively discussed in
previous works [8, 9, 10]. Meanwhile, differentiable soft-cloth coupling
remains a rarely explored topic in current literature, despite its prevalence
in robotic manipulation scenarios such as making taco. A line of works propose
methods for soft-rigid coupling. PlasticineLab [11] and FluidLab [4] deploys
Material Point Method (MPM) to simulate plastic objects, but they are limited
to one-way coupling, which suggests that force exerted on soft bodies cannot
react on rigid bodies. Maniskill2 [12] introduces two-way dynamics coupling,
but its pipeline is not differentiable. Besides, the contact models in these
works suffer from artifacts like penetration and unnatural rebound.

In this work, we present a differentiable pipeline to couple soft bodies with
articulated rigid bodies and clothes. We adopt MPM for soft bodies due to its
ability to simulate a large variety of deformable materials. Both rigid bodies
and clothes are represented as meshes in coupling. To achieve realistic
collision effects, we introduce a novel method called forecast-based contact
model to manage boundary conditions for MPM. Unlike previous methods, our
model takes a grid-to-particle transfer to look ahead in grid operation stage.
Specifically, the model computes particle velocities after transfer, imposes
constraints on particles within the contact region, and then adjust grid
velocity accordingly. Forecast-based contact model requires signed distance
fields (SDFs) of meshes to penalize particle penetration. While the definition
of SDF is straightforward for volumetric objects, it is hard to determine the
sign on non-volumetric meshes. To fix this problem, we propose a penetration
tracing algorithm that capitalizes on the localized motion of particles to
reconstruct SDF within confined zones. In this way, the contact model can be
applied to both soft-rigid and soft-cloth coupling.

In the complete simulation pipeline, we copy poses and velocities of rigid
bodies and clothes to the soft body simulator, and transfer force back to
rigid bodies and clothes. The whole process is differentiable, which allows us
to control the motion between different modalities by optimizing the force
acting on any point in the simulation. As a result, we implement a
differentiable simulator that enables manipulating soft bodies with
articulated rigid bodies and clothes. Code and documentation of the proposed
simulator, which we call SoftMAC, will be released at our project website.

Our main contributions can be summarized as follows:

  * •

We propose a novel contact model for MPM which reduces artifacts like
penetration and unnatural rebound.

  * •

We present a differentiable pipeline that couples soft bodies with articulated
rigid bodies and clothes, and show its efficacy under several manipulation
tasks.

  * •

To the best of our knowledge, SoftMAC is the first robotic simulator to
support two-way differentiable soft-rigid and soft-cloth coupling. Our work
lays a foundation for universal differentiable physics simulation.

##  II Related Work

###  II-A Differentiable Physics-based Simulation

Physics simulation provides an avenue to avoid real-world damages, accelerate
robotics-related problem solving, and establishes standardized benchmarks to
evaluate different algorithms. A branch of works provide simulation for
articulated rigid bodies [13, 14, 15, 16], which is fundamental in robotics.
With advances in deformable object manipulation, an increasing number of
physics engines are also proposed for clothes and soft bodies [17, 18].

Recent developments in automatic differentiation methods [19, 20, 21, 22]
boost the prosperity in differentiable physics simulators, which can convert
challenging control tasks into gradient-based optimization problems. Several
differentiable simulators are developed for rigid bodies [1, 2] and clothes
[6, 10, 23]. A line of works [3, 4, 11] simulate soft bodies with material
point method (MPM) to support various materials and dynamic processes. We also
deploy MPM for soft bodies in this work.

###  II-B Coupling Soft Bodies with Other Modalities

Unifying soft bodies with other modalities in a single framework requires the
design of contact mechanisms between different materials. A number of methods
have been proposed in robotics and computer graphics area. Du et al. [24] and
Harada et al. [25] provide simulation methods to couple cloth and fluid
computed by using smoothed particle hydrodynamics. MuJoCo [13] and Bullet [14]
enable manipulating soft bodies simulated with finite element method. However,
these methods only support a limited number of soft body materials. Recent
works based on MPM provide the opportunity to couple various types of soft
bodies with other modalities. PlasticineLab [11] and FluidLab [24] provide
interaction with rigid bodies by penalizing grid velocity directly, but their
contact models are one-way. Maniskill2 [12] transfers force back to rigid
bodies to enable bidirectional contact, and penalize particles in their
contact models to alleviate penetration. Nevertheless, their pipeline does not
provide gradient information. Our work is based on a novel contact model
better at preventing penetration. We also provide bidirectional contact for
both soft-rigid and soft-cloth coupling, and make the process differentiable.

###  II-C Robotic Manipulation with Differentiable Simulation

Differentiable physics simulation has been applied in several robotic
manipulation methods [26, 27]. DiffSkill [28] optimizes trajectories for
short-horizon skills with differentiable simulation and learns skill
abstractions. They then plan over the skills to solve long-horizon
manipulation tasks. DexDeform [29] trains a skill model from human
demonstrations, and uses a gradient optimizer to refine the trajectories
planned by the skill model to generate more demonstrations. SAGCI [30]
proposes a model-based learning method with differentiable simulation to
online verify and modify the environment model during interaction. SAM-RL [31]
combines differentiable physics simulation and rendering to propose a sensing-
aware learning pipeline that selects an informative viewpoint to monitor the
manipulation process.

##  III Contact Models

(a) Contact Models

(b) Penetration Tracing

Figure 2:  (a) Pour water into a thin glass. Grid-based model (left) leads to
severe penetration. Particle-based model (middle) also causes a few particles
to penetrate the glass. Forecast-based model (right) achieves the most robust
performance. (b) Drag four corners of a towel to squash a plasticine. Towel
goes through the plasticine without penetration tracing (left). Both the
plasticine and towel deform due to the contact after adding the algorithm
(right).

###  III-A Preliminary

We build our soft body simulator based on MLS-MPM [32]. For simplicity, we use
MPM for MLS-MPM in this section. In a simulation loop, the matter is
interpolated back and forth between particle and grid representations.
Consider a system that consists of $n_{p}$ particles and $n_{g}$ grid nodes.
Denote $x_{p},v_{p},m_{p}\in\mathbb{R}^{3n_{p}}$ as positions, velocities and
masses of particles, and $v_{g},p_{g},m_{g}\in\mathbb{R}^{3n_{g}}$ as
velocities, momentum and masses of grid nodes. We duplicate the masses to
higher dimension for the convenience of vector operations.
$W:\mathbb{R}^{3n_{p}}\rightarrow\mathbb{R}^{3n_{g}\times 3n_{p}}$ calculates
a sparse matrix that contains weights for the $3^{3}$ neighbouring grid nodes
surrounding each particle. Superscript denotes the time step of a variable.

#### Particle-to-Grid (P2G)

MPM first computes stress based on the constitutive model, and then transfer
particles’ momentum and masses to background grid using the interpolation
matrix $W(x_{p}^{n})$.

$\begin{split}p_{g}^{n+1}&=W(x^{n}_{p})(m_{p}^{T}v_{p}^{n}+p_{d}^{n})\\\
m_{g}^{n+1}&=W(x^{n}_{p})m_{p}^{n}\end{split}$ (1)

where $p_{d}^{n}$ is a momentum term that reflects internal forces.

#### Grid Operation

The discrete equations of momentum are solved on grid nodes. Boundary
conditions $BC(\cdot)$ can be enforced at this stage to set constraints on
grid velocity.

$\begin{split}\hat{v}_{g}^{n+1}&=(p_{g}^{n+1})^{T}(1/m_{g}^{n+1})\\\
v_{g}^{n+1}&=BC(\hat{v}_{g}^{n+1})\end{split}$ (2)

#### Grid-to-Particle (G2P)

The final stage transfers velocity back to particles, updates deformation
information, and advects particles based on the new velocities.

$\begin{split}v_{p}^{n+1}&=W(x_{p}^{n})^{T}v_{g}^{n+1}\\\
x_{p}^{n+1}&=x_{p}^{n}+v_{p}^{n+1}\Delta t\end{split}$ (3)

Grid-based and particle-based contact models are commonly used in previous
works [4, 11, 12] to couple MPM-simulated soft bodies with other modalities.
Grid-based model directly zeroes out the tangent component of grid velocity
through $BC(\cdot)$ in eq. 2. The reaction force can be computed via momentum
change $F=\Delta p/\Delta t$. The method leads to severe penetration when
coupling with delicate mesh-based objects. Scaling up grid resolution can
alleviate the problem to some extent at the cost of computation efficiency.
Particle-based model provides more fine-grained control by exerting penalty
force on particles. The contact surface is approximated as a spring: as a
particle crosses the contact boundary with distance $d>0$, a force $F=-kd$ is
exerted to push it back. The force also works as an indirect constraint for
grid velocity by adding a momentum term $F\Delta t$ to eq. 1. With a small
$k$, the contact model cannot effectively prevent penetration. However,
increasing $k$ will lead particles to rebound unnaturally near the boundary.
Alleviating one of the artifacts makes the other worse.

###  III-B Forecast-based Contact Model

We formulate the goal as reducing the number of penetrations at the end of
each simulation loop. Since particle positions $x_{p}^{n+1}$ are advected
based on velocities, we focus on how to set constraints on grid velocities to
obtain $v_{p}^{n+1}$ that can achieve the goal. The idea of looking ahead and
then adjusting grid velocity gives our method the name forecast-based contact
model. It also provides an intuition why the method has better performance in
preventing penetration.

We use $W$ as short for $W(x_{p}^{n})$ from this section. Our method first
takes a G2P transfer $v_{init}^{n+1}=W\hat{v}_{g}^{n+1}$. Then we apply a
boundary condition $BC_{p}(\cdot)$ on the particle velocities, which computes
constrained velocities $v_{tgt}^{n+1}$ that are supposed to avoid penetration.
Given $v_{tgt}^{n+1}$, we transform our goal into an optimization problem:

$v_{g}^{n+1}=\operatorname*{arg\,min}_{v_{g}}\|W^{T}v_{g}-\hat{v}^{n+1}_{tgt}\|^{2}$
(4)

The problem above can be tackled using iterative solvers like the conjugate
gradient method. However, the computation overhead will grow linearly with the
number of iterations. Instead, we resort to one-step gradient descent, which
is effective enough in practice.

$v_{g}^{n+1}=\hat{v}_{g}^{n+1}-\alpha W(W^{T}v_{g}-v_{tgt}^{n+1})$ (5)

where $\alpha$ is a predefined step size. The extra computation cost is
equivalent to interpolating a particle property to grid and transferring it
back. Besides, the derivatives of eq. 5 is simple enough to be directly
obtained through automatic differentiation methods.

Next, we introduce the details of $BC_{p}(\cdot)$. Given a particle with
velocity $v_{in}$, we first check whether it is within the contact region by
comparing signed distance $d$ with a threshold $\hat{d}$. If $d<\hat{d}$, the
contact point is approximated with the nearest point on the boundary, whose
velocity is $v_{c}$. For rigid bodies, $v_{c}$ is the linear velocity at the
contact point. For clothes, we find the three vertices that make up the
contact surface, and compute weighted average of their velocities based on the
barycentric coordinate of the contact point. We then decompose the relative
velocity $v_{rel}=v_{in}-v_{c}$ into a normal component $v_{n}$ and a
tangential component $v_{t}$. To penalize the velocity, we drop $v_{n}$ and
decay $v_{t}$ with friction:

$v_{out}=v_{t}\cdot\max(0,1-\mu{\|v_{n}\|}/{\|v_{t}\|})+v_{c}$ (6)

where $\mu$ is the friction coefficient. We also utilize two other techniques:
1) Blend the original and modified velocities with a smoothness factor
$s=\min\\{\exp(-\beta d),1\\}$, given by
$v^{\prime}_{out}=sv_{out}+(1-s)v_{in}$. $\beta$ is a predefined smoothness
factor. The method reduces drastic state changes and improves gradient
quality. 2) Advect the particle with $v_{out}$, and if penetration happens,
add a component to $v_{out}$ to ensure that the particles are moved to the
nearest legal position. Given velocity change, we can obtain impulse exerted
on the particle, thereby computing the reaction force $F$. The spatial force
is accumulated on each rigid body link. For clothes, we distribute $F$ to
three nodes of the nearest triangle based on the barycentric coordinate of the
contact point.

###  III-C Penetration Tracing for Soft-Cloth Coupling

Figure 3: Given a target face (yellow edge), we search for the triangles in
its neighboring area (red). If a particle is still in contact after system
state changes, its nearest face should be within the neighboring area due to
the small time interval.

The contact models require signed distance to couple MPM soft bodies with
objects represented by meshes. For rigid bodies, we pre-compute the signed
distance field on compact grid points within the bounding boxes, and look up
signed distance by interpolating the grid values during simulation. However,
the method is not applicable to soft-cloth coupling because meshes of clothes
varies by time. Besides, such meshes are non-volumetric, which means that both
sides can be considered as the outward surface. While it is feasible to
manually define a positive side for some simple shapes like a square, the idea
cannot generalize to special geometries such as a Möbius strip.

To fix the problem, we reconstruct signed distance fields for cloth meshes by
computing distance and sign separately. We first search the particle-face
pairs to find the nearest face for each particle, and compute absolute
distances accordingly. The process can be accelerated with spatial hashing.
The sign indicates whether a particle penetrated the mesh. Therefore, we
assign a binary penetration state $z^{n}$ for each particle and trace it
throughout the simulation process. Each time the position of particles or
meshes change, we check whether the state change will lead to penetration. If
a particle moves to the other side of the mesh, we update the penetration
state at the next time step, given by $z^{n+1}=1-z^{n}$. Otherwise, $z^{n+1}$
inherits the value of $z^{n}$.

The complication lies in how to check whether penetration happens. We provide
a simple method by utilizing the locality of motion in simulation. The small
time interval in physics simulation suggests that system state only changes a
bit at each time. If a particle is in contact with face $i$ at time $n$ and
face $j$ at time $n+1$, then we assume that faces $i$ and $j$ are within the
neighboring area of each other. Based on this idea, we pre-compute the
neighboring triangles around each face using breadth first search, and define
a consistent orientation for faces in this local area. Then we can directly
check whether a particle resides on the same side of the area during
simulation. An illustration can be found in fig. 3.

A few counter examples can be found for the algorithm. For instance, if two
pieces of fabric are stacked, a particle might penetrate a face not
neighboring to the previous contact face. The problem can be alleviated by the
barrier region left by Incremental Potential Contact [33], which is adopted by
the underlying cloth simulation in our system. Non-manifold geometry is
another counter example, but it is not supported in our cloth simulation. In
general, penetration tracing is effective in most manipulation scenarios.

Figure 4: Computation graph for soft-rigid coupled system (the same in soft-
cloth coupling). Blue nodes: state of MPM simulation. Orange nodes: state of
Articulated Rigid Body simulation. Green nodes: actions and external forces. A
line is connected between two nodes if and only if we directly use one state
or action to compute the resulting state. The gray box illustrates the details
of forecast-based contact model.

##  IV Differentiable Simulation Design

###  IV-A Forward Simulation

Denote $s_{M}^{n}$ as state of MPM object at time $n$. The forward simulation
for MPM is defined as $\mathcal{F}_{M}(s_{M}^{n})$. $s_{D}^{n}$ and
$a_{D}^{n}$ represent the state and action of the manipulator (articulated
rigid bodies or clothes). The forward simulation for manipulator is
$\mathcal{F}_{D}(s_{D}^{n},a_{D}^{n})$. While both simulations can be
conducted independently, interactions between the two modalities cannot be
directly simulated. Therefore, we introduce a mechanism to couple these
simulators.

Specifically, we modify the simulations as follows. The MPM simulator takes
poses and velocities of the manipulators as inputs. Based on the contact
models, it updates MPM state and computes the reaction force on the
manipulator. Manipulator simulation takes external force as an additional
input, which is involved in the computation of the next state:

$\begin{split}s_{M}^{n+1},F^{n+1}&=\mathcal{F}_{M}^{\prime}(s_{M}^{n},s_{D}^{n})\\\
s_{D}^{n+1}&=\mathcal{F}_{D}^{\prime}(s_{D}^{n},a_{D}^{n},F^{n+1})\end{split}$
(7)

In a simulation loop, we execute $\mathcal{F}_{M}^{\prime}$ first, with the
manipulators as boundary conditions. Then we transfer the force back and
execute $\mathcal{F}_{D}^{\prime}$. The computation graph is displayed in fig.
4. We define state of the coupled system as $s^{n}=(s^{n}_{M},s^{n}_{D})$, and
action as $a^{n}=a^{n}_{D}$. The forward simulation functions as:

$s^{n+1}=(s^{n+1}_{M},s^{n+1}_{D})=\mathcal{F}(s^{n},a^{n})$ (8)

###  IV-B Backward Gradients

The differentiable operations inside MPM simulation $\mathcal{F}^{\prime}_{M}$
help us gather the Jacobian matrices $\frac{\partial s_{M}^{n+1}}{\partial
s_{M}^{n}}$, $\frac{\partial s_{M}^{n+1}}{\partial s_{D}^{n}}$,
$\frac{\partial F^{n+1}}{\partial s_{M}^{n}}$, $\frac{\partial
F^{n+1}}{\partial s_{D}^{n}}$. The manipulator simulation
$\mathcal{F}^{\prime}_{D}$ computes $\frac{\partial s_{D}^{n+1}}{\partial
s_{D}^{n}}$, $\frac{\partial s_{D}^{n+1}}{\partial a^{n}}$, $\frac{\partial
s_{D}^{n+1}}{\partial F^{n+1}}$. Based on these Jacobian matrices, we apply
the chain rule to obtain gradients for every system state at each time step.

$s_{M}^{n}$ is directly involved in the computation of $s_{M}^{n+1}$, and
affects $s_{D}^{n+1}$ through the contact force $F^{n+1}$. Consequently, the
gradient of $s_{M}^{n}$ has two components:

$\frac{\partial\mathcal{L}}{\partial
s_{M}^{n}}=\frac{\partial\mathcal{L}}{\partial s_{M}^{n+1}}\frac{\partial
s_{M}^{n+1}}{\partial s_{M}^{n}}+\frac{\partial\mathcal{L}}{\partial
s_{D}^{n+1}}\frac{\partial s_{D}^{n+1}}{\partial F^{n+1}}\frac{\partial
F^{n+1}}{\partial s_{M}^{n}}$ (9)

$s_{D}^{n}$ influences $s_{D}^{n+1}$ in two ways. In addition to the one
inherent in $\mathcal{F}_{D}(s_{D}^{n},a_{D}^{n})$, $s_{D}^{n}$ affects the
contact force $F^{n}$ between MPM and the manipulator, and the force in turn
impacts on $s_{D}^{n+1}$. Another part of the gradient comes directly from
$s_{M}^{n+1}$, which takes $s_{D}^{n}$ as the boundary:

$\frac{\partial\mathcal{L}}{\partial
s_{D}^{n}}=\frac{\partial\mathcal{L}}{\partial s_{D}^{n+1}}(\frac{\partial
s_{D}^{n+1}}{\partial s_{D}^{n}}+\frac{\partial s_{D}^{n+1}}{\partial
F^{n+1}}\frac{\partial F^{n+1}}{\partial
s_{D}^{n}})+\frac{\partial\mathcal{L}}{\partial s_{M}^{n+1}}\frac{\partial
s_{M}^{n+1}}{\partial s_{D}^{n}}$ (10)

Finally, $a_{D}^{n}$ can be directly computed from
$\frac{\partial\mathcal{L}}{\partial s_{D}^{n+1}}$ inside the manipulator
simulation:

$\frac{\partial\mathcal{L}}{\partial
a_{D}^{n}}=\frac{\partial\mathcal{L}}{\partial s_{D}^{n+1}}\frac{\partial
s_{D}^{n+1}}{\partial a_{D}^{n}}$ (11)

It is worth noting that $\frac{\partial s_{M}^{n+2}}{\partial
a_{D}^{n}}=\frac{\partial s_{M}^{n+2}}{\partial s_{D}^{n+1}}\frac{\partial
s_{D}^{n+1}}{\partial a_{D}^{n}}$. The gradients propagated from $s_{M}^{n+2}$
to $a_{D}^{n}$ explain why we can define losses on soft bodies to optimize the
actions on the manipulators.

##  V Experiments

In this section, we introduce our system implementation, and conduct
quantitative and qualitative evaluations to answer the following questions: 1)
Does forecast-based contact model produce more realistic simulation results?
2) Can penetration tracing effectively support soft-cloth coupling? 3) How
accurate is the gradient information provided by the proposed two-way
differentiable simulation?

###  V-A System Implementation

We implement a differentiable soft body simulator using DiffTaichi [22].
Corotated constitutive models are used to simulate fluid, elastic and plastic
materials. We refer readers to FluidLab [24] for more details about simulating
different materials using MPM. We adopt two differentiable physics engines,
Jade [34] and DiffClothAI [10], for the simulation of articulated rigid bodies
and clothes, respectively. In each simulation loop of the coupled system, we
take actions, manipulator states, and MPM states as input, calculate contact
in the soft body simulator, and transfer external force to the rigid body /
clothes simulator. MPM typically requires a smaller time interval than the
other two simulators. In this case, we take several MPM substeps and average
the force on the manipulator. Gradients are calculated accordingly.

Figure 5: Shake a container with liquid. Grid-based contact model causes severe penetration. Particle-based model also faces the problem if the penalty coefficient $k$ is not high enough. Increasing $k$ alleviates penetration, but makes particles rebound more unnaturally near boundary. Forecast-based model avoids penetration without bringing obvious artifacts. Contact Model |  #Edge | #Penetration | Time (s)
---|---|---|---
Grid | 380.80 | 449 | 4.90
Particle-400 | 40.83 | 12 | 4.45
Particle-600 | 24.85 | 0 | 4.33
Forecast (Ours) | 27.69 | 0 | 5.24
TABLE I: Experiment results of shaking a container with liquid for 1100
frames. #Edge is the average number of particles inside the gray area over all
the frames. #Penetration is the number of particles that outside the container
in the final frame. Time is compared on an NVIDIA Tesla T4 GPU.

###  V-B Checking Forecast-based Contact Model

(a) Pour wine

(b) Squeeze plasticine

(c) Make taco

(d) Pull door

(e) Push towel

Figure 6: Experiment results of coupled differentiable simulation. (a) Control
glass to pour the liquid into a bowl. (b) Control rigid gripper to squeeze the
plasticine into target shape. (c) Control 2 selected points on the tortilla to
fold the taco into target shape. (d) Control soft gripper to pull the door
into target angle. (e) Control soft gripper to push the towel into target
pose. In all tasks, their respective training loss curves are plotted on the
right. Full trajectories are displayed in our video.

We first conduct a quantitative experiment to compare forecast-based contact
model with the baselines (grid-based and particle-based models). Specifically,
we initialize the 2D scene with fluid inside a circular rigid container, which
has a inner surface of radius $r_{1}$ and an outer surface of radius $r_{2}$.
The container shakes left and right at a constant speed. We denote distance to
the center of circle as $d$. Three metrics are introduced: 1) the number of
particles inside the container wall ($r_{1}<d\leq r_{2}$); 2) the number of
particles that penetrate the outer surface of the container ($d>r_{2}$); 3)
computation time. Since the contact models are not penetration-free, we
provide metric 1 as reference and focus on more metric 2.

We choose step size $\alpha=0.2$ for the onse-step gradient descent in
forecast-based contact model, and the objective in eq. 4 decreases by 83.1% on
average. As shown in table I, forecast-based model achieves the best
performance in reducing penetration. The computation efficiency lags behind
baselines as we introduce additional P2G and G2P transfers. Another empirical
observation is that while increasing the penalty coefficient $k$ in particle-
based contact model can alleviate penetration, it also exacerbates unnatural
rebound near the boundary. Meanwhile, our method does not have this problem.
Visualization for the last frame is provided in fig. 5. We also conduct a
qualitative experiment: pour liquid into a thin glass with high velocity.
Results shown in fig. 2a are consistent with our quantitative experiment.

###  V-C Checking Penetration Tracing

We drag the four corners of a towel to squash a plasticine. The control points
are moved down from their initial positions at a constant speed. We add and
remove the penetration tracing functions to demonstrate the effectiveness of
our method in soft-cloth coupling. The results are displayed in fig. 2b. We
find that lacking the penetration information causes the towel to go though
the plasticine directly. However, with penetration tracing, the plasticine is
squashed and the towel deforms due to the reaction force. We conduct the
comparison on particle-based contact model. With forecast-based model, the
towel cannot pass the plasticine even after removing the tracing algorithm in
this case.

###  V-D Trajectory Optimization

We check the quality of the gradient calculation under 5 robotic manipulation
tasks. In each task, we use Adam [35] to optimize a sequence of actions with
gradients provided by the differentiable simulator to accomplish the goal.

#### Rigid2MPM

We evaluate soft-rigid coupling on two tasks. In pour wine, we conduct 6 DoFs
control over a glass to pour liquid into a bowl. The state space combines
15000 DoFs liquid and a 6 DoFs glass. In squeeze plasticine, we conduct 2 DoFs
control over a gripper to squeeze the 16200 DoFs plasticine into target shape.
Losses are computed as the Chamfer distance between current positions and
target positions of the particles. In the pouring task, two additional loss
terms are calculated on poses and velocities of the glass to penalize it from
colliding with the bowl.

#### Cloth2MPM

We verify soft-cloth coupling with make taco. The tortilla and taco fillings
are simulated as clothes and MPM particles, respectively. We conduct 4 DoFs
control over two selected points of the tortilla and fold the taco into target
shape. The state space is combination of the 30000 DoFs taco fillings and the
651 DoFs tortilla. Loss is computed on taco fillings as the Chamfer distance
between current and target configurations. To avoid overly stretching the
tortilla, we control the particles in the perpendicular plane through circle
center and project the control points back to a confined region after each
epoch.

#### MPM2Rigid/Cloth

We check two-way differentiable coupling by controlling soft grippers to
manipulate articulated rigid bodies / clothes. We control MPM by exerting an
impulse on selected particles in P2G transfer. In pull door, we control the
soft gripper to pull the 1 DoF articulated door into target angle. In push
towel, we control the soft gripper to push the 432 DoFs towel into target
pose. We computed losses on the state of rigid bodies / clothes and optimize
the control over soft grippers.

The pour wine and make taco tasks are optimized for 40 and 25 epochs
respectively, costing 21.67s and 20.45s for each epoch on average on an NVIDIA
Tesla T4 GPU. Experiment results reported in fig. 6 demonstrate the
correctness and effectiveness of our differentiable simulation.

##  VI Conclusion

In this work, we introduce SoftMAC, a differentiable simulation that couples
soft bodies with articulated rigid bodies and clothes. We simulate soft bodies
with material point method, and provide a novel contact model that reduces
artifacts like penetration and unnatural rebound. To couple soft body
particles with deformable and non-volumetric clothes meshes, we propose a
method to reconstruct signed distance field in local area. We also provide a
differentiable coupling mechanism to unify the simulation of different
materials. Comprehensive experiments are conducted under robotic manipulation
scenarios and demonstrate the correctness and effectiveness of the
differentiable simulation.

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$\displaystyle\frac{C}{\log(N)}\exp\left(-c\log^{2}(N)\right).$

We deduce that

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n}\right]=\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(N),\log(N)]}\right\|^{2}_{n}\right]+\frac{C}{\log(N)}\exp\left(-c\log^{2}(N)\right).$
(58)

It remains to upper-bound the first term on the right hand side of Equation
(58).

##### Upper bound of
$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n}\mathds{1}_{[-\log(N),\log(N)]}\right]$.

For all $h\in\mathcal{S}_{m,L}$, we obtain from Equation (9),

$\gamma_{n,N}(\widehat{\sigma}^{2}_{m,L})-\gamma_{n,N}(\sigma^{2})\leq\gamma_{n,N}(h)-\gamma_{n,N}(\sigma^{2}).$
(59)

For all $h\in\mathcal{S}_{m,L}$,

$\gamma_{n,N}(h)-\gamma_{n,N}(\sigma^{2})=\left\|h-\sigma^{2}\right\|^{2}_{n,N}+2\nu_{1}(\sigma^{2}-h)+2\nu_{2}(\sigma^{2}-h)+2\nu_{3}(\sigma^{2}-h)+2\mu(\sigma^{2}-h)$

where

$\nu_{i}(h)=\frac{1}{nN}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{h(X^{j}_{k\Delta})\zeta^{j,i}_{k\Delta}}},\
\ i\in\\{1,2,3\\},\ \ \
\mu(h)=\frac{1}{nN}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{h(X^{j}_{k\Delta})R^{j}_{k\Delta}}},$
(60)

we deduce from Equation (59) that for all $h\in\mathcal{S}_{m,L}$,

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}\mathds{1}_{[-\log(N),\log(N)]}\right]\leq\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+2\sum_{i=1}^{3}{\mathbb{E}\left[\nu_{i}(\widehat{\sigma}^{2}_{m,L}-h)\right]}+2\mathbb{E}\left[\mu(\widehat{\sigma}^{2}_{m,L}-h)\right].$
(61)

For all $i\in\\{1,2,3\\}$ and for all $h\in\mathcal{S}_{m,L}$, one has

$\mathbb{E}\left[\nu_{i}\left(\widehat{\sigma}^{2}_{m,L}-h\right)\right]\leq\sqrt{2mL}\sqrt{\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{i}(\phi_{\ell})\right]}}.$
(62)

  1. 1.

Upper bound of
$\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{1}(\phi_{\ell})\right]}$

According to Equation (60), we have

$\forall\ell\in[\\![0,m-1]\\!],\
\nu_{1}(\phi_{\ell})=\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\phi_{\ell}(X^{j}_{k\Delta})\zeta^{j,1}_{k\Delta}}}$

where
$\zeta^{j,1}_{k\Delta}=\frac{1}{\Delta}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{j}_{s})dW^{j}_{s}}\right)^{2}-\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X^{j}_{s})ds}\right]$
is a martingale satisfying

$\mathbb{E}\left[\zeta^{1,1}_{k\Delta}|\mathcal{F}_{k\Delta}\right]=0\ \
\mathrm{and}\ \
\mathbb{E}\left[\left(\zeta^{1,1}_{k\Delta}\right)^{2}|\mathcal{F}_{k\Delta}\right]\leq\frac{1}{\Delta^{2}}\mathbb{E}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X^{1}_{s})ds}\right)^{2}\right]\leq
C\sigma^{4}_{1}$

with $C>0$ a constant, $W=W^{1}$ and $(\mathcal{F}_{t})_{t\geq 0}$ the natural
filtration of the martingale $(M_{t})_{t\in[0,1]}$ given for all $t\in[0,1]$
by $M_{t}=\int_{0}^{t}{\sigma(X^{1}_{s})dW_{s}}$. We derive that

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{1}(\phi_{\ell})\right]}=$
$\displaystyle\frac{1}{Nn^{2}}\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\phi_{\ell}(X^{j}_{k\Delta})\zeta^{1,1}_{k\Delta}}\right)^{2}\right]}=\frac{1}{Nn^{2}}\mathbb{E}\left[\sum_{k=0}^{n-1}{\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}(X^{1}_{k\Delta})\left(\zeta^{1,1}_{k\Delta}\right)^{2}}}\right].$

For each $k\in[\\![0,n-1]\\!]$, we have

$\begin{cases}\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}(X^{1}_{k\Delta})}=\sum_{\ell=-M}^{K-1}{B^{2}_{\ell}(X^{1}_{k\Delta})}=1\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\
spline\leavevmode\nobreak\ basis}\\\ \\\
\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}(X^{1}_{k\Delta})}\leq
Cm\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{For\leavevmode\nobreak\
an\leavevmode\nobreak\ orthonormal\leavevmode\nobreak\
basis\leavevmode\nobreak\ with}\leavevmode\nobreak\ C=\underset{0\leq\ell\leq
m-1}{\max}{\|\phi_{\ell}\|^{2}_{\infty}}.\end{cases}$

Finally, there exists a constant $C>0$ such that

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{1}(\phi_{\ell})\right]}\leq\begin{cases}\frac{C}{Nn}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\
spline\leavevmode\nobreak\ basis}\\\ \\\ C\frac{m}{Nn}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ an\leavevmode\nobreak\
orthonormal\leavevmode\nobreak\ basis}.\end{cases}$

  2. 2.

Upper bound of
$\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{2}(\phi_{\ell})\right]}$

For all $k\in[\\![0,n-1]\\!]$ and for all $s\in[0,1]$, set $\eta(s)=k\Delta$
if $s\in[k\Delta,(k+1)\Delta)$. We have:

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{2}(\phi_{\ell})\right]}$
$\displaystyle=\frac{4}{N}\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\int_{k\Delta}^{(k+1)\Delta}{\phi_{\ell}(X^{1}_{k\Delta})((k+1)\Delta-s)\sigma^{\prime}(X^{1}_{s})\sigma^{2}(X^{1}_{s})dW_{s}}}\right)^{2}\right]}$
$\displaystyle=\frac{4}{N}\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\int_{0}^{1}{\phi_{\ell}(X^{1}_{\eta(s)})(\eta(s)+\Delta-s)\sigma^{\prime}(X^{1}_{s})\sigma^{2}(X^{1}_{s})dW_{s}}\right)^{2}\right]}.$

We conclude that

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{2}(\phi_{\ell})\right]}\leq\begin{cases}\frac{C}{Nn^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\
spline\leavevmode\nobreak\ basis}\\\ \\\ C\frac{m}{Nn^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ an\leavevmode\nobreak\
orthonormal\leavevmode\nobreak\ basis}.\end{cases}$

where the constant $C>0$ depends on the diffusion coefficient and the upper
bound of the basis functions.

  3. 3.

Upper bound of
$\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{3}_{2}(\phi_{\ell})\right]}$

We have:

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{3}(\phi_{\ell})\right]}$
$\displaystyle=\frac{4}{Nn^{2}}\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\int_{k\Delta}^{(k+1)\Delta}{\phi_{\ell}(X^{1}_{k\Delta})b(X^{1}_{k\Delta})\sigma(X^{1}_{s})dW_{s}}}\right)^{2}\right]}$
$\displaystyle=\frac{4}{Nn^{2}}\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\int_{0}^{1}{\phi_{\ell}(X^{1}_{\eta(s)})b(X^{1}_{\eta(s)})\sigma(X^{1}_{s})dW_{s}}\right)^{2}\right]}$
$\displaystyle\leq\frac{4}{Nn^{2}}\mathbb{E}\left[\int_{0}^{1}{\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}(X^{1}_{\eta(s)})b(X^{1}_{\eta(s)})\sigma^{2}(X^{1}_{s})ds}}\right].$

Since for all $x\in\mathbb{R},\ b(x)\leq C_{0}(1+x^{2})$ and
$\underset{t\in[0,1]}{\sup}{\mathbb{E}\left(|X_{t}|^{2}\right)}<\infty$, there
exists a constant $C>0$ depending on the diffusion coefficient such that

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{3}(\phi_{\ell})\right]}\leq\begin{cases}\frac{C}{Nn^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\
spline\leavevmode\nobreak\ basis}\\\ \\\ C\frac{m}{Nn^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ an\leavevmode\nobreak\
orthonormal\leavevmode\nobreak\ basis}.\end{cases}$

We finally deduce that from Equations (61) and (62) that for all
$h\in\mathcal{S}_{m,L}$,

$\begin{cases}\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}\mathds{1}_{[-\log(N),\log(N)]}\right]\leq\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\sqrt{\frac{mL}{Nn}}+2\mathbb{E}\left[\mu(\widehat{\sigma}^{2}_{m,L}-h)\right]\leavevmode\nobreak\
\leavevmode\nobreak\ (1)\\\ \\\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}\mathds{1}_{[-\log(N),\log(N)]}\right]\leq\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\sqrt{\frac{m^{2}L}{Nn}}+2\mathbb{E}\left[\mu(\widehat{\sigma}^{2}_{m,L}-h)\right]\leavevmode\nobreak\
\leavevmode\nobreak\ (2)\end{cases}$ (63)

where $C>0$ is a constant, the result $(1)$ corresponds to the spline basis,
and the result (2) corresponds to any orthonormal basis. It remains to obtain
an upper bound of the term $\mu(\widehat{\sigma}^{2}_{m,L}-h)$. For all $a>0$
and for all $h\in\mathcal{S}_{m,L}$,

$\displaystyle 2\mu\left(\widehat{\sigma}^{2}_{m,L}-h\right)\leq$
$\displaystyle\frac{2}{a}\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}+\frac{2}{a}\left\|h-\sigma^{2}\right\|^{2}_{n,N}+\frac{a}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\left(R^{j}_{k\Delta}\right)^{2}}}$
$\displaystyle
2\mathbb{E}\left[\mu\left(\widehat{\sigma}^{2}_{m,L}-h\right)\right]\leq$
$\displaystyle\frac{2}{a}\mathbb{E}\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}+\frac{2}{a}\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+\frac{a}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\mathbb{E}\left[\left(R^{j}_{k\Delta}\right)^{2}\right]}}.$

Using Equations (36), (63) and setting $a=4$, we deduce that there exists
constant $C>0$ depending on $\sigma_{1}$ such that,

$\begin{cases}\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}\mathds{1}_{[-\log(N),\log(N)]}\right]\leq\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\left(\sqrt{\frac{mL}{Nn}}+\Delta^{2}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ [\mathbf{B}]\\\ \\\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}\mathds{1}_{[-\log(N),\log(N)]}\right]\leq\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\left(\sqrt{\frac{m^{2}L}{Nn}}+\Delta^{2}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ [\mathbf{H}].\end{cases}$ (64)

The final result is obtained from Equations (58) and (64). ∎

####  8.3.3 Proof of Lemma 4.5

###### Proof.

It is proven in Comte & Genon-Catalot (2020a) that for each dimension
$m\in\mathcal{M}$, the Gram matrix $\Psi_{m}$ built from the Hermite basis is
invertible. For the case of the B-spline basis, let us consider a vector
$\left(x_{-M},\cdots,x_{K-1}\right)\in\mathbb{R}^{m}$ such that
$x_{j}\in[u_{j+M},u_{j+M+1})$ and $B_{j}(x_{j})\neq 0$. Since
$[u_{j+M},u_{j+M+1})\cap[u_{j^{\prime}+M},u_{j^{\prime}+M+1})=\emptyset$ for
all $j,j^{\prime}\in\\{-M,\cdots,K-1\\}$ such that $j\neq j^{\prime}$, then
for all $j,j^{\prime}\in\\{-M,\cdots,K-1\\}$ such that $j\neq
j^{\prime},B_{j}(x_{j^{\prime}})=0$. Consequently, we obtain:

$\displaystyle\det\left(\left(B_{\ell}(x_{\ell^{\prime}})\right)_{-M\leq\ell,\ell^{\prime}\leq
K-1}\right)$
$\displaystyle=\det\left(\mathrm{diag}\left(B_{-M}(x_{M}),\cdots,B_{K-1}(x_{K-1})\right)\right)$
$\displaystyle=\prod_{\ell=-M}^{K-1}{B_{\ell}(x_{\ell})}\neq 0.$

Then, we deduce from Comte & Genon-Catalot (2020a), Lemma 1 that the matrix
$\Psi_{m}$ is invertible for all $m\in\mathcal{M}$, where the function $f_{T}$
are replaced by

$f_{n}:x\mapsto\frac{1}{n}\sum_{k=0}^{n-1}{p_{X}(k\Delta,x)}$

with $\lambda([-A_{N},A_{N}]\cap\mathrm{supp}(f_{n}))>0$, $\lambda$ being the
Lebesgue measure.

##### Case of the B-spline basis.

For all $w\in\mathbb{R}^{m}$ such that $\|w\|_{2,m}=1$, we have:

$w^{\prime}\Psi_{m}w=\|t_{w}\|^{2}_{n}=\int_{-A_{N}}^{A_{N}}{t^{2}_{w}(x)f_{n}(x)dx}+\frac{t^{2}_{w}(x_{0})}{n}\
\ \mathrm{with}\ \ t_{w}=\sum_{\ell=-M}^{K-1}{w_{\ell}B_{\ell}}.$

Under Assumption 2.1, the transition density $(t,x)\mapsto p_{X}(t,x)$ is
approximated as follows

$\forall(t,x)\in(0,1]\times\mathbb{R},\ \
\frac{1}{K_{*}\sqrt{t}}\exp\left(-c_{\sigma}\frac{x^{2}}{t}\right)\leq
p_{X}(t,x)\leq\frac{K_{*}}{\sqrt{t}}\exp\left(-\frac{x^{2}}{c_{\sigma}t}\right)$

where $K_{*}>1$ and $c_{\sigma}>1$. Since
$s\mapsto\exp\left(-c_{\sigma}x^{2}/s\right)$ is an increasing function, then
for $n$ large enough and for all $x\in[-A_{N},A_{N}]$,

$\displaystyle f_{n}(x)$
$\displaystyle\geq\frac{1}{K_{*}n}\sum_{k=1}^{n-1}{\exp\left(-c\frac{x^{2}}{k\Delta}\right)}\geq\frac{1}{K_{*}}\int_{0}^{1-\Delta}{\exp\left(-c_{\sigma}\frac{x^{2}}{s}\right)ds}$
$\displaystyle\geq\frac{1}{K_{*}}\int_{1-(\log(N))^{-1}}^{1-(2\log(N))^{-1}}{\exp\left(-c_{\sigma}\frac{x^{2}}{s}\right)ds}$
$\displaystyle\geq\frac{1}{2K_{*}\log(N)}\exp\left(-\frac{c_{\sigma}x^{2}}{1-\log^{-1}(N)}\right).$

Thus, the density function satisfies

$\forall x\in[-A_{N},A_{N}],\ \
f_{n}(x)\geq\dfrac{1}{2K_{*}\log(N)}\exp\left(-\frac{c_{\sigma}A^{2}_{N}}{1-\log^{-1}(N)}\right)\geq\dfrac{1}{2K_{*}\log(N)}\exp\left(-c_{\sigma}A^{2}_{N}\right).$
(65)

Finally, since there exists a constant $C_{1}>0$ such that $\|t_{w}\|^{2}\geq
C_{1}A_{N}K^{-1}_{N}$ (see Denis _et al._ (2021), Lemma 2.6), for all
$w\in\mathbb{R}^{m}$ ($m=K_{N}+M$) such that $\|w\|_{2,m}=1$, there exists a
constant $C>0$ such that,

$w^{\prime}\Psi_{m}w\geq\frac{CA_{N}}{m\log(N)}\exp\left(-c_{\sigma}A^{2}_{N}\right).$

##### Case of the Hermite basis.

For all $w\in\mathbb{R}^{m}$ such that $\|w\|_{2,m}=1$, we have

$\displaystyle
w^{\prime}\Psi_{m}w=\|t_{w}\|^{2}_{n}=\int_{-\infty}^{+\infty}{t^{2}_{w}(x)f_{n}(x)dx}+\frac{t^{2}_{w}(x_{0})}{n}\
\ \mathrm{with}\ \ t_{w}=\sum_{\ell=0}^{m-1}{w_{\ell}h_{\ell}}.$

Recall that for all $x\in\mathbb{R}$ such that $|x|\geq\sqrt{(3/2)(4m+3)},\
|h_{\ell}(x)|\leq c|x|\exp(-c_{0}x^{2})$ for all $\ell\geq 0$. Then we have

$\displaystyle w^{\prime}\Psi_{m}w\geq$ $\displaystyle\leavevmode\nobreak\
\int_{|x|\leq\sqrt{(3/2)(4m+3)}}{\left(\sum_{\ell=0}^{m-1}{w_{\ell}h_{\ell}(x)}\right)^{2}f_{n}(x)dx}$
$\displaystyle\geq$ $\displaystyle\leavevmode\nobreak\
\underset{x\in[-\sqrt{(3/2)(4m+3)},\sqrt{(3/2)(4m+3)}]}{\inf}{f_{n}(x)}\int_{|x|\leq\sqrt{(3/2)(4m+3)}}{\left(\sum_{\ell=0}^{m-1}{w_{\ell}h_{\ell}(x)}\right)^{2}dx}$
$\displaystyle\geq$ $\displaystyle\leavevmode\nobreak\
\frac{1}{2K_{*}\log(N)}\exp\left(-\frac{3c_{\sigma}(4m+3)}{2\left(1-\log^{-1}(N)\right)}\right)\int_{|x|\leq\sqrt{(3/2)(4m+3)}}{\left(\sum_{\ell=0}^{m-1}{w_{\ell}h_{\ell}(x)}\right)^{2}dx}$

since for all $x\in\mathbb{R},\ \
f_{n}(x)\geq(1/2K_{*}\log(N))\exp\left(-\frac{c_{\sigma}x^{2}}{1-\log^{-1}(N)}\right)$.
Set $a_{N}=\sqrt{(3/2)(4m+3)}$, then we obtain

$w^{\prime}\Psi_{m}w\geq\frac{\exp\left(-\frac{3c_{\sigma}(4m+3)}{2\left(1-\log^{-1}(N)\right)}\right)}{2K_{*}\log(N)}\left(\int_{-\infty}^{+\infty}{\left(\sum_{\ell=0}^{m-1}{w_{\ell}h_{\ell}(x)}\right)^{2}dx}-\int_{|x|>a_{N}}{\left(\sum_{\ell=0}^{m-1}{w_{\ell}h_{\ell}(x)}\right)^{2}dx}\right)\\\
\geq\frac{\exp\left(-\frac{3c_{\sigma}(4m+3)}{2\left(1-\log^{-1}(N)\right)}\right)}{2K_{*}\log(N)}\left(1-2c^{2}m\int_{a_{N}}^{+\infty}{x^{2}\exp(-8c_{0}x^{2})dx}\right)\\\
\geq\frac{\exp\left(-\frac{3c_{\sigma}(4m+3)}{2\left(1-\log^{-1}(N)\right)}\right)}{2K_{*}\log(N)}\left(1-\frac{c^{2}m}{8c_{0}}\sqrt{\frac{3}{2}(4m+3)}\exp\left(-12c_{0}(4m+3)\right)\right)$

where $c,c_{0}>0$ are constants depending on the Hermite basis. Finally, for
$N$ large enough,

$1-\frac{c^{2}m}{8c_{0}}\sqrt{\frac{3}{2}(4m+3)}\exp\left(-12c_{0}(4m+3)\right)\geq\frac{1}{2}.$

Finally, there exists a constant $C>0$ such that for all $w\in\mathbb{R}^{m}$
such that $\|w\|_{2,m}$,

$w^{\prime}\Psi_{m}w\geq\frac{C}{\log(N)}\exp\left(-\frac{3c_{\sigma}(4m+3)}{2\left(1-\log^{-1}(N)\right)}\right).$

∎

####  8.3.4 Proof of Theorem 4.6

The proof of Theorem 4.6 relies on the following lemma:

######  Lemma 8.9.

Under Assumptions 2.1 and for $\sigma^{2}\in\Sigma_{I}(\beta,R)$ with
$\beta\geq 1,\leavevmode\nobreak\ I=[-A_{N},A_{N}]$ and

$N\propto n,\leavevmode\nobreak\
A_{N}=o\left(\sqrt{\log(N)}\right),\leavevmode\nobreak\
K\propto\left((Nn)^{1/(2\beta+1)}A_{N}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ (m=K+M),$

the following holds:

$\displaystyle\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)\leq
C\exp\left(-c\log^{3/2}(N)\right)$

where $c,C>0$ are constants independent of $N$.

###### Proof of Theorem 4.6 .

According to Equations (55) in the proof of Theorem 4.1, for all dimension
$m=K+M$, with $K\in\mathcal{M}$, and for all $h\in\mathcal{S}_{K+M}$, there
exists a constant $C>0$ such that

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,m}}\right]\leq
C\left[\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\|h-\sigma^{2}_{A_{N}}\|^{2}_{n}}+\mathbb{E}\left(\underset{h\in\mathcal{S}_{K+M},\|h\|_{n}=1}{\sup}{\nu^{2}(h)}\right)+\Delta^{2}\right]$
(66)

where $\Omega_{n,N,m}$ is given in Equation (52) and
$\nu=\nu_{1}+\nu_{2}+\nu_{3}$ with the $\nu_{i}$ given in Equation (32) .

For all $h=\sum_{\ell=-M}^{K-1}{a_{\ell}B_{\ell}}\in\mathcal{S}_{K+M,L_{N}}$,

$\|h\|^{2}_{n}=\mathbb{E}\left[\frac{1}{n}\sum_{k=0}^{n-1}{h^{2}(X_{k\Delta})}\right]=\sum_{\ell=-M}^{K-1}{\sum_{\ell=-M}^{K-1}{a_{\ell}a_{\ell^{\prime}}\mathbb{E}\left[\frac{1}{n}\sum_{k=0}^{n-1}{B_{\ell}(X_{k\Delta})B_{\ell^{\prime}}(X_{k\Delta})}\right]}}=a^{\prime}\Psi_{m}a.$

The Gram matrix $\Psi_{m}$ is invertible for each $K\in\mathcal{M}$ (see proof
of Lemma 4.5). It follows that for all
$h=\sum_{\ell=-M}^{K-1}{a_{\ell}B_{\ell}}$ such that
$\|h\|^{2}_{n}=a^{\prime}\Psi_{m}a=1$, one has $a=\Psi^{-1/2}_{m}u$ where
$u\in\mathbb{R}^{m}$ and $\|u\|_{2,m}=1$. Furthermore, we have:

$h=\sum_{\ell=-M}^{K-1}{a_{\ell}B_{\ell}}=\sum_{\ell=-M}^{K-1}{u_{\ell}\sum_{\ell^{\prime}}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}B_{\ell^{\prime}}}}.$

Then for all $h\in\mathcal{S}_{K+M}$, we have $\nu^{2}(h)\leq
3(\nu^{2}_{1}(h)+\nu^{2}_{2}(h)+\nu^{2}_{3}(h))$ where,

$\forall i\in\\{1,2,3\\},\
\nu^{2}_{i}(h)\leq\sum_{\ell=-M}^{K-1}{\left(\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\sum_{\ell^{\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}B_{\ell^{\prime}}(X^{j}_{k\Delta})\zeta^{j,i}_{k\Delta}}}}\right)^{2}}.$

So we obtain,

$\forall i\in\\{1,2,3\\},\
\mathbb{E}\left[\underset{h\in\mathcal{S}_{K+M},\|h\|_{n}=1}{\sup}{\nu^{2}_{i}(h)}\right]\leq\frac{1}{Nn^{2}}\sum_{\ell=-M}^{K-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\sum_{\ell^{\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}B_{\ell^{\prime}}(X^{j}_{k\Delta})\zeta^{1,i}_{k\Delta}}}\right)^{2}\right]}$

For $i=1$, we have
$\zeta^{1,1}_{k\Delta}=\frac{1}{\Delta}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{1}_{s})}dW_{s}\right)^{2}-\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X^{1}_{s})ds}\right]$
and we obtained in the proof of Theorem 5.1 that there exists a constant $C>0$
such that for all $k\in[\\![0,n-1]\\!]$,

$\mathbb{E}\left[\zeta^{1,1}_{k\Delta}|\mathcal{F}_{k\Delta}\right]=0,\ \ \
\mathbb{E}\left[\left(\zeta^{1,1}_{k\Delta}\right)^{2}|\mathcal{F}_{k\Delta}\right]\leq
C\mathbb{E}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X_{u})du}\right)^{2}\right]\leq
C\sigma^{4}_{1}\Delta^{2}.$

We deduce that

$\displaystyle\mathbb{E}\left[\underset{h\in\mathcal{S}_{K+M},\|h\|_{n}=1}{\sup}{\nu^{2}_{1}(h)}\right]$
$\displaystyle=\frac{1}{Nn^{2}\Delta^{2}}\sum_{\ell=0}^{K-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\sum_{\ell^{\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}B_{\ell^{\prime}}(X^{1}_{k\Delta})\zeta^{1,1}_{k\Delta}}}\right)^{2}\right]}$
$\displaystyle\leq\frac{1}{N}\sum_{\ell=-M}^{K-1}{\sum_{k=0}^{n-1}{\mathbb{E}\left\\{\left(\sum_{\ell^{\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}B_{\ell^{\prime}}(X^{1}_{k\Delta})}\right)^{2}\left(\zeta^{1,1}_{k\Delta}\right)^{2}\right\\}}}$
$\displaystyle\leq\frac{4\sigma^{2}_{1}}{Nn}\sum_{\ell=-M}^{K-1}{\sum_{\ell^{\prime}=-M}^{K-1}{\sum_{\ell^{\prime\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime\prime},\ell}\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell^{\prime\prime}}}}}.$

We have:

$\displaystyle\sum_{\ell=-M}^{K-1}{\sum_{\ell^{\prime}=-M}^{K-1}{\sum_{\ell^{\prime\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime\prime},\ell}\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell^{\prime\prime}}}}}=\mathrm{Tr}\left(\Psi^{-1}_{m}\Psi_{m}\right)=\mathrm{Tr}(I_{m})=m.$

So we obtain

$\displaystyle\mathbb{E}\left[\underset{h\in\mathcal{S}_{K+M}}{\sup}{\nu^{2}_{1}(h)}\right]\leq\frac{4\sigma^{2}_{1}m}{Nn}.$

For $i=2$, we have
$\zeta^{1,2}_{k\Delta}=\frac{2}{\Delta}\int_{k\Delta}^{(k+1)\Delta}{\left((k+1)\Delta-s\right)\sigma^{\prime}(X^{1}_{s})\sigma^{2}(X^{1}_{s})dW_{s}}$
and

$\displaystyle\mathbb{E}\left[\underset{h\in\mathcal{S}_{K+M},\|h\|_{n}=1}{\sup}{\nu^{2}_{2}(h)}\right]$
$\displaystyle\leq\frac{1}{Nn^{2}}\sum_{\ell=-M}^{K-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\sum_{\ell^{\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}B_{\ell^{\prime}}(X^{j}_{k\Delta})\zeta^{1,2}_{k\Delta}}}\right)^{2}\right]}$
$\displaystyle\leq\frac{4\sigma^{4}_{1}\|\sigma^{\prime}\|^{2}_{\infty}\Delta}{Nn^{2}}\sum_{\ell=-M}^{K-1}{\sum_{k=0}^{n-1}{\mathbb{E}\left[\left(\sum_{\ell^{\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}B_{\ell^{\prime}}(X^{1}_{k\Delta})}\right)^{2}\right]}}$
$\displaystyle\leq\frac{4\sigma^{2}_{1}\|\sigma^{\prime}\|^{2}_{\infty}m}{Nn^{2}}.$

For $i=3$, we have
$\zeta^{1,3}_{k\Delta}=2b\left(X^{1}_{k\Delta}\right)\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{1}_{s})dW_{s}}$
and there exists constants $C_{1},C_{2}>0$ such that

$\displaystyle\mathbb{E}\left[\underset{h\in\mathcal{S}_{K+M},\|h\|_{n}=1}{\sup}{\nu^{2}_{3}(h)}\right]$
$\displaystyle\leq\frac{1}{Nn^{2}}\sum_{\ell=-M}^{K-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\sum_{\ell^{\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}B_{\ell^{\prime}}(X^{j}_{k\Delta})\zeta^{1,3}_{k\Delta}}}\right)^{2}\right]}$
$\displaystyle\leq
C_{1}\frac{\sigma^{2}_{1}\Delta}{Nn^{2}}\sum_{\ell=-M}^{K-1}{\sum_{k=0}^{n-1}{\mathbb{E}\left[\left(\sum_{\ell^{\prime}=-M}^{K-1}{\left[\Psi^{-1/2}_{m}\right]_{\ell^{\prime},\ell}B_{\ell^{\prime}}(X^{1}_{k\Delta})}\right)^{2}\right]}}$
$\displaystyle\leq C_{2}\frac{\sigma^{2}_{1}m}{Nn^{2}}.$

Finally, there exists a constant $C>0$ depending on $\sigma_{1}$ and $M$ such
that

$\mathbb{E}\left[\underset{h\in\mathcal{S}_{K+M},\|h\|_{n}=1}{\sup}{\nu^{2}(h)}\right]\leq
C\frac{m}{Nn}.$ (67)

From Equations (66) and (67) , we deduce that

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,m}}\right]\leq
C\left(\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\|h-\sigma^{2}_{A_{N}}\|^{2}_{n}}+\frac{m}{Nn}+\Delta^{2}\right)$

where $C>0$ is a constant depending on $\sigma_{1}$ and $M$. We obtain

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\right]\leq
C\left(\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\|h-\sigma^{2}_{A_{N}}\|^{2}_{n}}+\frac{m}{Nn}+\Delta^{2}\right)+\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\mathds{1}_{\Omega^{c}_{n,N,m}}\right]$

and for $N$ large enough,
$\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\leq
4mL$, and according to Lemma 8.9 ,

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\mathds{1}_{\Omega^{c}_{n,N,m}}\right]\leq
4mL\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)\leq
CmL\exp\left(-c\log^{3/2}(N)\right)$

where $c>0$ is a constant. Thus, there exists a constant $C>0$ such that

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\right]\leq$
$\displaystyle\leavevmode\nobreak\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,m}}\right]+\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\mathds{1}_{\Omega^{c}_{n,N,m}}\right]$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
C\left(\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\|h-\sigma^{2}_{A_{N}}\|^{2}_{n}}+\frac{m}{Nn}+mL\exp\left(-c\log^{3/2}(N)\right)+\Delta^{2}\right).$

Then, as $n\propto N$ and $L=\log(N)$, there exists a constant $C>0$ such that

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\right]\leq$
$\displaystyle\leavevmode\nobreak\
C\left(\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\|h-\sigma^{2}_{A_{N}}\|^{2}_{n}}+\frac{m}{Nn}\right).$

Finally, since $\sigma^{2}\in\Sigma_{I}(\beta,R)$ with $\beta\geq 1$ and
$I=[-A_{N},A_{N}]$, one has

$\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\|h-\sigma^{2}_{A_{N}}\|^{2}_{n}}\leq
CA^{2\beta}_{N}K^{-2\beta}$

where the constant $C>0$ depends on $\beta,R$ and $M$. Furthermore, as we
chose the inverval $[-A_{N},A_{N}]$ such that
$A_{N}=o\left(\sqrt{\log(N)}\right)$ and for
$K\propto\left((Nn)^{1/(2\beta+1)}A_{N}\right)$, we obtain

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\right]\leq$
$\displaystyle\leavevmode\nobreak\ C\log^{\beta}(N)(Nn)^{-2\beta/(2\beta+1)}.$

∎

###  8.4 Proof of Section 5

####  8.4.1 Proof of Theorem 5.1

Set for all $K,K^{\prime}\in\mathcal{K}=\left\\{2^{q},\leavevmode\nobreak\
q=0,\ldots,q_{\max},\leavevmode\nobreak\
2^{q_{\max}}\leq\sqrt{N}/\log(N)\right\\}\subset\mathcal{M}$,

$\mathcal{T}_{K,K^{\prime}}=\left\\{g\in\mathcal{S}_{K+M}+\mathcal{S}_{K^{\prime}+M},\
\|g\|_{n}=1,\leavevmode\nobreak\ \|g\|_{\infty}\leq\sqrt{L}\right\\}.$ (68)

Recall that for all $j\in[\\![1,N]\\!]$ and for all $k\in[\\![0,n]\\!]$,

$\zeta^{j,1}_{k\Delta}=\frac{1}{\Delta}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{j}_{s})dW^{j}_{s}}\right)^{2}-\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X^{j}_{s})ds}\right].$

The proof of Theorem 5.1 relies on the following lemma whose proof is in
Appendix.

######  Lemma 8.10.

Under Assumption 2.1, for all $\varepsilon,\ v>0$ and
$g\in\mathcal{T}_{K,K^{\prime}}$, there exists a real constant $C>0$ such
that,

$\displaystyle\mathbb{P}\left(\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{g(X^{j}_{k\Delta})\zeta^{j,1}_{k\Delta}}}\geq\varepsilon,\|g\|^{2}_{n,N}\leq
v^{2}\right)\leq\exp\left(-C\frac{Nn\varepsilon^{2}}{\sigma^{2}_{1}\left(\varepsilon\|g\|_{\infty}+4\sigma^{2}_{1}v^{2}\right)}\right)$

and for all $x>0$ such that
$x\leq\varepsilon^{2}/\sigma^{2}_{1}\left(\varepsilon\|g\|_{\infty}+4\sigma^{2}_{1}v^{2}\right)$,

$\displaystyle\mathbb{P}\left(\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{g(X^{j}_{k\Delta})\zeta^{j,1}_{k\Delta}}}\geq
2\sigma^{2}_{1}v\sqrt{x}+\sigma^{2}_{1}\|g\|_{\infty}x,\|g\|^{2}_{n,N}\leq
v^{2}\right)\leq\exp\left(-CNnx\right).$

###### Proof of Theorem 5.1.

From Equation (22), we have

$\displaystyle\widehat{K}:=\underset{K\in\mathcal{K}}{\arg\min}{\left\\{{\gamma}_{n,N}(\widehat{\sigma}^{2}_{K})+\mathrm{pen}(K)\right\\}}.$

For all $K\in\mathcal{K}$ and $h\in\mathcal{S}_{K+M,L}$,

$\gamma_{n,N}(\widehat{\sigma}^{2}_{\widehat{K}})+\mathrm{pen}(\widehat{K})\leq\gamma_{n,N}(h)+\mathrm{pen}(K),$

then, for all $K\in\mathcal{K}$ and for all $h\in\mathcal{S}_{K+M,L}$,

$\displaystyle\gamma_{n,N}(\widehat{\sigma}^{2}_{\widehat{K}})-\gamma_{n,N}(\sigma^{2}_{|I})\leq$
$\displaystyle\leavevmode\nobreak\
\gamma_{n,N}(h)-\gamma_{n,N}(\sigma^{2}_{|I})+\mathrm{pen}(K)-\mathrm{pen}(\widehat{K})$
$\displaystyle\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\leq$
$\displaystyle\leavevmode\nobreak\
\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n,N}+2\nu\left(\widehat{\sigma}^{2}_{\widehat{K}}-h\right)+2\mu\left(\widehat{\sigma}^{2}_{\widehat{K}}-h\right)+\mathrm{pen}(K)-\mathrm{pen}(\widehat{K})$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n,N}+\frac{1}{d}\left\|\widehat{\sigma}^{2}_{\widehat{K}}-t\right\|^{2}_{n}+d\underset{g\in\mathcal{T}_{K,\widehat{K}}}{\sup}{\nu^{2}(g)}+\frac{1}{d}\left\|\widehat{\sigma}^{2}_{\widehat{K}}-h\right\|^{2}_{n,N}$
$\displaystyle+\frac{d}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\left(R^{j}_{k\Delta}\right)^{2}}}+\mathrm{pen}(K)-\mathrm{pen}(\widehat{K})$

where $d>1$ and the space $\mathcal{T}_{K,\widehat{K}}$ is given in Equation
(68). On the set $\Omega_{n,N,K_{\max}}$ (given in Equation (52)): $\forall
h\in\mathcal{S}_{K+M},\ \
\frac{1}{2}\|h\|^{2}_{n}\leq\|h\|^{2}_{n,N}\leq\frac{3}{2}\|h\|^{2}_{n}$. Then
on $\Omega_{n,N,K_{\max}}$, for all $d>1$ and for all $h\in\mathcal{S}_{K+M}$
with $K\in\mathcal{K}$,

$\displaystyle\left(1-\frac{10}{d}\right)\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\leq$
$\displaystyle\leavevmode\nobreak\
\left(1+\frac{10}{d}\right)\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n,N}+d\underset{h\in\mathcal{T}_{K,\widehat{K}}}{\sup}{\nu^{2}(h)}+\frac{d}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\left(R^{j}_{k\Delta}\right)^{2}}}$
$\displaystyle+\mathrm{pen}(K)-\mathrm{pen}(\widehat{K}).$

We set $d=20$. Then, on $\Omega_{n,N,\max}$ and for all
$h\in\mathcal{S}_{K+M,L}$,

$\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\leq
3\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n,N}+20\underset{h\in\mathcal{T}_{K,\widehat{K}}}{\sup}{\nu^{2}(h)}+\frac{20}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\left(R^{j}_{k\Delta}\right)^{2}}}+2\left(\mathrm{pen}(K)-\mathrm{pen}(\widehat{K})\right).$
(69)

Let $q:\mathcal{K}^{2}\longrightarrow\mathbb{R}_{+}$ such that
$160q(K,K^{\prime})\leq 18\mathrm{pen}(K)+16\mathrm{pen}(K^{\prime})$. Thus,
on the set $\Omega_{n,N,K_{\max}}$, there exists a constant $C>0$ such that
for all $h\in\mathcal{S}_{K+M}$

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,K_{\max}}}\right]\leq$
$\displaystyle\leavevmode\nobreak\
34\left(\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+\mathrm{pen}(K)\right)$
$\displaystyle+160\mathbb{E}\left(\underset{h\in\mathcal{T}_{K,\widehat{K}}}{\sup}{\nu^{2}_{1}(h)}-q(K,\widehat{K})\right)+C\Delta^{2}$

where

$\nu_{1}(h):=\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{h(X^{j}_{k\Delta})\zeta^{j,1}_{k\Delta}}}$
(70)

with $\zeta^{j,1}_{k\Delta}$ the error term. We set for all
$K,K^{\prime}\in\mathcal{K}$,

$G_{K}(K^{\prime}):=\underset{h\in\mathcal{T}_{K,K^{\prime}}}{\sup}{\nu^{2}_{1}(h)}$
(71)

and for $N$ and $n$ large enough,
$\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\leq
4(K+M)L$. We deduce that,

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]\leq$
$\displaystyle\leavevmode\nobreak\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,K_{\max}}}\right]+\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\mathds{1}_{\Omega^{c}_{n,N,K_{\max}}}\right]$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
34\underset{K\in\mathcal{K}}{\inf}{\left(\underset{h\in\mathcal{S}_{K+M}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+\mathrm{pen}(K)\right)}$
$\displaystyle+C\Delta^{2}+4(K+M)L\mathbb{P}(\Omega^{c}_{n,N,K_{\max}})$
$\displaystyle+160\mathbb{E}\left[\left(G_{K}(\widehat{K})-q(K,\widehat{K})\right)_{+}\mathds{1}_{\Omega_{n,N,K_{\max}}}\right].$

In the sequel, we refer to the proof of Proposition 6.1 in Baraud _et al._
(2001). We known from Lorentz et al (see Lorentz _et al._ (1996)) that given
the unit ball $\overline{B}_{\|.\|_{n}}(0,1)$ of the approximation subspace
$\mathcal{S}_{K+M}$ with respect to norm $\|.\|_{n}$ defined as follows:

$\displaystyle\overline{B}_{\|.\|_{n}}(0,1)=\left\\{h\in\mathcal{S}_{K+M}:\|h\|_{n}\leq
1\right\\}=\left\\{h\in\mathcal{S}_{K+M}:\|h\|\leq\frac{1}{\tau_{0}}\right\\}=\overline{B}_{2}(0,1/\tau_{0}),$

we can find a $\varepsilon-$net $E_{\varepsilon}$ such that for each
$\varepsilon\in(0,1],\
\left|E_{\varepsilon}\right|\leq\left(\frac{3}{\varepsilon\tau_{0}}\right)^{K+M}$.

Recall that
$\mathcal{T}_{K,K^{\prime}}=\left\\{g\in\mathcal{S}_{K+M}+\mathcal{S}_{K^{\prime}+M},\|g\|_{n}=1,\|g\|_{\infty}\leq\sqrt{L}\right\\}$
and consider the sequence
$\left(E_{\varepsilon_{k}}\right)_{k\geq 1}$ of $\varepsilon-$net with
$\varepsilon_{k}=\varepsilon_{0}2^{-k}$ and $\varepsilon_{0}\in(0,1]$.
Moreover, set $N_{k}=\log(|E_{\varepsilon_{k}}|)$ for each $k\geq 0$. Then for
each $g\in\mathcal{S}_{K+M}+\mathcal{S}_{K^{\prime}+M}$ such that
$\|g\|_{\infty}\leq\sqrt{L}$, there exists a sequence
$\left(g_{k}\right)_{k\geq 0}$ with $g_{k}\in E_{\varepsilon_{k}}$ such that
$g=g_{0}+\sum_{k=1}^{\infty}{g_{k}-g_{k-1}}$. Set
$\widetilde{\mathbb{P}}:=\mathbb{P}\left(.\cap\Omega_{n,N,K_{\max}}\right)$
and

$\tau:=\sigma_{1}^{2}\sqrt{6x^{n,N}_{0}}+\sigma^{2}_{1}\sqrt{L}x^{n,N}_{0}+\sum_{k\geq
1}{\varepsilon_{k-1}\left\\{\sigma_{1}^{2}\sqrt{6x^{n,N}_{k}}+2\sigma^{2}_{1}\sqrt{L}x^{n,N}_{k}\right\\}}=y^{n,N}_{0}+\sum_{k\geq
0}{y^{n,N}_{k}}.$

For all $h\in\mathcal{T}_{K,K^{\prime}}$ and on the event
$\Omega_{n,N,K_{\max}}$, one has
$\|h\|^{2}_{n,N}\leq\frac{3}{2}\|h\|^{2}_{n}=\frac{3}{2}$. Then, using the
chaining technique of Baraud _et al._ (2001), we have

$\displaystyle\widetilde{\mathbb{P}}\left(\underset{h\in\mathcal{T}_{K,K^{\prime}}}{\sup}{\nu_{1}(h)}>\tau\right)$
$\displaystyle=\widetilde{\mathbb{P}}\left(\exists(h_{k})_{k\geq
0}\in\prod_{k\geq
0}{E_{\varepsilon_{k}}}/\nu_{1}(h)=\nu_{1}(h_{0})+\sum_{k=1}^{\infty}{\nu_{1}(h_{k}-h_{k-1})}>\tau\right)$
$\displaystyle\leq\sum_{h_{0}\in
E_{0}}{\widetilde{\mathbb{P}}\left(\nu_{1}(h_{0})>y^{n,N}_{0}\right)}+\sum_{k=1}^{\infty}{\sum_{\underset{h_{k-1}\in
E_{\varepsilon_{k-1}}}{h_{k}\in
E_{\varepsilon_{k}}}}{\widetilde{\mathbb{P}}\left(\nu_{1}(h_{k}-h_{k-1})>y^{n,N}_{k}\right)}}.$

According to Equation (70) and Lemma 8.10, there exists a constant $C>0$ such
that

$\displaystyle\widetilde{\mathbb{P}}\left(\nu_{1}(h_{0})>y^{n,N}_{0}\right)\leq$
$\displaystyle\leavevmode\nobreak\
\widetilde{\mathbb{P}}\left(\nu_{1}(h_{0})>\sigma_{1}\sqrt{6x^{n,N}_{0}}+\sigma^{2}_{1}\|h_{0}\|_{\infty}x^{n,N}_{0}\right)$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\ \exp(-CNnx^{n,N}_{0}),$
$\displaystyle\forall k\geq 1,\leavevmode\nobreak\
\widetilde{\mathbb{P}}\left(\nu_{1}(h_{k}-h_{k-1})>y^{n,N}_{0}\right)\leq$
$\displaystyle\leavevmode\nobreak\
\widetilde{\mathbb{P}}\left(\nu_{1}(h_{k}-h_{k-1})>\sigma_{1}\sqrt{6x^{n,N}_{k}}+\sigma^{2}_{1}\|h_{k}-h_{k-1}\|_{\infty}x^{n,N}_{k}\right)$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\ \exp(-CNnx^{n,N}_{k}).$

Finally, since $N_{k}=\log(\left|E_{\varepsilon_{k}}\right|)$ for all $k\geq
0$, we deduce that

$\displaystyle\widetilde{\mathbb{P}}\left(\underset{h\in\mathcal{T}_{K,K^{\prime}}}{\sup}{\nu_{1}(h)}>\tau\right)$
$\displaystyle\leq\left|E_{\varepsilon_{0}}\right|\exp(-CNnx^{n,N}_{0})+\sum_{k=1}^{\infty}{\left(\left|E_{\varepsilon_{k}}\right|+\left|E_{\varepsilon_{k-1}}\right|\right)\exp(-CNnx^{n,N}_{k})}$
$\displaystyle\leq\exp\left(N_{0}-CNnx^{n,N}_{0}\right)+\sum_{k=1}^{\infty}{\exp\left(N_{k}+N_{k-1}-CNnx^{n,N}_{k}\right)}.$

We choose $x^{n,N}_{0}$ and $x^{n,N}_{k},\ \ k\geq 1$ such that,

$\displaystyle N_{0}-CNnx^{n,N}_{0}=$
$\displaystyle-a\left(K+K^{\prime}+2M\right)-b$ $\displaystyle
N_{k}+N_{k-1}-CNnx^{n,N}_{k}=$
$\displaystyle-k(K+K^{\prime}+2M)-a\left(K+K^{\prime}+2M\right)-b$

where $a$ and $b$ are two positive real numbers. We deduce that

$x^{n,N}_{k}\leq C_{0}(1+k)\frac{K+K^{\prime}+2M}{Nn}\ \ \mathrm{and}\ \
\tau\leq C_{1}\sigma^{2}_{1}\sqrt{\sqrt{L}\frac{K+K^{\prime}+2M}{Nn}}$ (72)

with $C_{0}>0$ and $C_{1}$ two constants depending on $a$ and $b$. It comes
that

$\displaystyle\overset{\sim}{\mathbb{P}}\left(\underset{t\in\mathcal{T}_{K,K^{\prime}}}{\sup}{\nu(t)}>\tau\right)$
$\displaystyle\leq\frac{\mathrm{e}}{\mathrm{e}-1}\mathrm{e}^{-b}\exp\left\\{-a\left(K+K^{\prime}+2M\right)\right\\}.$

From Equation (72), we set

$q\left(K,K^{\prime}\right)=\kappa^{*}\sigma^{2}_{1}\sqrt{L}\frac{K+K^{\prime}+2M}{Nn}$

where $\kappa^{*}>0$ depends on $C_{1}>0$. Thus, for all
$K,K^{\prime}\in\mathcal{K}$,

$\mathbb{P}\left(\left\\{\underset{h\in\mathcal{T}_{K,K^{\prime}}}{\sup}{\nu^{2}(h)}>q\left(K,K^{\prime}\right)\right\\}\cap\Omega_{n,N,K_{\max}}\right)\leq\frac{\mathrm{e}^{-b+1}}{\mathrm{e}+1}\exp\left\\{-a\left(K+K^{\prime}+2M\right)\right\\}$

and there exists constants $c,C>0$ such that

$\mathbb{E}\left[\left(G_{K}\left(K^{\prime}\right)-q(K,K^{\prime})\right)_{+}\mathds{1}_{\Omega_{n,N,K_{\max}}}\right]\\\
\leq\frac{c(K+K^{\prime})}{Nn}\mathbb{P}\left(\left\\{\underset{t\in\mathcal{T}_{K,K^{\prime}}}{\sup}{\nu^{2}(t)}>q(K,K^{\prime})\right\\}\cap\Omega_{n,N,K_{\max}}\right)\\\
\leq\frac{C}{Nn}\exp\left\\{-\frac{a}{2}\left(K+K^{\prime}\right)\right\\}.$

Finally, there exists a real constant $C>0$ such that,

$\mathbb{E}\left[\left(G_{K}(\widehat{K})-q(K,\widehat{K})\right)_{+}\mathds{1}_{\Omega_{n,N,K_{\max}}}\right]\leq\sum_{K^{\prime}\in\mathcal{K}}{\mathbb{E}\left[\left(G_{K}\left(K^{\prime}\right)-q(K,K^{\prime})\right)_{+}\mathds{1}_{\Omega_{n,N,K_{\max}}}\right]}\leq\frac{C}{Nn}.$

We choose the penalty function $\mathrm{pen}$ such that for each
$K\in\mathcal{K}$,

$\mathrm{pen}(K)\geq\kappa\sigma^{2}_{1}\sqrt{L}\frac{K+M}{Nn}.$

For $N$ large enough, one has $\sigma^{2}_{1}\leq\sqrt{L}$. Thus, we finally
set $\mathrm{pen}(K)=\kappa\frac{(K+M)\log(N)}{Nn}$ with $L=\log(N)$. Then,
there exists a constant $C>0$ such that,

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]$
$\displaystyle\leq
34\underset{K\in\mathcal{K}}{\inf}{\left\\{\underset{h\in\mathcal{S}_{K+M}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+\mathrm{pen}(K)\right\\}}+\frac{C}{Nn}.$

∎

####  8.4.2 Proof of Theorem 5.2

###### Proof.

From Equation (22), we have

$\widehat{K}:=\underset{K\in\mathcal{K}}{\arg\min}{\gamma_{n,N}(\widehat{\sigma}^{2}_{K+M,L})+\mathrm{pen}(K)}.$

Then, for all $K\in\mathcal{K}$ and for all $h\in\mathcal{S}_{K+M,L}$, we have

$\displaystyle\gamma_{n,N}(\widehat{\sigma}^{2}_{\widehat{K},L})+\mathrm{pen}(\widehat{K})\leq\gamma_{n,N}(h)+\mathrm{pen}(K).$

Then, for all $K\in\mathcal{K}$ and for all $h\in\mathcal{S}_{K+M,L}$,

$\displaystyle\gamma_{n,N}(\widehat{\sigma}^{2}_{\widehat{K},L})-\gamma_{n,N}(\sigma^{2})\leq$
$\displaystyle\leavevmode\nobreak\
\gamma_{n,N}(h)-\gamma_{n,N}(\sigma^{2})+\mathrm{pen}(K)-\mathrm{pen}(\widehat{K})$
$\displaystyle\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}\right\|^{2}_{n,N}\leq$
$\displaystyle\leavevmode\nobreak\
\|h-\sigma^{2}\|^{2}_{n,N}+2\nu(\widehat{\sigma}^{2}_{\widehat{K},L}-h)+2\mu(\widehat{\sigma}^{2}_{\widehat{K},L}-h)+\mathrm{pen}(K)-\mathrm{pen}(\widehat{K}).$

We have for all $a>0$,

$\displaystyle
2\mathbb{E}\left[\mu\left(\widehat{\sigma}^{2}_{\widehat{K},L}-h\right)\right]\leq$
$\displaystyle\frac{2}{a}\mathbb{E}\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}\right\|^{2}_{n,N}+\frac{2}{a}\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+\frac{a}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\mathbb{E}\left[\left(R^{j}_{k\Delta}\right)^{2}\right]}}$

and since $\nu=\nu_{1}+\nu_{2}+\nu_{3}$, according to the proof of Theorem
4.3, there exists a constant $c>0$ such that

$\mathbb{E}\left[\nu(\widehat{\sigma}^{2}_{\widehat{K},L}-h)\right]\leq
c\mathbb{E}\left[\nu_{1}(\widehat{\sigma}^{2}_{\widehat{K},L}-h)\right]$

where the for $i\in\\{1,2,3\\}$ and for all
$h\in\mathcal{S}_{K+M,L},\leavevmode\nobreak\ K\in\mathcal{K}$,

$\nu_{i}(h)=\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{h(X^{j}_{k\Delta})\zeta^{j}_{k\Delta}}},$

and the $\zeta^{j}_{k\Delta}$ are given Then,

$\displaystyle\left(1-\frac{2}{a}\right)\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}\right\|^{2}_{n,N}\right]\leq$
$\displaystyle\leavevmode\nobreak\
\left(1+\frac{2}{a}\right)\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}\right\|^{2}_{n}}+2c\mathbb{E}\left[\nu_{1}(\widehat{\sigma}^{2}_{\widehat{K},L}-h)\right]$
$\displaystyle+\mathrm{pen}(K)-\mathrm{pen}(\widehat{K})+\frac{a}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\mathbb{E}\left[\left(R^{j}_{k\Delta}\right)^{2}\right]}}$

From Equation (36) and for $a=4$, there exists a constant $C>0$ such that

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}\right\|^{2}_{n,N}\right]\leq
3\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}\right\|^{2}_{n}}+4c\mathbb{E}\left[\nu_{1}(\widehat{\sigma}^{2}_{\widehat{K},L}-h)\right]+2\left(\mathrm{pen}(K)-\mathrm{pen}(\widehat{K})\right)+C\Delta^{2}.$
(73)

Since for all $K\in\mathcal{K},\leavevmode\nobreak\ \mathrm{pen}(K)\geq
2\kappa^{*}\sigma^{2}_{1}(K+M)\sqrt{2L}/(Nn)$, define the function
$q:(K,K^{\prime})\mapsto q(K,K^{\prime})$ such that

$q(K,K^{\prime})=2C^{*}\sigma^{2}_{1}\frac{(K+K^{\prime}+2M)\sqrt{2L}}{Nn}\geq
2\sigma^{2}_{1}v\sqrt{x^{n,N}}+\sigma^{2}_{1}vx^{n,N}$ (74)

where

$x^{n,N}\propto\left(\frac{K+K^{\prime}+2M}{Nn}\right)^{2}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\ \leavevmode\nobreak\
v=\sqrt{2L}.$

The constant $C^{*}>0$ depends on constants $\kappa^{*}>0$ and $c>0$ of
Equation (73) such that

$4cq(K,K^{\prime})\leq\mathrm{pen}(K)+2\mathrm{pen}(K^{\prime}).$

Then for all $K\in\mathcal{K}$ and for all $h\in\mathcal{S}_{K+M,L}$,

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}\right\|^{2}_{n,N}\right]\leq
3\left(\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}\right\|^{2}_{n}}+\mathrm{pen}(K)\right)+4c\mathbb{E}\left[\left(\nu_{1}(\widehat{\sigma}^{2}_{\widehat{m},L}-h)-q(K,\widehat{K})\right)_{+}\right]+C\Delta^{2}.$
(75)

For all $K\in\mathcal{K}$ and for all $h\in\mathcal{S}_{K+M,L}$ such that
$\|h\|_{\infty}\leq\sqrt{L}$, we have ,

$\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-h\right\|^{2}_{n,N}\leq\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-h\right\|^{2}_{\infty}\leq
2L=:v^{2}.$

Then, using Equation (74) and Lemma 8.10, there exists a constant $C>0$ such
that for all $K,K^{\prime}\in\mathcal{K}$ and for all
$h\in\mathcal{S}_{K+M,L}$,

$\mathbb{P}\left(\nu_{1}(\widehat{\sigma}^{2}_{K^{\prime},L}-h)\geq
q(K,K^{\prime}),\leavevmode\nobreak\
\|\widehat{\sigma}^{2}_{\widehat{K},L}-h\|^{2}_{n,N}\leq
v^{2}\right)\leq\exp\left(-CNnx^{n,N}\right).$ (76)

Since $L=\log(N)$, then for $N$ large enough,
$\sigma^{2}_{1}\leq\sqrt{\log(N)}$, we finally choose

$\mathrm{pen}(K)=\kappa\frac{(K+M)\log(N)}{Nn}$

where $\kappa>0$ is a new constant. Since
$\mathbb{E}\left[\nu_{1}(\widehat{\sigma}^{2}_{\widehat{K},L}-h)\right]\leq\mathrm{O}\left(\sqrt{\frac{(K_{\max}+M)\log^{2}(N)}{Nn}}\right)$
(see proof of Theorem 4.3), for all $K\in\mathcal{K}$ and
$h\in\mathcal{S}_{K+M,L}$, there exists a constant $c>0$ such that

$\displaystyle\mathbb{E}\left[\left(\nu_{1}(\widehat{\sigma}^{2}_{\widehat{K},L}-h)-q(K,\widehat{K})\right)_{+}\right]\leq$
$\displaystyle\leavevmode\nobreak\
\underset{K^{\prime}\in\mathcal{K}}{\max}{\left\\{\mathbb{E}\left[\left(\nu_{1}(\widehat{\sigma}^{2}_{K^{\prime},L}-h)-q(K,K^{\prime})\right)_{+}\right]\right\\}}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
cq(K,K_{\max})\underset{K^{\prime}\in\mathcal{K}}{\max}{\left\\{\mathbb{P}\left(\nu_{1}(\widehat{\sigma}^{2}_{K^{\prime},L}-h)\geq
q(K,K^{\prime})\right)\right\\}}.$

From Equation (76), we obtain that

$\mathbb{E}\left[\left(\nu_{1}(\widehat{\sigma}^{2}_{\widehat{K},L}-h)-q(K,\widehat{K})\right)_{+}\right]\leq
cq(K,K_{\max})\exp(-CNn)\leq\frac{C}{Nn}$ (77)

since $K$ and $K_{\max}$ increase with the size $N$ of the sample paths
$D_{N,n}$, and

$cNnq(K,K_{\max})\exp(-CNn)\rightarrow 0\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ \leavevmode\nobreak\
N\rightarrow\infty.$

Then, from Equations (77) and (75), there exists a constant $C>0$ such that

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}\right\|^{2}_{n,N}\right]\leq
3\underset{K\in\mathcal{K}}{\inf}{\left\\{\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}\right\|^{2}_{n}}+\mathrm{pen}(K)\right\\}}+\frac{C}{Nn}.$
(78)

∎

####  8.4.3 Proof of Theorem 5.3

###### Proof.

The proof of Theorem 5.3 is similar to the proof of Theorem 5.1. Then, from
Equation (69), for all $h\in\mathcal{S}_{K+M}$,

$\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\leq
3\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n,1}+20\underset{h\in\mathcal{T}_{K,\widehat{K}}}{\sup}{\nu^{2}(h)}+\frac{20}{n}\sum_{k=0}^{n-1}{\left(R^{1}_{k\Delta}\right)^{2}}+2\left(\mathrm{pen}(K)-\mathrm{pen}(\widehat{K})\right),$

where
$\mathcal{T}_{K,K^{\prime}}=\\{h\in\mathcal{S}_{K+M}+\mathcal{S}_{K^{\prime}+M},\leavevmode\nobreak\
\|h\|_{X}=1,\leavevmode\nobreak\ \|h\|_{\infty}\leq\sqrt{L}\\}$. Let
$q:\mathcal{K}^{2}\longrightarrow\mathbb{R}_{+}$ such that
$160q(K,K^{\prime})\leq 18\mathrm{pen}(K)+16\mathrm{pen}(K^{\prime})$.

Recall that the $\mathbb{L}^{2}-$norm $\|.\|$, the norm
$\mathbb{E}\left[\|.\|_{X}\right]$ and the empirical norm $\|.\|_{n}$ are
equivalent on $\mathbb{L}^{2}(I)$ since the transition density is bounded on
the compact interval $I$. Then, for all $K\in\mathcal{K}$ and
$h\in\mathcal{S}_{K+M,L},$ we have

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\right]\leq$
$\displaystyle\leavevmode\nobreak\
3\left(\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+\mathrm{pen}(K)\right)+20\mathbb{E}\left(\underset{h\in\mathcal{T}_{K,\widehat{K}}}{\sup}{\nu^{2}_{1}(h)}-q(K,\widehat{K})\right)+C\Delta^{2}$

where

$\nu_{1}(h):=\frac{1}{n}\sum_{k=0}^{n-1}{h(X^{1}_{k\Delta})\zeta^{1,1}_{k\Delta}}$

with $\zeta^{1,1}_{k\Delta}$ the error term. We set for all
$K,K^{\prime}\in\mathcal{K}$,
$G_{K}(K^{\prime}):=\underset{h\in\mathcal{T}_{K,K^{\prime}}}{\sup}{\nu^{2}_{1}(h)}$.
Then, there exists $C>0$ such that

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\right]\leq$
$\displaystyle\leavevmode\nobreak\
3\underset{K\in\mathcal{K}}{\inf}{\left\\{\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+\mathrm{pen}(K)\right\\}}+20\sum_{K^{\prime}\in\mathcal{K}}{\mathbb{E}\left[\left(G_{K}(K^{\prime})-q(K,K^{\prime})\right)_{+}\right]}$
$\displaystyle+C\Delta^{2}.$

Considering the unit ball $\overline{B}_{\|.\|_{X}}(0,1)$ of the approximation
subspace given by

$\overline{B}_{\|.\|_{X}}(0,1)=\left\\{h\in\mathcal{S}_{K+M},\leavevmode\nobreak\
\|h\|^{2}_{X}\leq
1\right\\}=\left\\{h\in\mathcal{S}_{K+M},\leavevmode\nobreak\
\|h\|^{2}\leq\frac{1}{\tau_{0}}\right\\}.$

We obtain from the proof of Theorem 5.1 with $N=1$ that,

$\sum_{K^{\prime}\in\mathcal{K}}{\mathbb{E}\left[\left(G_{K}(K^{\prime})-q(K,K^{\prime})\right)_{+}\right]}\leq\frac{C}{n},$

where $C>0$ is a constant,
$q(K,K^{\prime})\propto\sigma^{4}_{1}\frac{(K+K^{\prime}+2M)\sqrt{\log(n)}}{n}$
and $\mathrm{pen}(K)\propto\frac{(K+M)\log(n)}{n}$. Then we obtain

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\right]\leq\frac{3}{\tau_{0}}\underset{K\in\mathcal{K}}{\inf}{\left\\{\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+\mathrm{pen}(K)\right\\}}+\frac{C}{n}.$

∎

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Appendix

### Calibration

Fix the drift function $b(x)=1-x$, the time-horizon $T=1$ and at time
$t=0,\leavevmode\nobreak\ x_{0}=0$. Consider the following three models:

  * Model $1$: $\sigma(x)=1$

  * Model $2$: $\sigma(x)=0.1+0.9/\sqrt{1+x^{2}}$

  * Model $3$: $\sigma(x)=1/3+\sin^{2}(2\pi x)/\pi+1/(\pi+x^{2})$.

The three diffusion models satisfy Assumption 2.1 and are used to calibrate
the numerical constant $\kappa$ of the penalty function given in Theorem 5.1

As we already know, the adaptive estimator of $\sigma^{2}$ on the interval
$[-\sqrt{\log(N)},\sqrt{\log(N)}]$ necessitate a data-driven selection of an
optimal dimension through the minimization of the penalized least squares
contrast given in Equation (22) . Since the penalty function
$\mathrm{pen}(d_{N})=\kappa(K_{N}+M)\log^{2}(N)/N^{2}$ depends on the unknown
numerical constant $\kappa>0$, the goal is to select an optimal value of
$\kappa$ in the set $\mathcal{V}=\\{0.1,0.5,1,2,4,5,7,10\\}$ of its possible
values. To this end, we repeat $100$ times the following steps:

  1. 1.

Simulate learning samples $D_{N}$ and $D_{N^{\prime}}$ with
$N\in\\{50,100\\}$, $N^{\prime}=100$ and $n\in\\{100,250\\}$

  2. 2.

For each $\kappa\in\mathcal{V}$:

    1. (a)

For each $K_{N}\in\mathcal{K}$ and from $D_{N}$, compute
$\widehat{\sigma}^{2}_{d_{N},L_{N}}$ given in Equations (9) and (10).

    2. (b)

Select the optimal dimension $\widehat{K}_{N}\in\mathcal{K}$ using Equation
(22)

    3. (c)

Using the learning sample $D_{N^{\prime}}$, evaluate
$\left\|\widehat{\sigma}^{2}_{\widehat{d}_{N},L_{N}}-\sigma^{2}_{A}\right\|^{2}_{n,N^{\prime}}$
where $\widehat{d}_{N}=\widehat{K}_{N}+M$.

Then, we calculate average values of
$\left\|\widehat{\sigma}^{2}_{\widehat{d}_{N},L_{N}}-\sigma^{2}_{A}\right\|^{2}_{n,N^{\prime}}$
for each $\kappa\in\mathcal{V}$ and obtain the following results:

Figure 2: Calibration of the constant $\kappa\in\mathcal{V}$ of the penalty
function

We finally choose $5\in\mathcal{V}$ as the optimal value of $\kappa$ in
reference to the results of Figure 2 .

### Proof of Lemma 8.10

###### Proof.

We obtain from Comte,Genon-Catalot,Rozenholc (2007) proof of Lemma 3 that for
each $j\in[\\![1,N]\\!],\ \ k\in[\\![0,n-1]\\!]$ and
$p\in\mathbb{N}\setminus\\{0,1\\}$

$\displaystyle\mathbb{E}\left[\exp\left(ug(X^{j}_{k\Delta})\xi^{j,1}_{k\Delta}-\frac{au^{2}g^{2}(X^{j}_{k\Delta})}{1-bu}\right)|\mathcal{F}_{k\Delta}\right]\leq
1$

with $a=\mathrm{e}\left(4\sigma^{2}_{1}c^{2}\right)^{2},\
b=4\sigma^{2}_{1}c^{2}\mathrm{e}\|g\|_{\infty}$, $u\in\mathbb{R}$ such that
$bu<1$ and $c>0$ a real constant. Thus,

$\mathbb{P}\left(\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{g(X^{j}_{k\Delta})\xi^{j,1}_{k\Delta}}}\geq\varepsilon,\|g\|^{2}_{N,n}\leq
v^{2}\right)=\mathbb{E}\left(\mathbf{1}_{\left\\{\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{ug(X^{j}_{k\Delta})\xi^{(j,1)}_{k\Delta}\geq
Nnu\varepsilon}}\right\\}}\mathbf{1}_{\|g\|^{2}_{n,N}\leq v^{2}}\right)\\\
=\mathbb{E}\left(\mathbf{1}_{\left\\{\exp\left(\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{ug(X^{j}_{k\Delta})\xi^{(j,1)}_{k\Delta}}}\right)\mathrm{e}^{-Nnu\varepsilon}\geq
1\right\\}}\mathds{1}_{\|g\|^{2}_{N,n}\leq v^{2}}\right)\\\
\leq\mathrm{e}^{-Nnu\varepsilon}\mathbb{E}\left[\mathds{1}_{\|g\|^{2}_{n,N}\leq
v^{2}}\exp\left\\{\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{ug(X^{j}_{k\Delta})\xi^{j,1}_{k\Delta}}}\right\\}\right].$

It follows that,

$\displaystyle\mathbb{P}\left(\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{g(X^{j}_{k\Delta})\xi^{j,1}_{k\Delta}}}\geq\varepsilon,\|g\|^{2}_{n,N}\leq
v^{2}\right)\leq$ $\displaystyle\leavevmode\nobreak\
\exp\left\\{-Nnu\varepsilon+\frac{Nnau^{2}v^{2}}{1-bu}\right\\}.$

We set $u=\frac{\varepsilon}{\varepsilon b+2av^{2}}$. Then, we have
$-Nnu\varepsilon+Nnav^{2}u^{2}/(1-bu)=-Nn\varepsilon^{2}/2(\varepsilon
b+2av^{2})$ and,

$\displaystyle\mathbb{P}\left(\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{g(X^{j}_{k\Delta})\xi^{j,1}_{k\Delta}}}\geq\varepsilon,\|g\|^{2}_{n,N}\leq
v^{2}\right)$
$\displaystyle\leq\exp\left(-\frac{Nn\varepsilon^{2}}{2(\varepsilon
b+av^{2})}\right)$
$\displaystyle\leq\exp\left(-C\frac{Nn\varepsilon^{2}}{\sigma^{2}_{1}\left(\varepsilon\|g\|_{\infty}+4\sigma^{2}_{1}v^{2}\right)}\right)$

where $C>0$ is a constant depending on $c>0$. ∎

### Proof of Lemma 8.9

###### Proof.

Set $K_{n,N}=K_{N}$ since $N\propto n$. Let us remind the reader of the Gram
matrix $\Psi_{K_{N}}$ given in Equation (19),

$\Psi_{K_{N}}=\mathbb{E}\left[\frac{1}{Nn}\mathbf{F}^{\prime}_{K_{N}}\mathbf{F}_{K_{N}}\right]=\mathbb{E}\left(\widehat{\Psi}_{K_{N}}\right)$

where,

$\mathbf{F}_{K_{N}}:=\left((B_{\ell}(X^{j}_{0}),\ldots,(B_{\ell}(X^{j}_{(n-1)\Delta}))\right)_{\underset{1\leq
j\leq N}{0\leq\ell\leq K_{N}-1}}\in\mathbb{R}^{Nn\times(K_{N}+M)}$ (79)

The empirical counterpart $\widehat{\Psi}$ is the random matrix given by
$\widehat{\Psi}_{K_{N}}$ of size $(K_{N}+M)\times(K_{N}+M)$ is given by

$\widehat{\Psi}_{K_{N}}:=\frac{1}{Nn}\mathbf{F}^{\prime}_{K_{N}}\mathbf{F}_{K_{N}}=\left(\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{f_{\ell}(X^{j}_{k\Delta})f_{\ell^{\prime}}(X^{j}_{k\Delta})}}\right)_{\ell,\ell^{\prime}\in[-M,K_{N}-1]}.$
(80)

For all $t=\sum_{\ell=-M}^{K_{N}-1}a_{\ell}B_{\ell,M,{\bf u}}\in S_{K_{N},M}$
one has

$\|t\|_{n,N}^{2}=a^{\prime}\widehat{\Psi}_{K_{N}}a\ \ \mathrm{and}\ \
\|t\|_{n}^{2}=a^{\prime}\Psi_{K_{N}}a,\ \ \mathrm{with}\ \
a=\left(a_{-M},\cdots,a_{K_{N}-1}\right)^{\prime}.$

Under Assumption 2.1, we follow the lines of Comte _et al._ (2018) Proposition
2.3 and Lemma 6.2. Then,

$\displaystyle\sup_{t\in
S_{K_{N},M},\|t\|_{n}=1}\left|\|t\|_{n,N}^{2}-\|t\|_{n}^{2}\right|$
$\displaystyle=$
$\displaystyle\sup_{w\in\mathbb{R}^{K_{N}+M},\left\|\Phi_{K_{N}}^{1/2}w\right\|_{2,K_{N}+M}=1}\left|w^{\prime}\left(\widehat{\Psi}_{K_{N}}-\Psi_{K_{N}}\right)w\right|$
$\displaystyle=$
$\displaystyle\sup_{u\in\mathbb{R}^{K_{N}+M},\|u\|_{2,K_{N}+M}=1}\left|u^{\prime}\Psi_{K_{N}}^{-1/2}\left(\widehat{\Psi}_{K_{N}}-\Psi_{K_{N}}\right)\Psi_{K_{N}}^{-1/2}u\right|$
$\displaystyle=$
$\displaystyle\left\|\Psi_{K_{N}}^{-1/2}\widehat{\Psi}_{K_{N}}\Psi_{K_{N}}^{-1/2}-\operatorname{Id}_{K_{N}+M}\right\|_{\mathrm{op}}.$

Therefore,

$\Omega_{n,N,K_{N}}^{c}=\left\\{\left\|\Psi_{K_{N}}^{-1/2}\widehat{\Psi}_{K_{N}}\Psi_{K_{N}}^{-1/2}-\operatorname{Id}_{K_{N}+M}\right\|_{\mathrm{op}}>1/2\right\\}.$

Since $A_{N}=o\left(\sqrt{\log(N)}\right)$, we obtain from Denis _et al._
(2022), proof of Lemma 7.8, there exists a constant $C>0$ such that

$\mathbb{P}\left(\Omega^{c}_{n,N,K_{N}}\right)\leq
2(K_{N}+M)\exp\left(-C\log^{3/2}(N)\right).$ (81)

Finally, since $2(K_{N}+M)\exp\left(-(C/2)\log^{3/2}(N)\right)\longrightarrow
0$ as $N\longrightarrow+\infty$, one concludes from Equation (81) and for $N$
large enough,

$\mathbb{P}\left(\Omega^{c}_{n,N,K_{N}}\right)\leq
C\exp\left(-c\log^{3/2}(N)\right)$

where $c>0$ and $C>0$ are new constants. ∎