This document provides lecture notes on basic system properties, convolution, impulse response, difference equations, and their relationships. It defines key concepts such as linear, time-invariant, causal, and stable systems. Linear time-invariant systems have an output that can be computed using convolution of the input and impulse response. Difference equations provide an alternative representation of systems that allows for initial conditions and more efficient implementations using autoregressive or moving average models.
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In this talk, we discuss some recent advances in probabilistic schemes for high-dimensional PIDEs. It is known that traditional PDE solvers, e.g., finite element, finite difference methods, do not scale well with the increase of dimension. The idea of probabilistic schemes is to link a wide class of nonlinear parabolic PIDEs to stochastic Levy processes based on nonlinear version of the Feynman-Kac theory. As such, the solution of the PIDE can be represented by a conditional expectation (i.e., a high-dimensional integral) with respect to a stochastic dynamical system driven by Levy processes. In other words, we can solve the PIDEs by performing high-dimensional numerical integration. A variety of quadrature methods could be applied, including MC, QMC, sparse grids, etc. The probabilistic schemes have been used in many application problems, e.g., particle transport in plasmas (e.g., Vlasov-Fokker-Planck equations), nonlinear filtering (e.g., Zakai equations), and option pricing, etc.
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
In this slides, there is a description of the disjoint set representation (Union-Find Problem) with:
1.- Union by Rank
2.- Path Compression
heuristics.
In addition, the amortized complexity analysis is done by using the block idea.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
I am George P. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Perth, Australia. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
In this talk, we discuss some recent advances in probabilistic schemes for high-dimensional PIDEs. It is known that traditional PDE solvers, e.g., finite element, finite difference methods, do not scale well with the increase of dimension. The idea of probabilistic schemes is to link a wide class of nonlinear parabolic PIDEs to stochastic Levy processes based on nonlinear version of the Feynman-Kac theory. As such, the solution of the PIDE can be represented by a conditional expectation (i.e., a high-dimensional integral) with respect to a stochastic dynamical system driven by Levy processes. In other words, we can solve the PIDEs by performing high-dimensional numerical integration. A variety of quadrature methods could be applied, including MC, QMC, sparse grids, etc. The probabilistic schemes have been used in many application problems, e.g., particle transport in plasmas (e.g., Vlasov-Fokker-Planck equations), nonlinear filtering (e.g., Zakai equations), and option pricing, etc.
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
In this slides, there is a description of the disjoint set representation (Union-Find Problem) with:
1.- Union by Rank
2.- Path Compression
heuristics.
In addition, the amortized complexity analysis is done by using the block idea.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
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The (fast) component-by-component construction of lattice point sets and polynomial lattice point sets is a powerful method to obtain quadrature rules for approximating integrals over the dimensional unit cube.
In this talk, we present modifications of the component-by-component algorithm and of the more recent successive coordinate search algorithm, which yield savings of the construction cost for lattice rules and polynomial lattice rules in weighted function spaces. The idea is to reduce the size of the search space for coordinates which are associated with small weights and are therefore of less importance to the overall error compared to coordinates associated with large
weights. We analyze tractability conditions of the resulting quasi-Monte Carlo rules, and show some numerical results.
Optimal interval clustering: Application to Bregman clustering and statistica...Frank Nielsen
We present a generic dynamic programming method to compute the optimal clustering of n scalar elements into k pairwise disjoint intervals. This case includes 1D Euclidean k-means, k-medoids, k-medians, k-centers, etc. We extend the method to incorporate cluster size constraints and show how to choose the appropriate k by model selection. Finally, we illustrate and refine the method on two case studies: Bregman clustering and statistical mixture learning maximizing the complete likelihood.
http://arxiv.org/abs/1403.2485
Fixed Point Theorem in Fuzzy Metric Space Using (CLRg) Propertyinventionjournals
The object of this paper is to establish a common fixed point theorem for semi-compatible pair of self maps by using CLRg Property in fuzzy metric space.
The (fast) component-by-component construction of lattice point sets and polynomial lattice point sets is a powerful method to obtain quadrature rules for approximating integrals over the dimensional unit cube.
In this talk, we present modifications of the component-by-component algorithm and of the more recent successive coordinate search algorithm, which yield savings of the construction cost for lattice rules and polynomial lattice rules in weighted function spaces. The idea is to reduce the size of the search space for coordinates which are associated with small weights and are therefore of less importance to the overall error compared to coordinates associated with large
weights. We analyze tractability conditions of the resulting quasi-Monte Carlo rules, and show some numerical results.
Digital Signal Processing[ECEG-3171]-Ch1_L03Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
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This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
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Brief summary of signals
1. EECS 451 LECTURE NOTES
BASIC SYSTEM PROPERTIES
What: Input x[n] → |SYSTEM| → y[n] output.
Why: Design the system to filter input x[n].
DEF: A system is LINEAR if these two properties hold:
1. Scaling: If x[n] → |SYS| → y[n], then ax[n] → |SYS| → ay[n]
for: any constant a. NOT true if a varies with time (i.e., a[n]).
2. Superposition: If x1[n] → |SYS| → y1[n] and x2[n] → |SYS| → y2[n],
Then: (ax1[n] + bx2[n]) → |SYS| → (ay1[n] + by2[n])
for: any constants a, b. NOT true if a or b vary with time (i.e., a[n], b[n]).
EX: y[n] = 3x[n−2]; y[n] = x[n+1]−nx[n]+2x[n−1]; y[n] = sin(n)x[n].
NOT: y[n] = x2
[n]; y[n] = sin(x[n]); y[n] = |x[n]|; y[n] = x[n]/x[n − 1].
NOT: y[n] = x[n] + 1 (try it). This is called an affine system.
HOW: If any nonlinear function of x[n], not linear. Nonlinear of just n OK.
DEF: A system is TIME-INVARIANT if this property holds:
If x[n] → |SYS| → y[n], then x[n − N] → |SYS| → y[n − N]
for: any integer time delay N. NOT true if N varies with time (e.g., N(n)).
EX: y[n] = 3x[n − 2]; y[n] = sin(x[n]); y[n] = x[n]/x[n − 1].
NOT: y[n] = nx[n]; y[n] = x[n2
]; y[n] = x[2n]; y[n] = x[−n].
HOW: If n appears anywhere other than in x[n], not time-invariant. Else OK.
DEF: A system is CAUSAL if it has this form for some function F(·):
y[n] = F(x[n], x[n − 1], x[n − 2] . . .) (present and past input only).
Note: Physical systems must be causal. But DSP filters need not be causal!
DEF: A system is MEMORYLESS if y[n] = F(x[n]) (present input only).
DEF: A system is (BIBO) STABLE iff: Let x[n] → |SYSTEM| → y[n].
If |x[n]| < M for some constant M, then |y[n]| < N for some N.
i.e.: “Every bounded input (BI) produces a bounded output (BO).”
HOW: BIBO stable ⇔
∞
n=−∞ |h[n]| < L for some constant L
where: Impulse δ[n] → |SYS| → h[n]=impulse response.
EX: A time-invariant system is observed to have these two responses:
{0, 0, 3} → |SYS| → {0, 1, 0, 2} and {0, 0, 0, 1} → |SYS| → {1, 2, 1}.
Prove: The system is nonlinear.
Proof: By contradiction. Suppose the system is linear. But then:
{0, 0, 0, 1} → |SYS| → {1, 2, 1} implies {0, 0, 3} → |SYS| → {3, 6, 3}
since we know it is time-invariant. Then {0, 0, 3} produces two outputs!
2. EECS 451 LECTURE NOTES
CONVOLUTION AND IMPULSE RESPONSE
x[n] = {3, 1, 4, 6} ⇔ x[n] = 3δ[n + 1] + 1δ[n] + 4δ[n − 1] + 6δ[n − 2].
Note: x[n] = i x[i]δ[n − i] = x[n] ∗ δ[n] (sifting property of impulse).
Delay: x[n − D] is x[n] shifted right (later) if D > 0; left (earlier) if D < 0.
Fold: x[−n] is x[n] flipped/folded/reversed around n = 0.
Both: x[N − n] is x[−n] shifted right if N > 0 (since x[0] is now at n = N).
FOR LINEAR TIME-INVARIANT (LTI) SYSTEMS:
1. δ[n] → |LTI| → h[n] Definition of Impulse response h[n].
2. δ[n − i] → |LTI| → h[n − i] Time invariant: delay by i.
3. x[i]δ[n − i] → |LTI| → x[i]h[n − i] Linear: scale by x[i].
4. i x[i]δ[n − i] → |LTI| → i x[i]h[n − i] Linear: superposition.
5. x[n] → |LTI| → y[n] = i x[i]h[n − i] = h[n] ∗ x[n] Convolution.
Input x[n] into LTI system with no initial stored energy→output y[n].
PROPERTIES OF DISCRETE CONVOLUTION
1. y[n] = h[n] ∗ x[n] = x[n] ∗ h[n] = i h[i]x[n − i] = h[n − i]x[i].
2. h[n], x[n] both causal (h[n] = 0 for n < 0 and x[n] = 0 for n < 0)
→ y[n] =
n
i=0 h[i]x[n − i] =
n
i=0 h[n − i]x[i] also causal.
3. Suppose h[n] = 0 only for 0 ≤ n ≤ L (h[n] has length L + 1).
Suppose x[n] = 0 only for 0 ≤ n ≤ M (x[n] has length M + 1).
Then y[n] = 0 only for 0 ≤ n ≤ L + M (y[n] has length L + M + 1).
Note: Length[y[n]]=Length[h[n]]+Length[x[n]]-1.
Note: y[0] = h[0]x[0]; y[L + M] = h[L]x[M]; x[n] ∗ δ[n − D] = x[n − D].
4. x[n] → |h1[n]| → |h2[n]| → y[n] (cascade connection)
Equivalent to: x[n] → |h1[n] ∗ h2[n]| → y[n].
5. x[n] → ր
ց
→|h1[n]|→
→|h2[n]|→
ց
ր → y[n] (parallel connection)
Equivalent to: x[n] → |h1[n] + h2[n]| → y[n].
MA: y[n] = b0x[n] + b1x[n − 1] + . . . + bqx[n − q] (Moving Average)
Huh? Present output=weighted average of q most recent inputs.
Note: Equivalent to y[n] = b[n] ∗ x[n] where b[k] = bk, 0 ≤ k ≤ q.
FIR: Finite Impulse Response ⇔ h[n] has finite duration.
EX: Any MA system is also an FIR system, and vice-versa.
IIR: Infinite Impulse Response ⇔ h[n] not finite duration.
EX: h[n] = an
u[n] = an
for n ≥ 0 and |a| < 1 is stable and IIR.
4. EECS 451 DIFFERENCE EQUATIONS
MA: y(n) = b0x(n) + b1x(n − 1) + . . . + bqx(n − q) (Moving Average)
Huh? Present output=weighted average of q most recent inputs.
Note: Equivalent to y(n) = b(n) ∗ x(n) where b(k) = bk, 0 ≤ k ≤ q.
Why? So why bother? Because now can have initial conditions.
AR: y(n) + a1y(n − 1) + . . . + apy(n − p) = x(n) (AutoRegression)
Huh? Present output=weighted sum of p most recent outputs.
Note: Compute y(n) recursively from its p most recent values.
ARMA:
p
i=0
aiy(n − i)
AUTOREGRESSIVE
=
q
i=0
bix(n − i)
MOVING AVERAGE
(Combine AR and MA→ARMA).
PROPERTIES OF DIFFERENCE EQUATIONS
1. Clearly represent LTI system. Now allow initial conditions.
2. MA model⇔FIR (Finite Impulse Response) filter:
For MA model, h(n) has finite length q + 1.
3. AR model⇔IIR (Infinite Impulse Response) filter:
For AR model, h(n) does not have finite length.
But can implement using finite set of coefficients ai.
4. Analogous to differential equation in continuous time:
dp
dtp + a1
dp−1
dtp−1 + . . . + ap y(t) = dq
dtq + b1
dq−1
dtq−1 + . . . + bq x(t).
Note: Coefficients ai and bi are not directly analogous here.
Note: Time-invariant in both cases since coefficients indpt of time.
5. SKIP Section 2.4.3 in text: 1-sided z-transform is much easier.
READ Section 2.5 in text: signal flow graph implementations.
DIFFERENCE EQUATIONS VS. h(n)
Advantages of Difference Equations:
1. Analogous to differential equation in continuous time.
Often can write them down from model of physical system.
2. Can incorporate nonzero initial conditions.
3. AR model may be much more efficient implementation of system.
Disadvantages of Difference Equations:
1. There may not be a difference equation description of system!
Example: h(n) = 1
n
, n > 0 has no difference equation model.
Recall: h(t) = δ(t − 1) (delay) has no differential equation model.
2. May only want output y(N) for single time N (e.g., end of year).
Then convolution y(N) = h(i)x(N − i) more efficient computation.